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Verifiable Distributed Aggregation Functions
draft-irtf-cfrg-vdaf-08

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This is an older version of an Internet-Draft whose latest revision state is "Active".
Authors Richard Barnes , David Cook , Christopher Patton , Phillipp Schoppmann
Last updated 2023-11-20 (Latest revision 2023-08-31)
Replaces draft-patton-cfrg-vdaf
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draft-irtf-cfrg-vdaf-08
CFRG                                                        R. L. Barnes
Internet-Draft                                                     Cisco
Intended status: Informational                                   D. Cook
Expires: 23 May 2024                                                ISRG
                                                               C. Patton
                                                              Cloudflare
                                                           P. Schoppmann
                                                                  Google
                                                        20 November 2023

              Verifiable Distributed Aggregation Functions
                        draft-irtf-cfrg-vdaf-08

Abstract

   This document describes Verifiable Distributed Aggregation Functions
   (VDAFs), a family of multi-party protocols for computing aggregate
   statistics over user measurements.  These protocols are designed to
   ensure that, as long as at least one aggregation server executes the
   protocol honestly, individual measurements are never seen by any
   server in the clear.  At the same time, VDAFs allow the servers to
   detect if a malicious or misconfigured client submitted an
   measurement that would result in an invalid aggregate result.

Discussion Venues

   This note is to be removed before publishing as an RFC.

   Discussion of this document takes place on the Crypto Forum Research
   Group mailing list (cfrg@ietf.org), which is archived at
   https://mailarchive.ietf.org/arch/search/?email_list=cfrg.

   Source for this draft and an issue tracker can be found at
   https://github.com/cjpatton/vdaf.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

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   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on 23 May 2024.

Copyright Notice

   Copyright (c) 2023 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents (https://trustee.ietf.org/
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
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   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.1.  Change Log  . . . . . . . . . . . . . . . . . . . . . . .   8
   2.  Conventions and Definitions . . . . . . . . . . . . . . . . .  13
   3.  Overview  . . . . . . . . . . . . . . . . . . . . . . . . . .  14
   4.  Definition of DAFs  . . . . . . . . . . . . . . . . . . . . .  16
     4.1.  Sharding  . . . . . . . . . . . . . . . . . . . . . . . .  18
     4.2.  Preparation . . . . . . . . . . . . . . . . . . . . . . .  19
     4.3.  Validity of Aggregation Parameters  . . . . . . . . . . .  20
     4.4.  Aggregation . . . . . . . . . . . . . . . . . . . . . . .  20
     4.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . . . .  21
     4.6.  Execution of a DAF  . . . . . . . . . . . . . . . . . . .  22
   5.  Definition of VDAFs . . . . . . . . . . . . . . . . . . . . .  23
     5.1.  Sharding  . . . . . . . . . . . . . . . . . . . . . . . .  25
     5.2.  Preparation . . . . . . . . . . . . . . . . . . . . . . .  25
     5.3.  Validity of Aggregation Parameters  . . . . . . . . . . .  28
     5.4.  Aggregation . . . . . . . . . . . . . . . . . . . . . . .  28
     5.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . . . .  29
     5.6.  Execution of a VDAF . . . . . . . . . . . . . . . . . . .  29
     5.7.  Communication Patterns for Preparation  . . . . . . . . .  31
     5.8.  Ping-Pong Topology (Only Two Aggregators) . . . . . . . .  32
     5.9.  Star Topology (Any Number of Aggregators) . . . . . . . .  38
   6.  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . .  38
     6.1.  Finite Fields . . . . . . . . . . . . . . . . . . . . . .  38
       6.1.1.  Auxiliary Functions . . . . . . . . . . . . . . . . .  40
       6.1.2.  FFT-Friendly Fields . . . . . . . . . . . . . . . . .  41

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       6.1.3.  Parameters  . . . . . . . . . . . . . . . . . . . . .  41
     6.2.  Extendable Output Functions . . . . . . . . . . . . . . .  42
       6.2.1.  XofTurboShake128  . . . . . . . . . . . . . . . . . .  43
       6.2.2.  XofFixedKeyAes128 . . . . . . . . . . . . . . . . . .  44
       6.2.3.  The Domain Separation Tag and Binder String . . . . .  45
   7.  Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  46
     7.1.  Fully Linear Proof (FLP) Systems  . . . . . . . . . . . .  47
       7.1.1.  Encoding the Input  . . . . . . . . . . . . . . . . .  51
       7.1.2.  Multiple proofs . . . . . . . . . . . . . . . . . . .  51
     7.2.  Construction  . . . . . . . . . . . . . . . . . . . . . .  52
       7.2.1.  Sharding  . . . . . . . . . . . . . . . . . . . . . .  54
       7.2.2.  Preparation . . . . . . . . . . . . . . . . . . . . .  59
       7.2.3.  Validity of Aggregation Parameters  . . . . . . . . .  61
       7.2.4.  Aggregation . . . . . . . . . . . . . . . . . . . . .  61
       7.2.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . .  61
       7.2.6.  Auxiliary Functions . . . . . . . . . . . . . . . . .  62
       7.2.7.  Message Serialization . . . . . . . . . . . . . . . .  63
     7.3.  A General-Purpose FLP . . . . . . . . . . . . . . . . . .  66
       7.3.1.  Overview  . . . . . . . . . . . . . . . . . . . . . .  66
       7.3.2.  Validity Circuits . . . . . . . . . . . . . . . . . .  69
       7.3.3.  Construction  . . . . . . . . . . . . . . . . . . . .  71
     7.4.  Instantiations  . . . . . . . . . . . . . . . . . . . . .  74
       7.4.1.  Prio3Count  . . . . . . . . . . . . . . . . . . . . .  75
       7.4.2.  Prio3Sum  . . . . . . . . . . . . . . . . . . . . . .  76
       7.4.3.  Prio3SumVec . . . . . . . . . . . . . . . . . . . . .  78
       7.4.4.  Prio3Histogram  . . . . . . . . . . . . . . . . . . .  82
   8.  Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . .  84
     8.1.  Incremental Distributed Point Functions (IDPFs) . . . . .  86
     8.2.  Construction  . . . . . . . . . . . . . . . . . . . . . .  88
       8.2.1.  Client  . . . . . . . . . . . . . . . . . . . . . . .  90
       8.2.2.  Preparation . . . . . . . . . . . . . . . . . . . . .  92
       8.2.3.  Validity of Aggregation Parameters  . . . . . . . . .  95
       8.2.4.  Aggregation . . . . . . . . . . . . . . . . . . . . .  95
       8.2.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . .  96
       8.2.6.  Message Serialization . . . . . . . . . . . . . . . .  96
     8.3.  The IDPF scheme of BBCGGI21 . . . . . . . . . . . . . . .  99
       8.3.1.  Key Generation  . . . . . . . . . . . . . . . . . . . 100
       8.3.2.  Key Evaluation  . . . . . . . . . . . . . . . . . . . 102
       8.3.3.  Auxiliary Functions . . . . . . . . . . . . . . . . . 103
     8.4.  Instantiation . . . . . . . . . . . . . . . . . . . . . . 105
   9.  Security Considerations . . . . . . . . . . . . . . . . . . . 105
     9.1.  Requirements for the Verification Key . . . . . . . . . . 106
     9.2.  Requirements for the Nonce  . . . . . . . . . . . . . . . 107
     9.3.  Requirements for the Aggregation Parameters . . . . . . . 108
       9.3.1.  Additional Privacy Considerations . . . . . . . . . . 108
     9.4.  Requirements for XOFs . . . . . . . . . . . . . . . . . . 109
     9.5.  Choosing the Number of Proofs to Use for Prio3  . . . . . 109
   10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 109

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   11. References  . . . . . . . . . . . . . . . . . . . . . . . . . 110
     11.1.  Normative References . . . . . . . . . . . . . . . . . . 110
     11.2.  Informative References . . . . . . . . . . . . . . . . . 111
   Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 112
   Test Vectors  . . . . . . . . . . . . . . . . . . . . . . . . . . 112
     Prio3Count  . . . . . . . . . . . . . . . . . . . . . . . . . . 113
     Prio3Sum  . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
     Prio3SumVec . . . . . . . . . . . . . . . . . . . . . . . . . . 113
     Prio3Histogram  . . . . . . . . . . . . . . . . . . . . . . . . 113
     Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . . 113

1.  Introduction

   [TO BE REMOVED BY RFC EDITOR: The source for this draft and and the
   reference implementation can be found at https://github.com/cfrg/
   draft-irtf-cfrg-vdaf.]

   The ubiquity of the Internet makes it an ideal platform for
   measurement of large-scale phenomena, whether public health trends or
   the behavior of computer systems at scale.  There is substantial
   overlap, however, between information that is valuable to measure and
   information that users consider private.

   For example, consider an application that provides health information
   to users.  The operator of an application might want to know which
   parts of their application are used most often, as a way to guide
   future development of the application.  Specific users' patterns of
   usage, though, could reveal sensitive things about them, such as
   which users are researching a given health condition.

   In many situations, the measurement collector is only interested in
   aggregate statistics, e.g., which portions of an application are most
   used or what fraction of people have experienced a given disease.
   Thus systems that provide aggregate statistics while protecting
   individual measurements can deliver the value of the measurements
   while protecting users' privacy.

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   Most prior approaches to this problem fall under the rubric of
   "differential privacy (DP)" [Dwo06].  Roughly speaking, a data
   aggregation system that is differentially private ensures that the
   degree to which any individual measurement influences the value of
   the aggregate result can be precisely controlled.  For example, in
   systems like RAPPOR [EPK14], each user samples noise from a well-
   known distribution and adds it to their measurement before submitting
   to the aggregation server.  The aggregation server then adds up the
   noisy measurements, and because it knows the distribution from whence
   the noise was sampled, it can estimate the true sum with reasonable
   precision.

   Differentially private systems like RAPPOR are easy to deploy and
   provide a useful guarantee.  On its own, however, DP falls short of
   the strongest privacy property one could hope for.  Specifically,
   depending on the "amount" of noise a client adds to its measurement,
   it may be possible for a curious aggregator to make a reasonable
   guess of the measurement's true value.  Indeed, the more noise the
   clients add, the less reliable will be the server's estimate of the
   output.  Thus systems employing DP techniques alone must strike a
   delicate balance between privacy and utility.

   The ideal goal for a privacy-preserving measurement system is that of
   secure multi-party computation (MPC): No participant in the protocol
   should learn anything about an individual measurement beyond what it
   can deduce from the aggregate.  In this document, we describe
   Verifiable Distributed Aggregation Functions (VDAFs) as a general
   class of protocols that achieve this goal.

   VDAF schemes achieve their privacy goal by distributing the
   computation of the aggregate among a number of non-colluding
   aggregation servers.  As long as a subset of the servers executes the
   protocol honestly, VDAFs guarantee that no measurement is ever
   accessible to any party besides the client that submitted it.  At the
   same time, VDAFs are "verifiable" in the sense that malformed
   measurements that would otherwise garble the result of the
   computation can be detected and removed from the set of measurements.
   We refer to this property as "robustness".

   In addition to these MPC-style security goals of privacy and
   robustness, VDAFs can be composed with various mechanisms for
   differential privacy, thereby providing the added assurance that the
   aggregate result itself does not leak too much information about any
   one measurement.

      TODO(issue #94) Provide guidance for local and central DP and
      point to it here.

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   The cost of achieving these security properties is the need for
   multiple servers to participate in the protocol, and the need to
   ensure they do not collude to undermine the VDAF's privacy
   guarantees.  Recent implementation experience has shown that
   practical challenges of coordinating multiple servers can be
   overcome.  The Prio system [CGB17] (essentially a VDAF) has been
   deployed in systems supporting hundreds of millions of users: The
   Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure
   Notification Private Analytics collaboration among the Internet
   Security Research Group (ISRG), Google, Apple, and others [ENPA].

   The VDAF abstraction laid out in Section 5 represents a class of
   multi-party protocols for privacy-preserving measurement proposed in
   the literature.  These protocols vary in their operational and
   security requirements, sometimes in subtle but consequential ways.
   This document therefore has two important goals:

   1.  Providing higher-level protocols like [DAP] with a simple,
       uniform interface for accessing privacy-preserving measurement
       schemes, documenting relevant operational and security
       requirements, and specifying constraints for safe usage:

       1.  General patterns of communications among the various actors
           involved in the system (clients, aggregation servers, and the
           collector of the aggregate result);

       2.  Capabilities of a malicious coalition of servers attempting
           to divulge information about client measurements; and

       3.  Conditions that are necessary to ensure that malicious
           clients cannot corrupt the computation.

   2.  Providing cryptographers with design criteria that provide a
       clear deployment roadmap for new constructions.

   This document also specifies two concrete VDAF schemes, each based on
   a protocol from the literature.

   *  The aforementioned Prio system [CGB17] allows for the privacy-
      preserving computation of a variety aggregate statistics.  The
      basic idea underlying Prio is fairly simple:

      1.  Each client shards its measurement into a sequence of additive
          shares and distributes the shares among the aggregation
          servers.

      2.  Next, each server adds up its shares locally, resulting in an
          additive share of the aggregate.

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      3.  Finally, the aggregation servers send their aggregate shares
          to the data collector, who combines them to obtain the
          aggregate result.

      The difficult part of this system is ensuring that the servers
      hold shares of a valid, aggregatable value, e.g., the measurement
      is an integer in a specific range.  Thus Prio specifies a multi-
      party protocol for accomplishing this task.

      In Section 7 we describe Prio3, a VDAF that follows the same
      overall framework as the original Prio protocol, but incorporates
      techniques introduced in [BBCGGI19] that result in significant
      performance gains.

   *  More recently, Boneh et al.  [BBCGGI21] described a protocol
      called Poplar for solving the t-heavy-hitters problem in a
      privacy-preserving manner.  Here each client holds a bit-string of
      length n, and the goal of the aggregation servers is to compute
      the set of strings that occur at least t times.  The core
      primitive used in their protocol is a specialized Distributed
      Point Function (DPF) [GI14] that allows the servers to "query"
      their DPF shares on any bit-string of length shorter than or equal
      to n.  As a result of this query, each of the servers has an
      additive share of a bit indicating whether the string is a prefix
      of the client's string.  The protocol also specifies a multi-party
      computation for verifying that at most one string among a set of
      candidates is a prefix of the client's string.

      In Section 8 we describe a VDAF called Poplar1 that implements
      this functionality.

   Finally, perhaps the most complex aspect of schemes like Prio3 and
   Poplar1 is the process by which the client-generated measurements are
   prepared for aggregation.  Because these constructions are based on
   secret sharing, the servers will be required to exchange some amount
   of information in order to verify the measurement is valid and can be
   aggregated.  Depending on the construction, this process may require
   multiple round trips over the network.

   There are applications in which this verification step may not be
   necessary, e.g., when the client's software is run a trusted
   execution environment.  To support these applications, this document
   also defines Distributed Aggregation Functions (DAFs) as a simpler
   class of protocols that aim to provide the same privacy guarantee as
   VDAFs but fall short of being verifiable.

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      OPEN ISSUE Decide if we should give one or two example DAFs.
      There are natural variants of Prio3 and Poplar1 that might be
      worth describing.

   The remainder of this document is organized as follows: Section 3
   gives a brief overview of DAFs and VDAFs; Section 4 defines the
   syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6
   defines various functionalities that are common to our constructions;
   Section 7 describes the Prio3 construction; Section 8 describes the
   Poplar1 construction; and Section 9 enumerates the security
   considerations for VDAFs.

1.1.  Change Log

   (*) Indicates a change that breaks wire compatibility with the
   previous draft.

   08:

   *  Poplar1: Bind the report nonce to the authenticator vector
      programmed into the IDPF. (*)

   *  IdpfPoplar: Modify extend() by stealing each control bit from its
      corresponding seed.  This improves performance by reducing the
      number of AES calls per level from 3 to 2.  The cost is a slight
      reduction in the concrete privacy bound. (*)

   *  Prio3: Add support for generating and verifying mutliple proofs
      per measurement.  This enables a trade-off between communication
      cost and runtime: if more proofs are used, then a smaller field
      can be used without impacting robustness. (*)

   *  Replace SHAKE128 with TurboSHAKE128. (*)

   07:

   *  Rename PRG to XOF ("eXtendable Output Function").  Accordingly,
      rename PrgSha3 to XofShake128 and PrgFixedKeyAes128 to
      XofFixedKeyAes128.  "PRG" is a misnomer since we don't actually
      treat this object as a pseudorandom generator in existing security
      analysis.

   *  Replace cSHAKE128 with SHAKE128, re-implementing domain separation
      for the customization string using a simpler scheme.  This change
      addresses the reality that implementations of cSHAKE128 are less
      common. (*)

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   *  Define a new VDAF, called Prio3SumVec, that generalizes Prio3Sum
      to a vector of summands.

   *  Prio3Histogram: Update the codepoint and use the parallel sum
      optimization introduced by Prio3SumVec to reduce the proof size.
      (*)

   *  Daf, Vdaf: Rename interface methods to match verbiage in the
      draft.

   *  Daf: Align with Vdaf by adding a nonce to shard() and prep().

   *  Vdaf: Have prep_init() compute the first prep share.  This change
      is intended to simplify the interface by making the input to
      prep_next() not optional.

   *  Prio3: Split sharding into two auxiliary functions, one for
      sharding with joint randomness and another without.  This change
      is intended to improve readability.

   *  Fix bugs in the ping-pong interface discovered after implementing
      it.

   06:

   *  Vdaf: Define a wrapper interface for preparation that is suitable
      for the "ping-pong" topology in which two Aggregators exchange
      messages over a request/response protocol, like HTTP, and take
      turns executing the computation until input from the peer is
      required.

   *  Prio3Histogram: Generalize the measurement type so that the
      histogram can be used more easily with discrete domains. (*)

   *  Daf, Vdaf: Change the aggregation parameter validation algorithm
      to take the set of previous parameters rather than a list.  (The
      order of the parameters is irrelevant.)

   *  Daf, Vdaf, Idpf: Add parameter RAND_SIZE that specifies the number
      of random bytes consumed by the randomized algorithm (shard() for
      Daf and Vdaf and gen() for Idpf).

   05:

   *  IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key
      mode for AES-128 based on a construction from [GKWWY20].  This
      change is intended to improve performance of IDPF evaluation.
      Note that the new PRG is not suitable for all applications. (*)

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   *  Idpf: Add a binder string to the key-generation and evaluation
      algorithms.  This is used to plumb the nonce generated by the
      Client to the PRG.

   *  Plumb random coins through the interface of randomized algorithms.
      Specifically, add a random input to (V)DAF sharding algorithm and
      IDPF key-generation algorithm and require implementations to
      specify the length of the random input.  Accordingly, update
      Prio3, Poplar1, and IdpfPoplar to match the new interface.  This
      change is intended to improve coverage of test vectors.

   *  Use little-endian byte-order for field element encoding. (*)

   *  Poplar1: Move the last step of sketch evaluation from prep_next()
      to prep_shares_to_prep().

   04:

   *  Align security considerations with the security analysis of
      [DPRS23].

   *  Vdaf: Pass the nonce to the sharding algorithm.

   *  Vdaf: Rather than allow the application to choose the nonce
      length, have each implementation of the Vdaf interface specify the
      expected nonce length. (*)

   *  Prg: Split "info string" into two components: the "customization
      string", intended for domain separation; and the "binder string",
      used to bind the output to ephemeral values, like the nonce,
      associated with execution of a (V)DAF.

   *  Replace PrgAes128 with PrgSha3, an implementation of the Prg
      interface based on SHA-3, and use the new scheme as the default.
      Accordingly, replace Prio3Aes128Count with Prio3Count,
      Poplar1Aes128 with Poplar1, and so on.  SHA-3 is a safer choice
      for instantiating a random oracle, which is used in the analysis
      of Prio3 of [DPRS23]. (*)

   *  Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct
      customization string, as suggested by [DPRS23].  This is intended
      to make domain separation clearer, thereby simplifying security
      analysis. (*)

   *  Prio3: Replace "joint randomness hints" sent in each input share
      with "joint randomness parts" sent in the public share.  This
      reduces communication overhead when the number of shares exceeds
      two. (*)

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   *  Prio3: Bind nonce to joint randomness parts.  This is intended to
      address birthday attacks on robustness pointed out by [DPRS23].
      (*)

   *  Poplar1: Use different Prg invocations for producing the
      correlated randomness for inner and leaf nodes of the IDPF tree.
      This is intended to simplify implementations. (*)

   *  Poplar1: Don't bind the candidate prefixes to the verifier
      randomness.  This is intended to improve performance, while not
      impacting security.  According to the analysis of [DPRS23], it is
      necessary to restrict Poplar1 usage such that no report is
      aggregated more than once at a given level of the IDPF tree;
      otherwise, attacks on privacy may be possible.  In light of this
      restriction, there is no added benefit of binding to the prefixes
      themselves. (*)

   *  Poplar1: During preparation, assert that all candidate prefixes
      are unique and appear in order.  Uniqueness is required to avoid
      erroneously rejecting a valid report; the ordering constraint
      ensures the uniqueness check can be performed efficiently. (*)

   *  Poplar1: Increase the maximum candidate prefix count in the
      encoding of the aggregation parameter. (*)

   *  Poplar1: Bind the nonce to the correlated randomness derivation.
      This is intended to provide defense-in-depth by ensuring the
      Aggregators reject the report if the nonce does not match what the
      Client used for sharding. (*)

   *  Poplar1: Clarify that the aggregation parameter encoding is
      OPTIONAL.  Accordingly, update implementation considerations
      around cross-aggregation state.

   *  IdpfPoplar: Add implementation considerations around branching on
      the values of control bits.

   *  IdpfPoplar: When decoding the the control bits in the public
      share, assert that the trailing bits of the final byte are all
      zero. (*)

   03:

   *  Define codepoints for (V)DAFs and use them for domain separation
      in Prio3 and Poplar1. (*)

   *  Prio3: Align joint randomness computation with revised paper
      [BBCGGI19].  This change mitigates an attack on robustness. (*)

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   *  Prio3: Remove an intermediate PRG evaluation from query randomness
      generation. (*)

   *  Add additional guidance for choosing FFT-friendly fields.

   02:

   *  Complete the initial specification of Poplar1.

   *  Extend (V)DAF syntax to include a "public share" output by the
      Client and distributed to all of the Aggregators.  This is to
      accommodate "extractable" IDPFs as required for Poplar1.  (See
      [BBCGGI21], Section 4.3 for details.)

   *  Extend (V)DAF syntax to allow the unsharding step to take into
      account the number of measurements aggregated.

   *  Extend FLP syntax by adding a method for decoding the aggregate
      result from a vector of field elements.  The new method takes into
      account the number of measurements.

   *  Prio3: Align aggregate result computation with updated FLP syntax.

   *  Prg: Add a method for statefully generating a vector of field
      elements.

   *  Field: Require that field elements are fully reduced before
      decoding. (*)

   *  Define new field Field255.

   01:

   *  Require that VDAFs specify serialization of aggregate shares.

   *  Define Distributed Aggregation Functions (DAFs).

   *  Prio3: Move proof verifier check from prep_next() to
      prep_shares_to_prep(). (*)

   *  Remove public parameter and replace verification parameter with a
      "verification key" and "Aggregator ID".

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2.  Conventions and Definitions

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
   "OPTIONAL" in this document are to be interpreted as described in
   BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
   capitals, as shown here.

   Algorithms in this document are written in Python 3.  Type hints are
   used to define input and output types.  A fatal error in a program
   (e.g., failure to parse one of the function parameters) is usually
   handled by raising an exception.

   A variable with type Bytes is a byte string.  This document defines
   several byte-string constants.  When comprised of printable ASCII
   characters, they are written as Python 3 byte-string literals (e.g.,
   b'some constant string').

   A global constant VERSION of type Unsigned is defined, which
   algorithms are free to use as desired.  Its value SHALL be 8.

   This document describes algorithms for multi-party computations in
   which the parties typically communicate over a network.  Wherever a
   quantity is defined that must be be transmitted from one party to
   another, this document prescribes a particular encoding of that
   quantity as a byte string.

      OPEN ISSUE It might be better to not be prescriptive about how
      quantities are encoded on the wire.  See issue #58.

   Some common functionalities:

   *  zeros(len: Unsigned) -> Bytes returns an array of zero bytes.  The
      length of output MUST be len.

   *  gen_rand(len: Unsigned) -> Bytes returns an array of random bytes.
      The length of output MUST be len.

   *  byte(int: Unsigned) -> Bytes returns the representation of int as
      a byte string.  The value of int MUST be in [0,256).

   *  concat(parts: Vec[Bytes]) -> Bytes returns the concatenation of
      the input byte strings, i.e., parts[0] || ... || parts[len(parts)-
      1].

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   *  front(length: Unsigned, vec: Vec[Any]) -> (Vec[Any], Vec[Any])
      splits vec into two vectors, where the first vector is made up of
      the first length elements of the input.  I.e., (vec[:length],
      vec[length:]).

   *  xor(left: Bytes, right: Bytes) -> Bytes returns the bitwise XOR of
      left and right.  An exception is raised if the inputs are not the
      same length.

   *  to_be_bytes(val: Unsigned, length: Unsigned) -> Bytes converts val
      to big-endian bytes; its value MUST be in range [0, 2^(8*length)).
      Function from_be_bytes(encoded: Bytes) -> Unsigned computes the
      inverse.

   *  to_le_bytes(val: Unsigned, length: Unsigned) -> Bytes converts val
      to little-endian bytes; its value MUST be in range [0,
      2^(8*length)).  Function from_le_bytes(encoded: Bytes) -> Unsigned
      computes the inverse.

   *  next_power_of_2(n: Unsigned) -> Unsigned returns the smallest
      integer greater than or equal to n that is also a power of two.

   *  additive_secret_share(vec: Vec[Field], num_shares: Unsigned,
      field: type) -> Vec[Vec[Field]] takes a vector of field elements
      and returns multiple vectors of the same length, such that they
      all add up to the input vector, and each proper subset of the
      vectors are indistinguishable from random.

3.  Overview

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                    +--------------+
              +---->| Aggregator 0 |----+
              |     +--------------+    |
              |             ^           |
              |             |           |
              |             V           |
              |     +--------------+    |
              | +-->| Aggregator 1 |--+ |
              | |   +--------------+  | |
   +--------+-+ |           ^         | +->+-----------+
   | Client |---+           |         +--->| Collector |--> Aggregate
   +--------+-+                         +->+-----------+
              |            ...          |
              |                         |
              |             |           |
              |             V           |
              |    +----------------+   |
              +--->| Aggregator N-1 |---+
                   +----------------+

         Input shares           Aggregate shares

                  Figure 1: Overall data flow of a (V)DAF

   In a DAF- or VDAF-based private measurement system, we distinguish
   three types of actors: Clients, Aggregators, and Collectors.  The
   overall flow of the measurement process is as follows:

   *  To submit an individual measurement, the Client shards the
      measurement into "input shares" and sends one input share to each
      Aggregator.  We sometimes refer to this sequence of input shares
      collectively as the Client's "report".

   *  The Aggregators refine their input shares into "output shares".

      -  Output shares are in one-to-one correspondence with the input
         shares.

      -  Just as each Aggregator receives one input share of each
         measurement, if this process succeeds, then each aggregator
         holds one output share.

      -  In VDAFs, Aggregators will need to exchange information among
         themselves as part of the validation process.

   *  Each Aggregator combines the output shares in the batch to compute
      the "aggregate share" for that batch, i.e., its share of the
      desired aggregate result.

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   *  The Aggregators submit their aggregate shares to the Collector,
      who combines them to obtain the aggregate result over the batch.

   Aggregators are a new class of actor relative to traditional
   measurement systems where Clients submit measurements to a single
   server.  They are critical for both the privacy properties of the
   system and, in the case of VDAFs, the correctness of the measurements
   obtained.  The privacy properties of the system are assured by non-
   collusion among Aggregators, and Aggregators are the entities that
   perform validation of Client measurements.  Thus Clients trust
   Aggregators not to collude (typically it is required that at least
   one Aggregator is honest), and Collectors trust Aggregators to
   correctly run the protocol.

   Within the bounds of the non-collusion requirements of a given (V)DAF
   instance, it is possible for the same entity to play more than one
   role.  For example, the Collector could also act as an Aggregator,
   effectively using the other Aggregator(s) to augment a basic client-
   server protocol.

   In this document, we describe the computations performed by the
   actors in this system.  It is up to the higher-level protocol making
   use of the (V)DAF to arrange for the required information to be
   delivered to the proper actors in the proper sequence.  In general,
   we assume that all communications are confidential and mutually
   authenticated, with the exception that Clients submitting
   measurements may be anonymous.

4.  Definition of DAFs

   By way of a gentle introduction to VDAFs, this section describes a
   simpler class of schemes called Distributed Aggregation Functions
   (DAFs).  Unlike VDAFs, DAFs do not provide verifiability of the
   computation.  Clients must therefore be trusted to compute their
   input shares correctly.  Because of this fact, the use of a DAF is
   NOT RECOMMENDED for most applications.  See Section 9 for additional
   discussion.

   A DAF scheme is used to compute a particular "aggregation function"
   over a set of measurements generated by Clients.  Depending on the
   aggregation function, the Collector might select an "aggregation
   parameter" and disseminates it to the Aggregators.  The semantics of
   this parameter is specific to the aggregation function, but in
   general it is used to represent the set of "queries" that can be made
   on the measurement set.  For example, the aggregation parameter is
   used to represent the candidate prefixes in Poplar1 Section 8.

   Execution of a DAF has four distinct stages:

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   *  Sharding - Each Client generates input shares from its measurement
      and distributes them among the Aggregators.

   *  Preparation - Each Aggregator converts each input share into an
      output share compatible with the aggregation function.  This
      computation involves the aggregation parameter.  In general, each
      aggregation parameter may result in a different an output share.

   *  Aggregation - Each Aggregator combines a sequence of output shares
      into its aggregate share and sends the aggregate share to the
      Collector.

   *  Unsharding - The Collector combines the aggregate shares into the
      aggregate result.

   Sharding and Preparation are done once per measurement.  Aggregation
   and Unsharding are done over a batch of measurements (more precisely,
   over the recovered output shares).

   A concrete DAF specifies an algorithm for the computation needed in
   each of these stages.  The interface of each algorithm is defined in
   the remainder of this section.  In addition, a concrete DAF defines
   the associated constants and types enumerated in the following table.

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            +=============+===================================+
            | Parameter   | Description                       |
            +=============+===================================+
            | ID          | Algorithm identifier for this     |
            |             | DAF.  A 32-bit, unsigned integer. |
            +-------------+-----------------------------------+
            | SHARES      | Number of input shares into which |
            |             | each measurement is sharded.      |
            +-------------+-----------------------------------+
            | NONCE_SIZE  | Size of the nonce passed by the   |
            |             | application.                      |
            +-------------+-----------------------------------+
            | RAND_SIZE   | Size of the random byte string    |
            |             | passed to sharding algorithm.     |
            +-------------+-----------------------------------+
            | Measurement | Type of each measurement.         |
            +-------------+-----------------------------------+
            | PublicShare | Type of each public share.        |
            +-------------+-----------------------------------+
            | InputShare  | Type of each input share.         |
            +-------------+-----------------------------------+
            | AggParam    | Type of aggregation parameter.    |
            +-------------+-----------------------------------+
            | OutShare    | Type of each output share.        |
            +-------------+-----------------------------------+
            | AggShare    | Type of the aggregate share.      |
            +-------------+-----------------------------------+
            | AggResult   | Type of the aggregate result.     |
            +-------------+-----------------------------------+

                Table 1: Constants and types defined by each
                               concrete DAF.

   These types define the inputs and outputs of DAF methods at various
   stages of the computation.  Some of these values need to be written
   to the network in order to carry out the computation.  In particular,
   it is RECOMMENDED that concrete instantiations of the Daf interface
   specify a method of encoding the PublicShare, InputShare, and
   AggShare.

   Each DAF is identified by a unique, 32-bit integer ID.  Identifiers
   for each (V)DAF specified in this document are defined in Table 17.

4.1.  Sharding

   In order to protect the privacy of its measurements, a DAF Client
   shards its measurements into a sequence of input shares.  The shard
   method is used for this purpose.

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   *  Daf.shard(measurement: Measurement, nonce: bytes[Daf.NONCE_SIZE],
      rand: bytes[Daf.RAND_SIZE]) -> tuple[PublicShare,
      list[InputShare]] is the randomized sharding algorithm run by each
      Client.  The input rand consists of the random bytes consumed by
      the algorithm.  This value MUST be generated using a
      cryptographically secure pseudorandom number generator (CSPRNG).
      It consumes the measurement and produces a "public share",
      distributed to each of the Aggregators, and a corresponding
      sequence of input shares, one for each Aggregator.  The length of
      the output vector MUST be SHARES.

       Client
       ======

       measurement
         |
         V
       +----------------------------------------------+
       | shard                                        |
       +----------------------------------------------+
         |              |              |     |
         |              |         ...  |    public_share
         |              |              |     |
         |    +---------|-----+--------|-----+
         |    |         |     |        |     |
         V    |         V     |        V     |
        input_share_0  input_share_1  input_share_[SHARES-1]
         |    |         |     |   ...  |     |
         V    V         V     V        V     V
       Aggregator 0   Aggregator 1    Aggregator SHARES-1

       Figure 2: The Client divides its measurement into input shares
       and distributes them to the Aggregators.  The public share is
                       broadcast to all Aggregators.

4.2.  Preparation

   Once an Aggregator has received the public share and one of the input
   shares, the next step is to prepare the input share for aggregation.
   This is accomplished using the following algorithm:

   *  Daf.prep(agg_id: Unsigned, agg_param: AggParam, nonce:
      bytes[NONCE_SIZE], public_share: PublicShare, input_share:
      InputShare) -> OutShare is the deterministic preparation
      algorithm.  It takes as input the public share and one of the
      input shares generated by a Client, the Aggregator's unique
      identifier, the aggregation parameter selected by the Collector,
      and a nonce and returns an output share.

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   The protocol in which the DAF is used MUST ensure that the
   Aggregator's identifier is equal to the integer in range [0, SHARES)
   that matches the index of input_share in the sequence of input shares
   output by the Client.

4.3.  Validity of Aggregation Parameters

   Concrete DAFs implementations MAY impose certain restrictions for
   input shares and aggregation parameters.  Protocols using a DAF MUST
   ensure that for each input share and aggregation parameter agg_param,
   Daf.prep is only called if Daf.is_valid(agg_param,
   previous_agg_params) returns True, where previous_agg_params contains
   all aggregation parameters that have previously been used with the
   same input share.

   DAFs MUST implement the following function:

   *  Daf.is_valid(agg_param: AggParam, previous_agg_params:
      set[AggParam]) -> Bool: Checks if the agg_param is compatible with
      all elements of previous_agg_params.

4.4.  Aggregation

   Once an Aggregator holds output shares for a batch of measurements
   (where batches are defined by the application), it combines them into
   a share of the desired aggregate result:

   *  Daf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) ->
      AggShare is the deterministic aggregation algorithm.  It is run by
      each Aggregator a set of recovered output shares.

       Aggregator 0    Aggregator 1        Aggregator SHARES-1
       ============    ============        ===================

       out_share_0_0   out_share_1_0       out_share_[SHARES-1]_0
       out_share_0_1   out_share_1_1       out_share_[SHARES-1]_1
       out_share_0_2   out_share_1_2       out_share_[SHARES-1]_2
            ...             ...                     ...
       out_share_0_B   out_share_1_B       out_share_[SHARES-1]_B
         |               |                   |
         V               V                   V
       +-----------+   +-----------+       +-----------+
       | aggregate |   | aggregate |   ... | aggregate |
       +-----------+   +-----------+       +-----------+
         |               |                   |
         V               V                   V
       agg_share_0     agg_share_1         agg_share_[SHARES-1]

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    Figure 3: Aggregation of output shares. `B` indicates the number of
                         measurements in the batch.

   For simplicity, we have written this algorithm in a "one-shot" form,
   where all output shares for a batch are provided at the same time.
   Many DAFs may also support a "streaming" form, where shares are
   processed one at a time.

   Implementation note: For most natural DAFs (and VDAFs) it is not
   necessary for an Aggregator to store all output shares individually
   before aggregating.  Typically it is possible to merge output shares
   into aggregate shares as they arrive, merge these into other
   aggregate shares, and so on.  In particular, this is the case when
   the output shares are vectors over some finite field and aggregating
   them involves merely adding up the vectors element-wise.  Such is the
   case for Prio3 Section 7 and Poplar1 Section 8.

4.5.  Unsharding

   After the Aggregators have aggregated a sufficient number of output
   shares, each sends its aggregate share to the Collector, who runs the
   following algorithm to recover the following output:

   *  Daf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
      num_measurements: Unsigned) -> AggResult is run by the Collector
      in order to compute the aggregate result from the Aggregators'
      shares.  The length of agg_shares MUST be SHARES. num_measurements
      is the number of measurements that contributed to each of the
      aggregate shares.  This algorithm is deterministic.

       Aggregator 0    Aggregator 1        Aggregator SHARES-1
       ============    ============        ===================

       agg_share_0     agg_share_1         agg_share_[SHARES-1]
         |               |                   |
         V               V                   V
       +-----------------------------------------------+
       | unshard                                       |
       +-----------------------------------------------+
         |
         V
       agg_result

       Collector
       =========

          Figure 4: Computation of the final aggregate result from
                             aggregate shares.

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      QUESTION Maybe the aggregation algorithms should be randomized in
      order to allow the Aggregators (or the Collector) to add noise for
      differential privacy.  (See the security considerations of [DAP].)
      Or is this out-of-scope of this document?  See https://github.com/
      ietf-wg-ppm/ppm-specification/issues/19.

4.6.  Execution of a DAF

   Securely executing a DAF involves emulating the following procedure.

   def run_daf(Daf,
               agg_param: Daf.AggParam,
               measurements: list[Daf.Measurement],
               nonces: list[bytes[Daf.NONCE_SIZE]]):
       out_shares = [[] for j in range(Daf.SHARES)]
       for (measurement, nonce) in zip(measurements, nonces):
           # Each Client shards its measurement into input shares and
           # distributes them among the Aggregators.
           rand = gen_rand(Daf.RAND_SIZE)
           (public_share, input_shares) = \
               Daf.shard(measurement, nonce, rand)

           # Each Aggregator prepares its input share for aggregation.
           for j in range(Daf.SHARES):
               out_shares[j].append(
                   Daf.prep(j, agg_param, nonce,
                            public_share, input_shares[j]))

       # Each Aggregator aggregates its output shares into an aggregate
       # share and sends it to the Collector.
       agg_shares = []
       for j in range(Daf.SHARES):
           agg_share_j = Daf.aggregate(agg_param,
                                       out_shares[j])
           agg_shares.append(agg_share_j)

       # Collector unshards the aggregate result.
       num_measurements = len(measurements)
       agg_result = Daf.unshard(agg_param, agg_shares,
                                num_measurements)
       return agg_result

                       Figure 5: Execution of a DAF.

   The inputs to this procedure are the same as the aggregation function
   computed by the DAF: An aggregation parameter and a sequence of
   measurements.  The procedure prescribes how a DAF is executed in a
   "benign" environment in which there is no adversary and the messages

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   are passed among the protocol participants over secure point-to-point
   channels.  In reality, these channels need to be instantiated by some
   "wrapper protocol", such as [DAP], that realizes these channels using
   suitable cryptographic mechanisms.  Moreover, some fraction of the
   Aggregators (or Clients) may be malicious and diverge from their
   prescribed behaviors.  Section 9 describes the execution of the DAF
   in various adversarial environments and what properties the wrapper
   protocol needs to provide in each.

5.  Definition of VDAFs

   Like DAFs described in the previous section, a VDAF scheme is used to
   compute a particular aggregation function over a set of Client-
   generated measurements.  Evaluation of a VDAF involves the same four
   stages as for DAFs: Sharding, Preparation, Aggregation, and
   Unsharding.  However, the Preparation stage will require interaction
   among the Aggregators in order to facilitate verifiability of the
   computation's correctness.  Accommodating this interaction will
   require syntactic changes.

   Overall execution of a VDAF comprises the following stages:

   *  Sharding - Computing input shares from an individual measurement

   *  Preparation - Conversion and verification of input shares to
      output shares compatible with the aggregation function being
      computed

   *  Aggregation - Combining a sequence of output shares into an
      aggregate share

   *  Unsharding - Combining a sequence of aggregate shares into an
      aggregate result

   In contrast to DAFs, the Preparation stage for VDAFs now performs an
   additional task: Verification of the validity of the recovered output
   shares.  This process ensures that aggregating the output shares will
   not lead to a garbled aggregate result.

   The remainder of this section defines the VDAF interface.  The
   attributes are listed in Table 2 are defined by each concrete VDAF.

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      +=================+==========================================+
      | Parameter       | Description                              |
      +=================+==========================================+
      | ID              | Algorithm identifier for this VDAF.      |
      +-----------------+------------------------------------------+
      | VERIFY_KEY_SIZE | Size (in bytes) of the verification key  |
      |                 | (Section 5.2).                           |
      +-----------------+------------------------------------------+
      | RAND_SIZE       | Size of the random byte string passed to |
      |                 | sharding algorithm.                      |
      +-----------------+------------------------------------------+
      | NONCE_SIZE      | Size (in bytes) of the nonce.            |
      +-----------------+------------------------------------------+
      | ROUNDS          | Number of rounds of communication during |
      |                 | the Preparation stage (Section 5.2).     |
      +-----------------+------------------------------------------+
      | SHARES          | Number of input shares into which each   |
      |                 | measurement is sharded (Section 5.1).    |
      +-----------------+------------------------------------------+
      | Measurement     | Type of each measurement.                |
      +-----------------+------------------------------------------+
      | PublicShare     | Type of each public share.               |
      +-----------------+------------------------------------------+
      | InputShare      | Type of each input share.                |
      +-----------------+------------------------------------------+
      | AggParam        | Type of aggregation parameter.           |
      +-----------------+------------------------------------------+
      | OutShare        | Type of each output share.               |
      +-----------------+------------------------------------------+
      | AggShare        | Type of the aggregate share.             |
      +-----------------+------------------------------------------+
      | AggResult       | Type of the aggregate result.            |
      +-----------------+------------------------------------------+
      | PrepState       | Aggregator's state during preparation.   |
      +-----------------+------------------------------------------+
      | PrepShare       | Type of each prep share.                 |
      +-----------------+------------------------------------------+
      | PrepMessage     | Type of each prep message.               |
      +-----------------+------------------------------------------+

       Table 2: Constants and types defined by each concrete VDAF.

   Some of these values need to be written to the network in order to
   carry out the computation.  In particular, it is RECOMMENDED that
   concrete instantiations of the Vdaf interface specify a method of
   encoding the PublicShare, InputShare, AggShare, PrepShare, and
   PrepMessage.

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   Each VDAF is identified by a unique, 32-bit integer ID.  Identifiers
   for each (V)DAF specified in this document are defined in Table 17.
   The following method is defined for every VDAF:

   def domain_separation_tag(Vdaf, usage: Unsigned) -> Bytes:
       """
       Format domain separation tag for this VDAF with the given usage.
       """
       return format_dst(0, Vdaf.ID, usage)

   It is used to construct a domain separation tag for an instance of
   Xof used by the VDAF.  (See Section 6.2.)

5.1.  Sharding

   Sharding transforms a measurement into input shares as it does in
   DAFs (cf.  Section 4.1); in addition, it takes a nonce as input and
   produces a public share:

   *  Vdaf.shard(measurement: Measurement, nonce:
      bytes[Vdaf.NONCE_SIZE], rand: bytes[Vdaf.RAND_SIZE]) ->
      tuple[PublicShare, list[InputShare]] is the randomized sharding
      algorithm run by each Client.  Input rand consists of the random
      bytes consumed by the algorithm.  It consumes the measurement and
      the nonce and produces a public share, distributed to each of
      Aggregators, and the corresponding sequence of input shares, one
      for each Aggregator.  Depending on the VDAF, the input shares may
      encode additional information used to verify the recovered output
      shares (e.g., the "proof shares" in Prio3 Section 7).  The length
      of the output vector MUST be SHARES.

   In order to ensure privacy of the measurement, the Client MUST
   generate the random bytes and nonce using a CSPRNG.  (See Section 9
   for details.)

5.2.  Preparation

   To recover and verify output shares, the Aggregators interact with
   one another over ROUNDS rounds.  Prior to each round, each Aggregator
   constructs an outbound message.  Next, the sequence of outbound
   messages is combined into a single message, called a "preparation
   message", or "prep message" for short.  (Each of the outbound
   messages are called "preparation-message shares", or "prep shares"
   for short.)  Finally, the preparation message is distributed to the
   Aggregators to begin the next round.

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   An Aggregator begins the first round with its input share and it
   begins each subsequent round with the previous prep message.  Its
   output in the last round is its output share and its output in each
   of the preceding rounds is a prep share.

   This process involves a value called the "aggregation parameter" used
   to map the input shares to output shares.  The Aggregators need to
   agree on this parameter before they can begin preparing the
   measurement shares for aggregation.

       Aggregator 0   Aggregator 1        Aggregator SHARES-1
       ============   ============        ===================

       input_share_0  input_share_1       input_share_[SHARES-1]
         |              |              ...  |
         V              V                   V
       +-----------+  +-----------+       +-----------+
       | prep_init |  | prep_init |       | prep_init |
       +-----------+  +------------+      +-----------+
         |              |              ...  |
         V              V                   V
       +----------------------------------------------+   \
       | prep_shares_to_prep                          |   |
       +----------------------------------------------+   |
         |              |              ...  |             |
         V              V                   V             | x ROUNDS
       +-----------+  +-----------+       +-----------+   |
       | prep_next |  | prep_next |       | prep_next |   |
       +-----------+  +-----------+       +-----------+   |
         |              |              ...  |             |
         V              V                   V             /
        ...            ...                 ...
         |              |              ...  |
         V              V                   V
       out_share_0    out_share_1         out_share_[SHARES-1]

    Figure 6: VDAF preparation process on the input shares for a single
   measurement.  At the end of the computation, each Aggregator holds an
                         output share or an error.

   To facilitate the preparation process, a concrete VDAF implements the
   following methods:

   *  Vdaf.prep_init(verify_key: bytes[Vdaf.VERIFY_KEY_SIZE], agg_id:
      Unsigned, agg_param: AggParam, nonce: bytes[Vdaf.NONCE_SIZE],
      public_share: PublicShare, input_share: InputShare) ->
      tuple[PrepState, PrepShare] is the deterministic preparation-state
      initialization algorithm run by each Aggregator to begin

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      processing its input share into an output share.  Its inputs are
      the shared verification key (verify_key), the Aggregator's unique
      identifier (agg_id), the aggregation parameter (agg_param), the
      nonce provided by the environment (nonce, see Figure 7), the
      public share (public_share), and one of the input shares generated
      by the Client (input_share).  Its output is the Aggregator's
      initial preparation state and initial prep share.

      It is up to the high level protocol in which the VDAF is used to
      arrange for the distribution of the verification key prior to
      generating and processing reports.  (See Section 9 for details.)

      Protocols using the VDAF MUST ensure that the Aggregator's
      identifier is equal to the integer in range [0, SHARES) that
      matches the index of input_share in the sequence of input shares
      output by the Client.

      Protocols MUST ensure that public share consumed by each of the
      Aggregators is identical.  This is security critical for VDAFs
      such as Poplar1.

   *  Vdaf.prep_next(prep_state: PrepState, prep_msg: PrepMessage) ->
      Union[tuple[PrepState, PrepShare], OutShare] is the deterministic
      preparation-state update algorithm run by each Aggregator.  It
      updates the Aggregator's preparation state (prep_state) and
      returns either its next preparation state and its message share
      for the current round or, if this is the last round, its output
      share.  An exception is raised if a valid output share could not
      be recovered.  The input of this algorithm is the inbound
      preparation message.

   *  Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares:
      list[PrepShare]) -> PrepMessage is the deterministic preparation-
      message pre-processing algorithm.  It combines the prep shares
      generated by the Aggregators in the previous round into the prep
      message consumed by each in the next round.

   In effect, each Aggregator moves through a linear state machine with
   ROUNDS states.  The Aggregator enters the first state on using the
   initialization algorithm, and the update algorithm advances the
   Aggregator to the next state.  Thus, in addition to defining the
   number of rounds (ROUNDS), a VDAF instance defines the state of the
   Aggregator after each round.

      TODO Consider how to bake this "linear state machine" condition
      into the syntax.  Given that Python 3 is used as our pseudocode,
      it's easier to specify the preparation state using a class.

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   The preparation-state update accomplishes two tasks: recovery of
   output shares from the input shares and ensuring that the recovered
   output shares are valid.  The abstraction boundary is drawn so that
   an Aggregator only recovers an output share if it is deemed valid (at
   least, based on the Aggregator's view of the protocol).  Another way
   to draw this boundary would be to have the Aggregators recover output
   shares first, then verify that they are valid.  However, this would
   allow the possibility of misusing the API by, say, aggregating an
   invalid output share.  Moreover, in protocols like Prio+ [AGJOP21]
   based on oblivious transfer, it is necessary for the Aggregators to
   interact in order to recover aggregatable output shares at all.

   Note that it is possible for a VDAF to specify ROUNDS == 0, in which
   case each Aggregator runs the preparation-state update algorithm once
   and immediately recovers its output share without interacting with
   the other Aggregators.  However, most, if not all, constructions will
   require some amount of interaction in order to ensure validity of the
   output shares (while also maintaining privacy).

      OPEN ISSUE accommodating 0-round VDAFs may require syntax changes
      if, for example, public keys are required.  On the other hand, we
      could consider defining this class of schemes as a different
      primitive.  See issue#77.

5.3.  Validity of Aggregation Parameters

   Similar to DAFs (see Section 4.3), VDAFs MAY impose restrictions for
   input shares and aggregation parameters.  Protocols using a VDAF MUST
   ensure that for each input share and aggregation parameter agg_param,
   the preparation phase (including Vdaf.prep_init, Vdaf.prep_next, and
   Vdaf.prep_shares_to_prep; see Section 5.2) is only called if
   Vdaf.is_valid(agg_param, previous_agg_params) returns True, where
   previous_agg_params contains all aggregation parameters that have
   previously been used with the same input share.

   VDAFs MUST implement the following function:

   *  Vdaf.is_valid(agg_param: AggParam, previous_agg_params:
      set[AggParam]) -> Bool: Checks if the agg_param is compatible with
      all elements of previous_agg_params.

5.4.  Aggregation

   VDAF Aggregation is identical to DAF Aggregation (cf.  Section 4.4):

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   *  Vdaf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) ->
      AggShare is the deterministic aggregation algorithm.  It is run by
      each Aggregator over the output shares it has computed for a batch
      of measurements.

   The data flow for this stage is illustrated in Figure 3.  Here again,
   we have the aggregation algorithm in a "one-shot" form, where all
   shares for a batch are provided at the same time.  VDAFs typically
   also support a "streaming" form, where shares are processed one at a
   time.

5.5.  Unsharding

   VDAF Unsharding is identical to DAF Unsharding (cf.  Section 4.5):

   *  Vdaf.unshard(agg_param: AggParam, agg_shares: list[AggShare],
      num_measurements: Unsigned) -> AggResult is run by the Collector
      in order to compute the aggregate result from the Aggregators'
      shares.  The length of agg_shares MUST be SHARES. num_measurements
      is the number of measurements that contributed to each of the
      aggregate shares.  This algorithm is deterministic.

   The data flow for this stage is illustrated in Figure 4.

5.6.  Execution of a VDAF

   Secure execution of a VDAF involves simulating the following
   procedure.

 def run_vdaf(Vdaf,
              verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
              agg_param: Vdaf.AggParam,
              nonces: list[bytes[Vdaf.NONCE_SIZE]],
              measurements: list[Vdaf.Measurement]):
     out_shares = []
     for (nonce, measurement) in zip(nonces, measurements):
         # Each Client shards its measurement into input shares.
         rand = gen_rand(Vdaf.RAND_SIZE)
         (public_share, input_shares) = \
             Vdaf.shard(measurement, nonce, rand)

         # Each Aggregator initializes its preparation state.
         prep_states = []
         outbound = []
         for j in range(Vdaf.SHARES):
             (state, share) = Vdaf.prep_init(verify_key, j,
                                             agg_param,
                                             nonce,

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                                             public_share,
                                             input_shares[j])
             prep_states.append(state)
             outbound.append(share)

         # Aggregators recover their output shares.
         for i in range(Vdaf.ROUNDS-1):
             prep_msg = Vdaf.prep_shares_to_prep(agg_param,
                                                 outbound)
             outbound = []
             for j in range(Vdaf.SHARES):
                 out = Vdaf.prep_next(prep_states[j], prep_msg)
                 (prep_states[j], out) = out
                 outbound.append(out)

         # The final outputs of the prepare phase are the output shares.
         prep_msg = Vdaf.prep_shares_to_prep(agg_param,
                                             outbound)
         outbound = []
         for j in range(Vdaf.SHARES):
             out_share = Vdaf.prep_next(prep_states[j], prep_msg)
             outbound.append(out_share)
         out_shares.append(outbound)

     # Each Aggregator aggregates its output shares into an
     # aggregate share. In a distributed VDAF computation, the
     # aggregate shares are sent over the network.
     agg_shares = []
     for j in range(Vdaf.SHARES):
         out_shares_j = [out[j] for out in out_shares]
         agg_share_j = Vdaf.aggregate(agg_param, out_shares_j)
         agg_shares.append(agg_share_j)

     # Collector unshards the aggregate.
     num_measurements = len(measurements)
     agg_result = Vdaf.unshard(agg_param, agg_shares,
                               num_measurements)
     return agg_result

                     Figure 7: Execution of a VDAF.

   The inputs to this algorithm are the aggregation parameter, a list of
   measurements, and a nonce for each measurement.  This document does
   not specify how the nonces are chosen, but security requires that the
   nonces be unique.  See Section 9 for details.  As explained in
   Section 4.6, the secure execution of a VDAF requires the application
   to instantiate secure channels between each of the protocol
   participants.

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5.7.  Communication Patterns for Preparation

   In each round of preparation, each Aggregator writes a prep share to
   some broadcast channel, which is then processed into the prep message
   using the public prep_shares_to_prep() algorithm and broadcast to the
   Aggregators to start the next round.  In this section we describe
   some approaches to realizing this broadcast channel functionality in
   protocols that use VDAFs.

   The state machine of each Aggregator is shown in Figure 8.

                  +----------------+
                  |                |
                  v                |
   Start ------> Continued(prep_state) --> Finished(out_share)
    |                |
    |                |
    +--> Rejected <--+

               Figure 8: State machine for VDAF preparation.

   State transitions are made when the state is acted upon by the host's
   local inputs and/or messages sent by the peers.  The initial state is
   Start.  The terminal states are Rejected, which indicates that the
   report cannot be processed any further, and Finished(out_share),
   which indicates that the Aggregator has recovered an output share
   out_share.

   class State:
       pass

   class Start(State):
       pass

   class Continued(State):
       def __init__(self, prep_state):
           self.prep_state = prep_state

   class Finished(State):
       def __init__(self, output_share):
           self.output_share = output_share

   class Rejected(State):
       def __init__(self):
           pass

   Note that there is no representation of the Start state as it is
   never instantiated in the ping-pong topology.

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   For convenience, the methods described in this section are defined in
   terms of opaque byte strings.  A compatible Vdaf MUST specify methods
   for encoding public shares, input shares, prep shares, prep messages,
   and aggregation parameters.  Minimally:

   *  Vdaf.decode_public_share(encoded: bytes) -> Vdaf.PublicShare
      decodes a public share.

   *  Vdaf.decode_input_share(agg_id: Unsigned, encoded: bytes) ->
      Vdaf.InputShare decodes an input share, using the aggregator ID as
      optional context.

   *  Vdaf.encode_prep_share(prep_share: Vdaf.PrepShare) -> bytes
      encodes a prep share.

   *  Vdaf.decode_prep_share(prep_state: Vdaf.PrepState, encoded: bytes)
      -> Vdaf.PrepShare decodes a prep share, using the prep state as
      optional context.

   *  Vdaf.encode_prep_msg(prep_msg: Vdaf.PrepMessage) -> bytes encodes
      a prep message.

   *  Vdaf.decode_prep_msg(prep_state: Vdaf.PrepState, encoded: bytes)
      -> Vdaf.PrepMessage decodes a prep message, using the prep state
      as optional decoding context.

   *  Vdaf.decode_agg_param(encoded: bytes) -> Vdaf.AggParam decodes an
      aggregation parameter.

   *  Vdaf.encode_agg_param(agg_param: Vdaf.AggParam) -> bytes encodes
      an aggregation parameter.

   Implementations of Prio3 and Poplar1 MUST use the encoding scheme
   specified in Section 7.2.7 and Section 8.2.6 respectively.

5.8.  Ping-Pong Topology (Only Two Aggregators)

   For VDAFs with precisely two Aggregators (i.e., Vdaf.SHARES == 2),
   the following "ping pong" communication pattern can be used.  It is
   compatible with any request/response transport protocol, such as
   HTTP.

   Let us call the initiating party the "Leader" and the responding
   party the "Helper".  The high-level idea is that the Leader and
   Helper will take turns running the computation locally until input
   from their peer is required:

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   *  For a 1-round VDAF (e.g., Prio3 (Section 7)), the Leader sends its
      prep share to the Helper, who computes the prep message locally,
      computes its output share, then sends the prep message to the
      Leader.  Preparation requires just one round trip between the
      Leader and the Helper.

   *  For a 2-round VDAF (e.g., Poplar1 (Section 8)), the Leader sends
      its first-round prep share to the Helper, who replies with the
      first-round prep message and its second-round prep share.  In the
      next request, the Leader computes its second-round prep share
      locally, computes its output share, and sends the second-round
      prep message to the Helper.  Finally, the Helper computes its own
      output share.

   *  In general, each request includes the Leader's prep share for the
      previous round and/or the prep message for the current round;
      correspondingly, each response consists of the prep message for
      the current round and the Helper's prep share for the next round.

   The Aggregators proceed in this ping-ponging fashion until a step of
   the computation fails (indicating the report is invalid and should be
   rejected) or preparation is completed.  All told there there are
   ceil((Vdaf.ROUNDS+1)/2) requests sent.

   Each message in the ping-pong protocol is structured as follows
   (expressed in TLS syntax as defined in Section 3 of [RFC8446]):

   enum {
     initialize(0),
     continue(1),
     finish(2),
     (255)
   } MessageType;

   struct {
     MessageType type;
     select (Message.type) {
       case initialize:
         opaque prep_share<0..2^32-1>;
       case continue:
         opaque prep_msg<0..2^32-1>;
         opaque prep_share<0..2^32-1>;
       case finish:
         opaque prep_msg<0..2^32-1>;
     };
   } Message;

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   These messages are used to transition between the states described in
   Section 5.7.  They are encoded and decoded to or from byte buffers as
   described Section 3 of [RFC8446]) using the following routines:

   *  encode_ping_pong_message(message: Message) -> bytes encodes a
      Message into an opaque byte buffer.

   *  decode_pong_pong_message(encoded: bytes) -> Message decodes an
      opaque byte buffer into a Message, raising an error if the bytes
      are not a valid encoding.

   The Leader's initial transition is computed with the following
   procedure:

   def ping_pong_leader_init(
               Vdaf,
               vdaf_verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
               agg_param: bytes,
               nonce: bytes[Vdaf.NONCE_SIZE],
               public_share: bytes,
               input_share: bytes,
           ) -> tuple[State, bytes]:
       try:
           (prep_state, prep_share) = Vdaf.prep_init(
               vdaf_verify_key,
               0,
               Vdaf.decode_agg_param(agg_param),
               nonce,
               Vdaf.decode_public_share(public_share),
               Vdaf.decode_input_share(0, input_share),
           )
           outbound = Message.initialize(
               Vdaf.encode_prep_share(prep_share))
           return (Continued(prep_state), encode_ping_pong_message(outbound))
       except:
           return (Rejected(), None)

   The output is the State to which the Leader has transitioned and an
   encoded Message.  If the Leader's state is Rejected, then processing
   halts.  Otherwise, if the state is Continued, then processing
   continues.

   The Leader sends the outbound message to the Helper.  The Helper's
   initial transition is computed using the following procedure:

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   def ping_pong_helper_init(
               Vdaf,
               vdaf_verify_key: bytes[Vdaf.VERIFY_KEY_SIZE],
               agg_param: bytes,
               nonce: bytes[Vdaf.NONCE_SIZE],
               public_share: bytes,
               input_share: bytes,
               inbound_encoded: bytes,
           ) -> tuple[State, bytes]:
       try:
           (prep_state, prep_share) = Vdaf.prep_init(
               vdaf_verify_key,
               1,
               Vdaf.decode_agg_param(agg_param),
               nonce,
               Vdaf.decode_public_share(public_share),
               Vdaf.decode_input_share(1, input_share),
           )

           inbound = decode_ping_pong_message(inbound_encoded)

           if inbound.type != 0: # initialize
               return (Rejected(), None)

           prep_shares = [
               Vdaf.decode_prep_share(prep_state, inbound.prep_share),
               prep_share,
           ]
           return Vdaf.ping_pong_transition(
               agg_param,
               prep_shares,
               prep_state,
           )
       except:
           return (Rejected(), None)

   Procedure ping_pong_transition() takes in the prep shares, combines
   them into the prep message, and computes the next prep state of the
   caller:

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  def ping_pong_transition(
              Vdaf,
              agg_param: Vdaf.AggParam,
              prep_shares: list[Vdaf.PrepShare],
              prep_state: Vdaf.PrepState,
           ) -> (State, bytes):
      prep_msg = Vdaf.prep_shares_to_prep(agg_param,
                                          prep_shares)
      out = Vdaf.prep_next(prep_state, prep_msg)
      if type(out) == Vdaf.OutShare:
          outbound = Message.finish(Vdaf.encode_prep_msg(prep_msg))
          return (Finished(out), encode_ping_pong_message(outbound))
      (prep_state, prep_share) = out
      outbound = Message.continue(
          Vdaf.encode_prep_msg(prep_msg),
          Vdaf.encode_prep_share(prep_share),
      )
      return (Continued(prep_state), encode_ping_pong_message(outbound))

   The output is the State to which the Helper has transitioned and an
   encoded Message.  If the Helper's state is Finished or Rejected, then
   processing halts.  Otherwise, if the state is Continued, then
   processing continues.

   Next, the Helper sends the outbound message to the Leader.  The
   Leader computes its next state transition using the function
   ping_pong_leader_continued:

   def ping_pong_leader_continued(
               Vdaf,
               agg_param: bytes,
               state: State,
               inbound_encoded: bytes,
           ) -> (State, Optional[bytes]):
       return Vdaf.ping_pong_continued(
           True,
           agg_param,
           state,
           inbound_encoded,
       )

   def ping_pong_continued(
               Vdaf,
               is_leader: bool,
               agg_param: bytes,
               state: State,
               inbound_encoded: bytes,
           ) -> (State, Optional[bytes]):

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       try:
           inbound = decode_ping_pong_message(inbound_encoded)

           if inbound.type == 0: # initialize
               return (Rejected(), None)

           if !isinstance(state, Continued):
               return (Rejected(), None)

           prep_msg = Vdaf.decode_prep_msg(state.prep_state, inbound.prep_msg)
           out = Vdaf.prep_next(state.prep_state, prep_msg)
           if type(out) == tuple[Vdaf.PrepState, Vdaf.PrepShare] \
                   and inbound.type == 1:
               # continue
               (prep_state, prep_share) = out
               prep_shares = [
                   Vdaf.decode_prep_share(prep_state, inbound.prep_share),
                   prep_share,
               ]
               if is_leader:
                   prep_shares.reverse()
               return Vdaf.ping_pong_transition(
                   Vdaf.decode_agg_param(agg_param),
                   prep_shares,
                   prep_state,
               )
           elif type(out) == Vdaf.OutShare and inbound.type == 2:
               # finish
               return (Finished(out), None)
           else:
               return (Rejected(), None)

       except:
           return (Rejected(), None)

   If the Leader's state is Finished or Rejected, then processing halts.
   Otherwise, the Leader sends the outbound message to the Helper.  The
   Helper computes its next state transition using the function
   ping_pong_helper_continued:

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   def ping_pong_helper_continued(
               Vdaf,
               agg_param: bytes,
               state: State,
               inbound_encoded: bytes,
           ) -> (State, Optional[bytes]):
       return Vdaf.ping_pong_continued(
           False,
           agg_param,
           state,
           inbound_encoded,
       )

   They continue in this way until processing halts.  Note that,
   depending on the number of rounds of preparation that are required,
   there may be one more message to send before the peer can also finish
   processing (i.e., outbound != None).

5.9.  Star Topology (Any Number of Aggregators)

   The ping-pong topology of the previous section is only suitable for
   VDAFs involving exactly two Aggregators.  If more Aggregators are
   required, the star topology described in this section can be used
   instead.

      TODO Describe the Leader-emulated broadcast channel architecture
      that was originally envisioned for DAP.  (As of DAP-05 we are
      going with the ping pong architecture described in the previous
      section.)

6.  Preliminaries

   This section describes the primitives that are common to the VDAFs
   specified in this document.

6.1.  Finite Fields

   Both Prio3 and Poplar1 use finite fields of prime order.  Finite
   field elements are represented by a class Field with the following
   associated parameters:

   *  MODULUS: Unsigned is the prime modulus that defines the field.

   *  ENCODED_SIZE: Unsigned is the number of bytes used to encode a
      field element as a byte string.

   A concrete Field also implements the following class methods:

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   *  Field.zeros(length: Unsigned) -> output: Vec[Field] returns a
      vector of zeros.  The length of output MUST be length.

   *  Field.rand_vec(length: Unsigned) -> output: Vec[Field] returns a
      vector of random field elements.  The length of output MUST be
      length.

   A field element is an instance of a concrete Field.  The concrete
   class defines the usual arithmetic operations on field elements.  In
   addition, it defines the following instance method for converting a
   field element to an unsigned integer:

   *  elem.as_unsigned() -> Unsigned returns the integer representation
      of field element elem.

   Likewise, each concrete Field implements a constructor for converting
   an unsigned integer into a field element:

   *  Field(integer: Unsigned) returns integer represented as a field
      element.  The value of integer MUST be less than Field.MODULUS.

   Each concrete Field has two derived class methods, one for encoding a
   vector of field elements as a byte string and another for decoding a
   vector of field elements.

   def encode_vec(Field, data: Vec[Field]) -> Bytes:
       encoded = Bytes()
       for x in data:
           encoded += to_le_bytes(x.as_unsigned(), Field.ENCODED_SIZE)
       return encoded

   def decode_vec(Field, encoded: Bytes) -> Vec[Field]:
       L = Field.ENCODED_SIZE
       if len(encoded) % L != 0:
           raise ERR_DECODE

       vec = []
       for i in range(0, len(encoded), L):
           encoded_x = encoded[i:i+L]
           x = from_le_bytes(encoded_x)
           if x >= Field.MODULUS:
               raise ERR_DECODE # Integer is larger than modulus
           vec.append(Field(x))
       return vec

             Figure 9: Derived class methods for finite fields.

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   Finally, Field implements the following methods for representing a
   value as a sequence of field elements, each of which represents a bit
   of the input.

   def encode_into_bit_vector(Field,
                              val: Unsigned,
                              bits: Unsigned) -> Vec[Field]:
       """
       Encode the bit representation of `val` with at most `bits` number
       of bits, as a vector of field elements.
       """
       if val >= 2 ** bits:
           # Sanity check we are able to represent `val` with `bits`
           # number of bits.
           raise ValueError("Number of bits is not enough to represent "
                            "the input integer.")
       encoded = []
       for l in range(bits):
           encoded.append(Field((val >> l) & 1))
       return encoded

   def decode_from_bit_vector(Field, vec: Vec[Field]) -> Field:
       """
       Decode the field element from the bit representation, expressed
       as a vector of field elements `vec`.
       """
       bits = len(vec)
       if Field.MODULUS >> bits == 0:
           raise ValueError("Number of bits is too large to be "
                            "represented by field modulus.")
       decoded = Field(0)
       for (l, bit) in enumerate(vec):
           decoded += Field(1 << l) * bit
       return decoded

    Figure 10: Derived class methods to encode integers into bit vector
                              representation.

6.1.1.  Auxiliary Functions

   The following auxiliary functions on vectors of field elements are
   used in the remainder of this document.  Note that an exception is
   raised by each function if the operands are not the same length.

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   def vec_sub(left: Vec[Field], right: Vec[Field]):
       """
       Subtract the right operand from the left and return the result.
       """
       return list(map(lambda x: x[0] - x[1], zip(left, right)))

   def vec_add(left: Vec[Field], right: Vec[Field]):
       """Add the right operand to the left and return the result."""
       return list(map(lambda x: x[0] + x[1], zip(left, right)))

               Figure 11: Common functions for finite fields.

6.1.2.  FFT-Friendly Fields

   Some VDAFs require fields that are suitable for efficient computation
   of the discrete Fourier transform, as this allows for fast polynomial
   interpolation.  (One example is Prio3 (Section 7) when instantiated
   with the generic FLP of Section 7.3.3.)  Specifically, a field is
   said to be "FFT-friendly" if, in addition to satisfying the interface
   described in Section 6.1, it implements the following method:

   *  Field.gen() -> Field returns the generator of a large subgroup of
      the multiplicative group.  To be FFT-friendly, the order of this
      subgroup MUST be a power of 2.  In addition, the size of the
      subgroup dictates how large interpolated polynomials can be.  It
      is RECOMMENDED that a generator is chosen with order at least
      2^20.

   FFT-friendly fields also define the following parameter:

   *  GEN_ORDER: Unsigned is the order of a multiplicative subgroup
      generated by Field.gen().

6.1.3.  Parameters

   The tables below define finite fields used in the remainder of this
   document.

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   +==============+================+=======================+==========+
   | Parameter    | Field64        | Field128              | Field255 |
   +==============+================+=======================+==========+
   | MODULUS      | 2^32 *         | 2^66 *                | 2^255 -  |
   |              | 4294967295 + 1 | 4611686018427387897 + | 19       |
   |              |                | 1                     |          |
   +--------------+----------------+-----------------------+----------+
   | ENCODED_SIZE | 8              | 16                    | 32       |
   +--------------+----------------+-----------------------+----------+
   | Generator    | 7^4294967295   | 7^4611686018427387897 | n/a      |
   +--------------+----------------+-----------------------+----------+
   | GEN_ORDER    | 2^32           | 2^66                  | n/a      |
   +--------------+----------------+-----------------------+----------+

     Table 3: Parameters for the finite fields used in this document.

6.2.  Extendable Output Functions

   VDAFs in this specification use extendable output functions (XOFs) to
   extract short, fixed-length strings we call "seeds" from long input
   strings and expand seeds into long output strings.  We specify a
   single interface that is suitable for both purposes.

   XOFs are defined by a class Xof with the following associated
   parameter and methods:

   *  SEED_SIZE: Unsigned is the size (in bytes) of a seed.

   *  Xof(seed: Bytes[Xof.SEED_SIZE], dst: Bytes, binder: Bytes)
      constructs an instance of Xof from the given seed, domain
      separation tag, and binder string.  (See below for definitions of
      these.)  The seed MUST be of length SEED_SIZE and MUST be
      generated securely (i.e., it is either the output of gen_rand or a
      previous invocation of the XOF).

   *  xof.next(length: Unsigned) returns the next length bytes of output
      of xof.

   Each Xof has two derived methods.  The first is used to derive a
   fresh seed from an existing one.  The second is used to compute a
   sequence of field elements.

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   def derive_seed(Xof,
                   seed: Bytes[Xof.SEED_SIZE],
                   dst: Bytes,
                   binder: Bytes):
       """Derive a new seed."""
       xof = Xof(seed, dst, binder)
       return xof.next(Xof.SEED_SIZE)

   def next_vec(self, Field, length: Unsigned):
       """Output the next `length` elements of `Field`."""
       m = next_power_of_2(Field.MODULUS) - 1
       vec = []
       while len(vec) < length:
           x = from_le_bytes(self.next(Field.ENCODED_SIZE))
           x &= m
           if x < Field.MODULUS:
               vec.append(Field(x))
       return vec

   def expand_into_vec(Xof,
                       Field,
                       seed: Bytes[Xof.SEED_SIZE],
                       dst: Bytes,
                       binder: Bytes,
                       length: Unsigned):
       """
       Expand the input `seed` into vector of `length` field elements.
       """
       xof = Xof(seed, dst, binder)
       return xof.next_vec(Field, length)

                    Figure 12: Derived methods for XOFs.

6.2.1.  XofTurboShake128

   This section describes XofTurboShake128, an XOF based on the
   TurboSHAKE128 [TurboSHAKE].  This XOF is RECOMMENDED for all use
   cases within VDAFs.  The length of the domain separation string dst
   passed to XofTurboShake128 MUST NOT exceed 255 bytes.

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class XofTurboShake128(Xof):
    """XOF wrapper for TurboSHAKE128."""

    # Associated parameters
    SEED_SIZE = 16

    def __init__(self, seed, dst, binder):
        self.l = 0
        self.m = to_le_bytes(len(dst), 1) + dst + seed + binder

    def next(self, length: Unsigned) -> Bytes:
        self.l += length

        # Function `TurboSHAKE128(M, D, L)` is as defined in
        # Section 2.2 of [TurboSHAKE].
        #
        # Implementation note: Rather than re-generate the output
        # stream each time `next()` is invoked, most implementations
        # of TurboSHAKE128 will expose an "absorb-then-squeeze" API that
        # allows stateful handling of the stream.
        stream = TurboSHAKE128(self.m, 1, self.l)
        return stream[-length:]

            Figure 13: Definition of XOF XofTurboShake128.

6.2.2.  XofFixedKeyAes128

   While XofTurboShake128 as described above can be securely used in all
   cases where a XOF is needed in the VDAFs described in this document,
   there are some cases where a more efficient instantiation based on
   fixed-key AES is possible.  For now, this is limited to the XOF used
   inside the Idpf Section 8.1 implementation in Poplar1 Section 8.3.
   It is NOT RECOMMENDED to use this XOF anywhere else.  The length of
   the domain separation string dst passed to XofFixedKeyAes128 MUST NOT
   exceed 255 bytes.  See Security Considerations Section 9 for a more
   detailed discussion.

class XofFixedKeyAes128(Xof):
    """
    XOF based on a circular collision-resistant hash function from
    fixed-key AES.
    """

    # Associated parameters
    SEED_SIZE = 16

    def __init__(self, seed, dst, binder):
        self.length_consumed = 0

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        # Use TurboSHAKE128 to derive a key from the binder string and
        # domain separation tag. Note that the AES key does not need
        # to be kept secret from any party. However, when used with
        # IdpfPoplar, we require the binder to be a random nonce.
        #
        # Implementation note: This step can be cached across XOF
        # evaluations with many different seeds.
        dst_length = to_le_bytes(len(dst), 1)
        self.fixed_key = TurboSHAKE128(dst_length + dst + binder, 2, 16)
        self.seed = seed

    def next(self, length: Unsigned) -> Bytes:
        offset = self.length_consumed % 16
        new_length = self.length_consumed + length
        block_range = range(
            int(self.length_consumed / 16),
            int(new_length / 16) + 1)
        self.length_consumed = new_length

        hashed_blocks = [
            self.hash_block(xor(self.seed, to_le_bytes(i, 16))) \
                         for i in block_range
        ]
        return concat(hashed_blocks)[offset:offset+length]

    def hash_block(self, block):
        """
        The multi-instance tweakable circular correlation-robust hash
        function of [GKWWY20] (Section 4.2). The tweak here is the key
        that stays constant for all XOF evaluations of the same Client,
        but differs between Clients.

        Function `AES128(key, block)` is the AES-128 blockcipher.
        """
        lo, hi = block[:8], block[8:]
        sigma_block = concat([hi, xor(hi, lo)])
        return xor(AES128(self.fixed_key, sigma_block), sigma_block)

6.2.3.  The Domain Separation Tag and Binder String

   XOFs are used to map a seed to a finite domain, e.g., a fresh seed or
   a vector of field elements.  To ensure domain separation, the
   derivation is needs to be bound to some distinguished domain
   separation tag.  The domain separation tag encodes the following
   values:

   1.  The document version (i.e.,VERSION);

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   2.  The "class" of the algorithm using the output (e.g., VDAF);

   3.  A unique identifier for the algorithm; and

   4.  Some indication of how the output is used (e.g., for deriving the
       measurement shares in Prio3 Section 7).

   The following algorithm is used in the remainder of this document in
   order to format the domain separation tag:

   def format_dst(algo_class: Unsigned,
                  algo: Unsigned,
                  usage: Unsigned) -> Bytes:
       """Format XOF domain separation tag for use within a (V)DAF."""
       return concat([
           to_be_bytes(VERSION, 1),
           to_be_bytes(algo_class, 1),
           to_be_bytes(algo, 4),
           to_be_bytes(usage, 2),
       ])

   It is also sometimes necessary to bind the output to some ephemeral
   value that multiple parties need to agree on.  We call this input the
   "binder string".

7.  Prio3

   This section describes Prio3, a VDAF for Prio [CGB17].  Prio is
   suitable for a wide variety of aggregation functions, including (but
   not limited to) sum, mean, standard deviation, estimation of
   quantiles (e.g., median), and linear regression.  In fact, the scheme
   described in this section is compatible with any aggregation function
   that has the following structure:

   *  Each measurement is encoded as a vector over some finite field.

   *  Measurement validity is determined by an arithmetic circuit
      evaluated over the encoded measurement.  (An "arithmetic circuit"
      is a function comprised of arithmetic operations in the field.)
      The circuit's output is a single field element: if zero, then the
      measurement is said to be "valid"; otherwise, if the output is
      non-zero, then the measurement is said to be "invalid".

   *  The aggregate result is obtained by summing up the encoded
      measurement vectors and computing some function of the sum.

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   At a high level, Prio3 distributes this computation as follows.  Each
   Client first shards its measurement by first encoding it, then
   splitting the vector into secret shares and sending a share to each
   Aggregator.  Next, in the preparation phase, the Aggregators carry
   out a multi-party computation to determine if their shares correspond
   to a valid measurement (as determined by the arithmetic circuit).
   This computation involves a "proof" of validity generated by the
   Client.  Next, each Aggregator sums up its shares locally.  Finally,
   the Collector sums up the aggregate shares and computes the aggregate
   result.

   This VDAF does not have an aggregation parameter.  Instead, the
   output share is derived from the measurement share by applying a
   fixed map.  See Section 8 for an example of a VDAF that makes
   meaningful use of the aggregation parameter.

   As the name implies, Prio3 is a descendant of the original Prio
   construction.  A second iteration was deployed in the [ENPA] system,
   and like the VDAF described here, the ENPA system was built from
   techniques introduced in [BBCGGI19] that significantly improve
   communication cost.  That system was specialized for a particular
   aggregation function; the goal of Prio3 is to provide the same level
   of generality as the original construction.

   The core component of Prio3 is a "Fully Linear Proof (FLP)" system.
   Introduced by [BBCGGI19], the FLP encapsulates the functionality
   required for encoding and validating measurements.  Prio3 can be
   thought of as a transformation of a particular class of FLPs into a
   VDAF.

   The remainder of this section is structured as follows.  The syntax
   for FLPs is described in Section 7.1.  The generic transformation of
   an FLP into Prio3 is specified in Section 7.2.  Next, a concrete FLP
   suitable for any validity circuit is specified in Section 7.3.
   Finally, instantiations of Prio3 for various types of measurements
   are specified in Section 7.4.  Test vectors can be found in
   Appendix "Test Vectors".

7.1.  Fully Linear Proof (FLP) Systems

   Conceptually, an FLP is a two-party protocol executed by a prover and
   a verifier.  In actual use, however, the prover's computation is
   carried out by the Client, and the verifier's computation is
   distributed among the Aggregators.  The Client generates a "proof" of
   its measurement's validity and distributes shares of the proof to the
   Aggregators.  Each Aggregator then performs some computation on its
   measurement share and proof share locally and sends the result to the
   other Aggregators.  Combining the exchanged messages allows each

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   Aggregator to decide if it holds a share of a valid measurement.
   (See Section 7.2 for details.)

   As usual, we will describe the interface implemented by a concrete
   FLP in terms of an abstract base class Flp that specifies the set of
   methods and parameters a concrete FLP must provide.

   The parameters provided by a concrete FLP are listed in Table 4.

       +================+==========================================+
       | Parameter      | Description                              |
       +================+==========================================+
       | PROVE_RAND_LEN | Length of the prover randomness, the     |
       |                | number of random field elements consumed |
       |                | by the prover when generating a proof    |
       +----------------+------------------------------------------+
       | QUERY_RAND_LEN | Length of the query randomness, the      |
       |                | number of random field elements consumed |
       |                | by the verifier                          |
       +----------------+------------------------------------------+
       | JOINT_RAND_LEN | Length of the joint randomness, the      |
       |                | number of random field elements consumed |
       |                | by both the prover and verifier          |
       +----------------+------------------------------------------+
       | MEAS_LEN       | Length of the encoded measurement        |
       |                | (Section 7.1.1)                          |
       +----------------+------------------------------------------+
       | OUTPUT_LEN     | Length of the aggregatable output        |
       |                | (Section 7.1.1)                          |
       +----------------+------------------------------------------+
       | PROOF_LEN      | Length of the proof                      |
       +----------------+------------------------------------------+
       | VERIFIER_LEN   | Length of the verifier message generated |
       |                | by querying the measurement and proof    |
       +----------------+------------------------------------------+
       | Measurement    | Type of the measurement                  |
       +----------------+------------------------------------------+
       | AggResult      | Type of the aggregate result             |
       +----------------+------------------------------------------+
       | Field          | As defined in (Section 6.1)              |
       +----------------+------------------------------------------+

          Table 4: Constants and types defined by a concrete FLP.

   An FLP specifies the following algorithms for generating and
   verifying proofs of validity (encoding is described below in
   Section 7.1.1):

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   *  Flp.prove(meas: Vec[Field], prove_rand: Vec[Field], joint_rand:
      Vec[Field]) -> Vec[Field] is the deterministic proof-generation
      algorithm run by the prover.  Its inputs are the encoded
      measurement, the "prover randomness" prove_rand, and the "joint
      randomness" joint_rand.  The prover randomness is used only by the
      prover, but the joint randomness is shared by both the prover and
      verifier.

   *  Flp.query(meas: Vec[Field], proof: Vec[Field], query_rand:
      Vec[Field], joint_rand: Vec[Field], num_shares: Unsigned) ->
      Vec[Field] is the query-generation algorithm run by the verifier.
      This is used to "query" the measurement and proof.  The result of
      the query (i.e., the output of this function) is called the
      "verifier message".  In addition to the measurement and proof,
      this algorithm takes as input the query randomness query_rand and
      the joint randomness joint_rand.  The former is used only by the
      verifier. num_shares specifies how many shares were generated.

   *  Flp.decide(verifier: Vec[Field]) -> Bool is the deterministic
      decision algorithm run by the verifier.  It takes as input the
      verifier message and outputs a boolean indicating if the
      measurement from which it was generated is valid.

   Our application requires that the FLP is "fully linear" in the sense
   defined in [BBCGGI19].  As a practical matter, what this property
   implies is that, when run on a share of the measurement and proof,
   the query-generation algorithm outputs a share of the verifier
   message.  Furthermore, the privacy property of the FLP system ensures
   that the verifier message reveals nothing about the measurement other
   than whether it is valid.  Therefore, to decide if a measurement is
   valid, the Aggregators will run the query-generation algorithm
   locally, exchange verifier shares, combine them to recover the
   verifier message, and run the decision algorithm.

   The query-generation algorithm includes a parameter num_shares that
   specifies the number of shares that were generated.  If these data
   are not secret shared, then num_shares == 1.  This parameter is
   useful for certain FLP constructions.  For example, the FLP in
   Section 7.3 is defined in terms of an arithmetic circuit; when the
   circuit contains constants, it is sometimes necessary to normalize
   those constants to ensure that the circuit's output, when run on a
   valid measurement, is the same regardless of the number of shares.

   An FLP is executed by the prover and verifier as follows:

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  def run_flp(flp, meas: Vec[Flp.Field], num_shares: Unsigned):
      joint_rand = flp.Field.rand_vec(flp.JOINT_RAND_LEN)
      prove_rand = flp.Field.rand_vec(flp.PROVE_RAND_LEN)
      query_rand = flp.Field.rand_vec(flp.QUERY_RAND_LEN)

      # Prover generates the proof.
      proof = flp.prove(meas, prove_rand, joint_rand)

      # Shard the measurement and the proof.
      meas_shares = additive_secret_share(meas, num_shares, flp.Field)
      proof_shares = additive_secret_share(proof, num_shares, flp.Field)

      # Verifier queries the meas shares and proof shares.
      verifier_shares = [
          flp.query(
              meas_share,
              proof_share,
              query_rand,
              joint_rand,
              num_shares,
          )
          for meas_share, proof_share in zip(meas_shares, proof_shares)
      ]

      # Combine the verifier shares into the verifier.
      verifier = flp.Field.zeros(len(verifier_shares[0]))
      for verifier_share in verifier_shares:
          verifier = vec_add(verifier, verifier_share)

      # Verifier decides if the measurement is valid.
      return flp.decide(verifier)

                     Figure 14: Execution of an FLP.

   The proof system is constructed so that, if meas is valid, then
   run_flp(Flp, meas, 1) always returns True.  On the other hand, if
   meas is invalid, then as long as joint_rand and query_rand are
   generated uniform randomly, the output is False with overwhelming
   probability.

   We remark that [BBCGGI19] defines a much larger class of fully linear
   proof systems than we consider here.  In particular, what is called
   an "FLP" here is called a 1.5-round, public-coin, interactive oracle
   proof system in their paper.

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7.1.1.  Encoding the Input

   The type of measurement being aggregated is defined by the FLP.
   Hence, the FLP also specifies a method of encoding raw measurements
   as a vector of field elements:

   *  Flp.encode(measurement: Measurement) -> Vec[Field] encodes a raw
      measurement as a vector of field elements.  The return value MUST
      be of length MEAS_LEN.

   For some FLPs, the encoded measurement also includes redundant field
   elements that are useful for checking the proof, but which are not
   needed after the proof has been checked.  An example is the "integer
   sum" data type from [CGB17] in which an integer in range [0, 2^k) is
   encoded as a vector of k field elements, each representing a bit of
   the integer (this type is also defined in Section 7.4.2).  After
   consuming this vector, all that is needed is the integer it
   represents.  Thus the FLP defines an algorithm for truncating the
   encoded measurement to the length of the aggregated output:

   *  Flp.truncate(meas: Vec[Field]) -> Vec[Field] maps an encoded
      measurement (e.g., the bit-encoding of the measurement) to an
      aggregatable output (e.g., the singleton vector containing the
      measurement).  The length of the input MUST be MEAS_LEN and the
      length of the output MUST be OUTPUT_LEN.

   Once the aggregate shares have been computed and combined together,
   their sum can be converted into the aggregate result.  This could be
   a projection from the FLP's field to the integers, or it could
   include additional post-processing.

   *  Flp.decode(output: Vec[Field], num_measurements: Unsigned) ->
      AggResult maps a sum of aggregate shares to an aggregate result.
      The length of the input MUST be OUTPUT_LEN. num_measurements is
      the number of measurements that contributed to the aggregated
      output.

   We remark that, taken together, these three functionalities
   correspond roughly to the notion of "Affine-aggregatable encodings
   (AFEs)" from [CGB17].

7.1.2.  Multiple proofs

   To improve soundness, the prover can construct multiple unique proofs
   for its measurement such that the verifier will only accept the
   measurement once all proofs have been verified.  Notably, several
   proofs using a smaller field can offer the same level of soundness as
   a single proof using a large field.

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   To generate these proofs for a specific measurement, the prover calls
   Flp.prove multiple times, each time using an independently generated
   prover and joint randomness string.  The verifier checks each proof
   independently, each time with an independently generated query
   randomness string.  It accepts the measurement only if all the
   decision algorithm accepts on each proof.

   See Section 9.5 below for discussions on choosing the right number of
   proofs.

7.2.  Construction

   This section specifies Prio3, an implementation of the Vdaf interface
   (Section 5).  It has two generic parameters: an Flp (Section 7.1) and
   a Xof (Section 6.2).  It also has an associated constant, PROOFS,
   with a value within the range of [1, 256), denoting the number of
   FLPs generated by the Client (Section 7.1.2).  The value of PROOFS is
   1 unless explicitly specified.

   The associated constants and types required by the Vdaf interface are
   defined in Table 5.  The methods required for sharding, preparation,
   aggregation, and unsharding are described in the remaining
   subsections.  These methods refer to constants enumerated in Table 6.

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     +=================+============================================+
     | Parameter       | Value                                      |
     +=================+============================================+
     | VERIFY_KEY_SIZE | Xof.SEED_SIZE                              |
     +-----------------+--------------------------------------------+
     | RAND_SIZE       | Xof.SEED_SIZE * (1 + 2 * (SHARES - 1)) if  |
     |                 | Flp.JOINT_RAND_LEN == 0 else Xof.SEED_SIZE |
     |                 | * (1 + 2 * (SHARES - 1) + SHARES)          |
     +-----------------+--------------------------------------------+
     | NONCE_SIZE      | 16                                         |
     +-----------------+--------------------------------------------+
     | ROUNDS          | 1                                          |
     +-----------------+--------------------------------------------+
     | SHARES          | in [2, 256)                                |
     +-----------------+--------------------------------------------+
     | Measurement     | Flp.Measurement                            |
     +-----------------+--------------------------------------------+
     | AggParam        | None                                       |
     +-----------------+--------------------------------------------+
     | PublicShare     | Optional[list[bytes]]                      |
     +-----------------+--------------------------------------------+
     | InputShare      | Union[tuple[list[Flp.Field],               |
     |                 | list[Flp.Field], Optional[bytes]],         |
     |                 | tuple[bytes, bytes, Optional[bytes]]]      |
     +-----------------+--------------------------------------------+
     | OutShare        | list[Flp.Field]                            |
     +-----------------+--------------------------------------------+
     | AggShare        | list[Flp.Field]                            |
     +-----------------+--------------------------------------------+
     | AggResult       | Flp.AggResult                              |
     +-----------------+--------------------------------------------+
     | PrepState       | tuple[list[Flp.Field], Optional[Bytes]]    |
     +-----------------+--------------------------------------------+
     | PrepShare       | tuple[list[Flp.Field], Optional[Bytes]]    |
     +-----------------+--------------------------------------------+
     | PrepMessage     | Optional[bytes]                            |
     +-----------------+--------------------------------------------+

                   Table 5: VDAF parameters for Prio3.

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               +==================================+=======+
               | Variable                         | Value |
               +==================================+=======+
               | USAGE_MEAS_SHARE: Unsigned       | 1     |
               +----------------------------------+-------+
               | USAGE_PROOF_SHARE: Unsigned      | 2     |
               +----------------------------------+-------+
               | USAGE_JOINT_RANDOMNESS: Unsigned | 3     |
               +----------------------------------+-------+
               | USAGE_PROVE_RANDOMNESS: Unsigned | 4     |
               +----------------------------------+-------+
               | USAGE_QUERY_RANDOMNESS: Unsigned | 5     |
               +----------------------------------+-------+
               | USAGE_JOINT_RAND_SEED: Unsigned  | 6     |
               +----------------------------------+-------+
               | USAGE_JOINT_RAND_PART: Unsigned  | 7     |
               +----------------------------------+-------+

                    Table 6: Constants used by Prio3.

7.2.1.  Sharding

   Recall from Section 7.1 that the FLP syntax calls for "joint
   randomness" shared by the prover (i.e., the Client) and the verifier
   (i.e., the Aggregators).  VDAFs have no such notion.  Instead, the
   Client derives the joint randomness from its measurement in a way
   that allows the Aggregators to reconstruct it from their shares.
   (This idea is based on the Fiat-Shamir heuristic and is described in
   Section 6.2.3 of [BBCGGI19].)

   The sharding algorithm involves the following steps:

   1.  Encode the Client's measurement for the FLP

   2.  Shard the measurement into a sequence of measurement shares

   3.  Derive the joint randomness from the measurement shares and nonce

   4.  Run the FLP proof-generation algorithm using the derived joint
       randomness

   5.  Shard the proof into a sequence of proof shares

   6.  Return the public share, consisting of the joint randomness
       parts, and the input shares, each consisting of the measurement
       share, proof share, and blind of one of the Aggregators

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   As described in Section 7.1.2, the soundness of the FLP can be
   amplified by generating and verifying multiple FLPs.  (This in turn
   improves the robustness of Prio3.)  To support this, in Prio3:

   *  In step 3, derive as much joint randomness as required by PROOFS
      proofs

   *  Repeat step 4 PROOFS times, each time with a unique joint
      randomness

   Depending on the FLP, joint randomness may not be required.  In
   particular, when Flp.JOINT_RAND_LEN == 0, the Client does not derive
   the joint randomness (Step 3).  The sharding algorithm is specified
   below.

   def shard(Prio3, measurement, nonce, rand):
       l = Prio3.Xof.SEED_SIZE
       seeds = [rand[i:i+l] for i in range(0, Prio3.RAND_SIZE, l)]

       meas = Prio3.Flp.encode(measurement)
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           return Prio3.shard_with_joint_rand(meas, nonce, seeds)
       else:
           return Prio3.shard_without_joint_rand(meas, seeds)

             Figure 15: Input-distribution algorithm for Prio3.

   It starts by splitting the randomness into seeds.  It then encodes
   the measurement as prescribed by the FLP and calls one of two
   methods, depending on whether joint randomness is required by the
   FLP.  The methods are defined in the subsections below.

7.2.1.1.  FLPs without joint randomness

   The following method is used for FLPs that do not require joint
   randomness, i.e., when Flp.JOINT_RAND_LEN == 0:

   def shard_without_joint_rand(Prio3, meas, seeds):
     k_helper_seeds, seeds = front((Prio3.SHARES-1) * 2, seeds)
     k_helper_meas_shares = [
         k_helper_seeds[i]
         for i in range(0, (Prio3.SHARES-1) * 2, 2)
     ]
     k_helper_proofs_shares = [
         k_helper_seeds[i]
         for i in range(1, (Prio3.SHARES-1) * 2, 2)
     ]
     (k_prove,), seeds = front(1, seeds)

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     # Shard the encoded measurement into shares.
     leader_meas_share = meas
     for j in range(Prio3.SHARES-1):
         leader_meas_share = vec_sub(
             leader_meas_share,
             Prio3.helper_meas_share(j+1, k_helper_meas_shares[j]),
         )

     # Generate and shard each proof into shares.
     prove_rands = Prio3.prove_rands(k_prove)
     leader_proofs_share = []
     for _ in range(Prio3.PROOFS):
         prove_rand, prove_rands = front(
             Prio3.Flp.PROVE_RAND_LEN, prove_rands)
         leader_proofs_share += Prio3.Flp.prove(meas, prove_rand, [])
     for j in range(Prio3.SHARES-1):
         leader_proofs_share = vec_sub(
             leader_proofs_share,
             Prio3.helper_proofs_share(j+1, k_helper_proofs_shares[j]),
         )

     # Each Aggregator's input share contains its measurement share
     # and share of proof(s).
     input_shares = []
     input_shares.append((
         leader_meas_share,
         leader_proofs_share,
         None,
     ))
     for j in range(Prio3.SHARES-1):
         input_shares.append((
             k_helper_meas_shares[j],
             k_helper_proofs_shares[j],
             None,
         ))
     return (None, input_shares)

    Figure 16: Sharding an encoded measurement without joint randomness.

   The steps in this method are as follows:

   1.  Shard the encoded measurement into shares

   2.  Generate and shard each proof into shares

   3.  Encode each measurement and shares of each proof into an input
       share

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   Notice that only one pair of measurement and proof(s) share (called
   the "leader" shares above) are vectors of field elements.  The other
   shares (called the "helper" shares) are represented instead by XOF
   seeds, which are expanded into vectors of field elements.

   The methods on Prio3 for deriving the prover randomness, measurement
   shares, and proof shares and the methods for encoding the input
   shares are defined in Section 7.2.6.

7.2.1.2.  FLPs with joint randomness

   The following method is used for FLPs that require joint randomness,
   i.e., for which Flp.JOINT_RAND_LEN > 0:

   def shard_with_joint_rand(Prio3, meas, nonce, seeds):
     k_helper_seeds, seeds = front((Prio3.SHARES-1) * 3, seeds)
     k_helper_meas_shares = [
         k_helper_seeds[i]
         for i in range(0, (Prio3.SHARES-1) * 3, 3)
     ]
     k_helper_proofs_shares = [
         k_helper_seeds[i]
         for i in range(1, (Prio3.SHARES-1) * 3, 3)
     ]
     k_helper_blinds = [
         k_helper_seeds[i]
         for i in range(2, (Prio3.SHARES-1) * 3, 3)
     ]
     (k_leader_blind,), seeds = front(1, seeds)
     (k_prove,), seeds = front(1, seeds)

     # Shard the encoded measurement into shares and compute the
     # joint randomness parts.
     leader_meas_share = meas
     k_joint_rand_parts = []
     for j in range(Prio3.SHARES-1):
         helper_meas_share = Prio3.helper_meas_share(
             j+1, k_helper_meas_shares[j])
         leader_meas_share = vec_sub(leader_meas_share,
                                     helper_meas_share)
         k_joint_rand_parts.append(Prio3.joint_rand_part(
             j+1, k_helper_blinds[j], helper_meas_share, nonce))
     k_joint_rand_parts.insert(0, Prio3.joint_rand_part(
         0, k_leader_blind, leader_meas_share, nonce))

     # Generate and shard each proof into shares.
     prove_rands = Prio3.prove_rands(k_prove)
     joint_rands = Prio3.joint_rands(

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         Prio3.joint_rand_seed(k_joint_rand_parts))
     leader_proofs_share = []
     for _ in range(Prio3.PROOFS):
         prove_rand, prove_rands = front(
             Prio3.Flp.PROVE_RAND_LEN, prove_rands)
         joint_rand, joint_rands = front(
             Prio3.Flp.JOINT_RAND_LEN, joint_rands)
         leader_proofs_share += Prio3.Flp.prove(meas,
                                                prove_rand, joint_rand)
     for j in range(Prio3.SHARES-1):
         leader_proofs_share = vec_sub(
             leader_proofs_share,
             Prio3.helper_proofs_share(j+1, k_helper_proofs_shares[j]),
         )

     # Each Aggregator's input share contains its measurement share,
     # share of proof(s), and blind. The public share contains the
     # Aggregators' joint randomness parts.
     input_shares = []
     input_shares.append((
         leader_meas_share,
         leader_proofs_share,
         k_leader_blind,
     ))
     for j in range(Prio3.SHARES-1):
         input_shares.append((
             k_helper_meas_shares[j],
             k_helper_proofs_shares[j],
             k_helper_blinds[j],
         ))
     return (k_joint_rand_parts, input_shares)

     Figure 17: Sharding an encoded measurement with joint randomness.

   The difference between this procedure and previous one is that here
   we compute joint randomnesses joint_rands, split it into multiple
   joint_rand, and pass each joint_rand to the proof generationg
   algorithm.  (In Figure 16 the joint randomness is the empty vector,
   [].)  This requires generating an additional value, called the
   "blind", that is incorporated into each input share.

   The joint randomness computation involves the following steps:

   1.  Compute a "joint randomness part" from each measurement share and
       blind

   2.  Compute a "joint randomness seed" from the joint randomness parts

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   3.  Compute the joint randomness for each proof evaluation from the
       joint randomness seed

   This three-step process is designed to ensure that the joint
   randomness does not leak the measurement to the Aggregators while
   preventing a malicious Client from tampering with the joint
   randomness in a way that allows it to break robustness.  To bootstrap
   the required check, the Client encodes the joint randomness parts in
   the public share.  (See Section 7.2.2 for details.)

   The methods used in this computation are defined in Section 7.2.6.

7.2.2.  Preparation

   This section describes the process of recovering output shares from
   the input shares.  The high-level idea is that each Aggregator first
   queries its measurement and share of proof(s) locally, then exchanges
   its share of verifier(s) with the other Aggregators.  The shares of
   verifier(s) are then combined into the verifier message(s) used to
   decide whether to accept.

   In addition, for FLPs that require joint randomness, the Aggregators
   must ensure that they have all used the same joint randomness for the
   query-generation algorithm.  To do so, they collectively re-derive
   the joint randomness from their measurement shares just as the Client
   did during sharding.

   In order to avoid extra round of communication, the Client sends each
   Aggregator a "hint" consisting of the joint randomness parts.  This
   leaves open the possibility that the Client cheated by, say, forcing
   the Aggregators to use joint randomness that biases the proof check
   procedure some way in its favor.  To mitigate this, the Aggregators
   also check that they have all computed the same joint randomness seed
   before accepting their output shares.  To do so, they exchange their
   parts of the joint randomness along with their shares of verifier(s).

   The definitions of constants and a few auxiliary functions are
   defined in Section 7.2.6.

  def prep_init(Prio3, verify_key, agg_id, _agg_param,
              nonce, public_share, input_share):
    k_joint_rand_parts = public_share
    (meas_share, proofs_share, k_blind) = \
        Prio3.expand_input_share(agg_id, input_share)
    out_share = Prio3.Flp.truncate(meas_share)

    # Compute the joint randomness.
    joint_rand = []

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    k_corrected_joint_rand, k_joint_rand_part = None, None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        k_joint_rand_part = Prio3.joint_rand_part(
            agg_id, k_blind, meas_share, nonce)
        k_joint_rand_parts[agg_id] = k_joint_rand_part
        k_corrected_joint_rand = Prio3.joint_rand_seed(
            k_joint_rand_parts)
        joint_rands = Prio3.joint_rands(k_corrected_joint_rand)

    # Query the measurement and proof share.
    query_rands = Prio3.query_rands(verify_key, nonce)
    verifiers_share = []
    for _ in range(Prio3.PROOFS):
        proof_share, proofs_share = front(
            Prio3.Flp.PROOF_LEN, proofs_share)
        query_rand, query_rands = front(
            Prio3.Flp.QUERY_RAND_LEN, query_rands)
        if Prio3.Flp.JOINT_RAND_LEN > 0:
            joint_rand, joint_rands = front(
                Prio3.Flp.JOINT_RAND_LEN, joint_rands)
        verifiers_share += Prio3.Flp.query(meas_share,
                                            proof_share,
                                            query_rand,
                                            joint_rand,
                                            Prio3.SHARES)

    prep_state = (out_share, k_corrected_joint_rand)
    prep_share = (verifiers_share, k_joint_rand_part)
    return (prep_state, prep_share)

  def prep_next(Prio3, prep, prep_msg):
    k_joint_rand = prep_msg
    (out_share, k_corrected_joint_rand) = prep

    # If joint randomness was used, check that the value computed by the
    # Aggregators matches the value indicated by the Client.
    if k_joint_rand != k_corrected_joint_rand:
        raise ERR_VERIFY  # joint randomness check failed

    return out_share

  def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
    # Unshard the verifier shares into the verifier message.
    verifiers = Prio3.Flp.Field.zeros(
        Prio3.Flp.VERIFIER_LEN * Prio3.PROOFS)
    k_joint_rand_parts = []
    for (verifiers_share, k_joint_rand_part) in prep_shares:
        verifiers = vec_add(verifiers, verifiers_share)

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        if Prio3.Flp.JOINT_RAND_LEN > 0:
            k_joint_rand_parts.append(k_joint_rand_part)

    # Verify that each proof is well-formed and the input is valid
    for _ in range(Prio3.PROOFS):
        verifier, verifiers = front(Prio3.Flp.VERIFIER_LEN, verifiers)
        if not Prio3.Flp.decide(verifier):
            raise ERR_VERIFY  # proof verifier check failed

    # Combine the joint randomness parts computed by the
    # Aggregators into the true joint randomness seed. This is
    # used in the last step.
    k_joint_rand = None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        k_joint_rand = Prio3.joint_rand_seed(k_joint_rand_parts)
    return k_joint_rand

                 Figure 18: Preparation state for Prio3.

7.2.3.  Validity of Aggregation Parameters

   Every input share MUST only be used once, regardless of the
   aggregation parameters used.

   def is_valid(agg_param, previous_agg_params):
       return len(previous_agg_params) == 0

          Figure 19: Validity of aggregation parameters for Prio3.

7.2.4.  Aggregation

   Aggregating a set of output shares is simply a matter of adding up
   the vectors element-wise.

   def aggregate(Prio3, _agg_param, out_shares):
       agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
       for out_share in out_shares:
           agg_share = vec_add(agg_share, out_share)
       return agg_share

                Figure 20: Aggregation algorithm for Prio3.

7.2.5.  Unsharding

   To unshard a set of aggregate shares, the Collector first adds up the
   vectors element-wise.  It then converts each element of the vector
   into an integer.

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   def unshard(Prio3, _agg_param,
               agg_shares, num_measurements):
       agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
       for agg_share in agg_shares:
           agg = vec_add(agg, agg_share)
       return Prio3.Flp.decode(agg, num_measurements)

         Figure 21: Computation of the aggregate result for Prio3.

7.2.6.  Auxiliary Functions

   This section defines a number of auxiliary functions referenced by
   the main algorithms for Prio3 in the preceding sections.

   The following methods are called by the sharding and preparation
   algorithms.

  def helper_meas_share(Prio3, agg_id, k_share):
      return Prio3.Xof.expand_into_vec(
          Prio3.Flp.Field,
          k_share,
          Prio3.domain_separation_tag(USAGE_MEAS_SHARE),
          byte(agg_id),
          Prio3.Flp.MEAS_LEN,
      )

  def helper_proofs_share(Prio3, agg_id, k_share):
    return Prio3.Xof.expand_into_vec(
        Prio3.Flp.Field,
        k_share,
        Prio3.domain_separation_tag(USAGE_PROOF_SHARE),
        byte(Prio3.PROOFS) + byte(agg_id),
        Prio3.Flp.PROOF_LEN * Prio3.PROOFS,
    )

  def expand_input_share(Prio3, agg_id, input_share):
    (meas_share, proofs_share, k_blind) = input_share
    if agg_id > 0:
        meas_share = Prio3.helper_meas_share(agg_id, meas_share)
        proofs_share = Prio3.helper_proofs_share(agg_id, proofs_share)
    return (meas_share, proofs_share, k_blind)

  def prove_rands(Prio3, k_prove):
    return Prio3.Xof.expand_into_vec(
        Prio3.Flp.Field,
        k_prove,
        Prio3.domain_separation_tag(USAGE_PROVE_RANDOMNESS),
        byte(Prio3.PROOFS),

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        Prio3.Flp.PROVE_RAND_LEN * Prio3.PROOFS,
    )

  def query_rands(Prio3, verify_key, nonce):
    return Prio3.Xof.expand_into_vec(
        Prio3.Flp.Field,
        verify_key,
        Prio3.domain_separation_tag(USAGE_QUERY_RANDOMNESS),
        byte(Prio3.PROOFS) + nonce,
        Prio3.Flp.QUERY_RAND_LEN * Prio3.PROOFS,
    )

  def joint_rand_part(Prio3, agg_id, k_blind, meas_share, nonce):
      return Prio3.Xof.derive_seed(
          k_blind,
          Prio3.domain_separation_tag(USAGE_JOINT_RAND_PART),
          byte(agg_id) + nonce + Prio3.Flp.Field.encode_vec(meas_share),
      )

  def joint_rand_seed(Prio3, k_joint_rand_parts):
      """Derive the joint randomness seed from its parts."""
      return Prio3.Xof.derive_seed(
          zeros(Prio3.Xof.SEED_SIZE),
          Prio3.domain_separation_tag(USAGE_JOINT_RAND_SEED),
          concat(k_joint_rand_parts),
      )

  def joint_rands(Prio3, k_joint_rand_seed):
    """Derive the joint randomness from its seed."""
    return Prio3.Xof.expand_into_vec(
        Prio3.Flp.Field,
        k_joint_rand_seed,
        Prio3.domain_separation_tag(USAGE_JOINT_RANDOMNESS),
        byte(Prio3.PROOFS),
        Prio3.Flp.JOINT_RAND_LEN * Prio3.PROOFS,
    )

7.2.7.  Message Serialization

   This section defines serialization formats for messages exchanged
   over the network while executing Prio3.  It is RECOMMENDED that
   implementations provide serialization methods for them.

   Message structures are defined following Section 3 of [RFC8446]).  In
   the remainder we use S as an alias for Prio3.Xof.SEED_SIZE and F as
   an alias for Prio3.Field.ENCODED_SIZE.  XOF seeds are represented as
   follows:

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   opaque Prio3Seed[S];

   Field elements are encoded in little-endian byte order (as defined in
   Section 6.1) and represented as follows:

   opaque Prio3Field[F];

7.2.7.1.  Public Share

   The encoding of the public share depends on whether joint randomness
   is required for the underlying FLP (i.e., Prio3.Flp.JOINT_RAND_LEN >
   0).  If joint randomness is not used, then the public share is the
   empty string.  If joint randomness is used, then the public share
   encodes the joint randomness parts as follows:

   struct {
       Prio3Seed k_joint_rand_parts[S * Prio3.SHARES];
   } Prio3PublicShareWithJointRand;

7.2.7.2.  Input share

   Just as for the public share, the encoding of the input shares
   depends on whether joint randomness is used.  If so, then each input
   share includes the Aggregator's blind for generating its joint
   randomness part.

   In addition, the encoding of the input shares depends on which
   aggregator is receiving the message.  If the aggregator ID is 0, then
   the input share includes the full measurement and share of proof(s).
   Otherwise, if the aggregator ID is greater than 0, then the
   measurement and shares of proof(s) are represented by XOF seeds.  We
   shall call the former the "Leader" and the latter the "Helpers".

   In total there are four variants of the input share.  When joint
   randomness is not used, the Leader's share is structured as follows:

   struct {
       Prio3Field meas_share[F * Prio3.Flp.MEAS_LEN];
       Prio3Field proofs_share[F * Prio3.Flp.PROOF_LEN * Prio3.PROOFS];
   } Prio3LeaderShare;

   When joint randomness is not used, the Helpers' shares are structured
   as follows:

   struct {
       Prio3Seed k_meas_share;
       Prio3Seed k_proofs_share;
   } Prio3HelperShare;

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   When joint randomness is used, the Leader's input share is structured
   as follows:

   struct {
       Prio3LeaderShare inner;
       Prio3Seed k_blind;
   } Prio3LeaderShareWithJointRand;

   Finally, when joint randomness is used, the Helpers' shares are
   structured as follows:

   struct {
       Prio3HelperShare inner;
       Prio3Seed k_blind;
   } Prio3HelperShareWithJointRand;

7.2.7.3.  Prep Share

   When joint randomness is not used, the prep share is structured as
   follows:

struct {
    Prio3Field verifiers_share[F * Prio3.Flp.VERIFIER_LEN * Prio3.PROOFS];
} Prio3PrepShare;

   When joint randomness is used, the prep share includes the
   Aggregator's joint randomness part and is structured as follows:

struct {
    Prio3Field verifiers_share[F * Prio3.Flp.VERIFIER_LEN * Prio3.PROOFS];
    Prio3Seed k_joint_rand_part;
} Prio3PrepShareWithJointRand;

7.2.7.4.  Prep Message

   When joint randomness is not used, the prep message is the empty
   string.  Otherwise the prep message consists of the joint randomness
   seed computed by the Aggregators:

   struct {
       Prio3Seed k_joint_rand;
   } Prio3PrepMessageWithJointRand;

7.2.7.5.  Aggregation

   Aggregate shares are structured as follows:

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   struct {
       Prio3Field agg_share[F * Prio3.Flp.OUTPUT_LEN];
   } Prio3AggShare;

7.3.  A General-Purpose FLP

   This section describes an FLP based on the construction from in
   [BBCGGI19], Section 4.2.  We begin in Section 7.3.1 with an overview
   of their proof system and the extensions to their proof system made
   here.  The construction is specified in Section 7.3.3.

      OPEN ISSUE We're not yet sure if specifying this general-purpose
      FLP is desirable.  It might be preferable to specify specialized
      FLPs for each data type that we want to standardize, for two
      reasons.  First, clear and concise specifications are likely
      easier to write for specialized FLPs rather than the general one.
      Second, we may end up tailoring each FLP to the measurement type
      in a way that improves performance, but breaks compatibility with
      the general-purpose FLP.

      In any case, we can't make this decision until we know which data
      types to standardize, so for now, we'll stick with the general-
      purpose construction.  The reference implementation can be found
      at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc.

      OPEN ISSUE Chris Wood points out that the this section reads more
      like a paper than a standard.  Eventually we'll want to work this
      into something that is readily consumable by the CFRG.

7.3.1.  Overview

   In the proof system of [BBCGGI19], validity is defined via an
   arithmetic circuit evaluated over the encoded measurement: If the
   circuit output is zero, then the measurement is deemed valid;
   otherwise, if the circuit output is non-zero, then the measurement is
   deemed invalid.  Thus the goal of the proof system is merely to allow
   the verifier to evaluate the validity circuit over the measurement.
   For our application (Section 7), this computation is distributed
   among multiple Aggregators, each of which has only a share of the
   measurement.

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   Suppose for a moment that the validity circuit C is affine, meaning
   its only operations are addition and multiplication-by-constant.  In
   particular, suppose the circuit does not contain a multiplication
   gate whose operands are both non-constant.  Then to decide if a
   measurement x is valid, each Aggregator could evaluate C on its share
   of x locally, broadcast the output share to its peers, then combine
   the output shares locally to recover C(x).  This is true because for
   any SHARES-way secret sharing of x it holds that

   C(x_shares[0] + ... + x_shares[SHARES-1]) =
       C(x_shares[0]) + ... + C(x_shares[SHARES-1])

   (Note that, for this equality to hold, it may be necessary to scale
   any constants in the circuit by SHARES.)  However this is not the
   case if C is not-affine (i.e., it contains at least one
   multiplication gate whose operands are non-constant).  In the proof
   system of [BBCGGI19], the proof is designed to allow the
   (distributed) verifier to compute the non-affine operations using
   only linear operations on (its share of) the measurement and proof.

   To make this work, the proof system is restricted to validity
   circuits that exhibit a special structure.  Specifically, an
   arithmetic circuit with "G-gates" (see [BBCGGI19], Definition 5.2) is
   composed of affine gates and any number of instances of a
   distinguished gate G, which may be non-affine.  We will refer to this
   class of circuits as 'gadget circuits' and to G as the "gadget".

   As an illustrative example, consider a validity circuit C that
   recognizes the set L = set([0], [1]).  That is, C takes as input a
   length-1 vector x and returns 0 if x[0] is in [0,2) and outputs
   something else otherwise.  This circuit can be expressed as the
   following degree-2 polynomial:

   C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]

   This polynomial recognizes L because x[0]^2 = x[0] is only true if
   x[0] == 0 or x[0] == 1.  Notice that the polynomial involves a non-
   affine operation, x[0]^2.  In order to apply [BBCGGI19], Theorem 4.3,
   the circuit needs to be rewritten in terms of a gadget that subsumes
   this non-affine operation.  For example, the gadget might be
   multiplication:

   Mul(left, right) = left * right

   The validity circuit can then be rewritten in terms of Mul like so:

   C(x[0]) = Mul(x[0], x[0]) - x[0]

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   The proof system of [BBCGGI19] allows the verifier to evaluate each
   instance of the gadget (i.e., Mul(x[0], x[0]) in our example) using a
   linear function of the measurement and proof.  The proof is
   constructed roughly as follows.  Let C be the validity circuit and
   suppose the gadget is arity-L (i.e., it has L input wires.).  Let
   wire[j-1,k-1] denote the value of the jth wire of the kth call to the
   gadget during the evaluation of C(x).  Suppose there are M such calls
   and fix distinct field elements alpha[0], ..., alpha[M-1].  (We will
   require these points to have a special property, as we'll discuss in
   Section 7.3.1.1; but for the moment it is only important that they
   are distinct.)

   The prover constructs from wire and alpha a polynomial that, when
   evaluated at alpha[k-1], produces the output of the kth call to the
   gadget.  Let us call this the "gadget polynomial".  Polynomial
   evaluation is linear, which means that, in the distributed setting,
   the Client can disseminate additive shares of the gadget polynomial
   that the Aggregators then use to compute additive shares of each
   gadget output, allowing each Aggregator to compute its share of C(x)
   locally.

   There is one more wrinkle, however: It is still possible for a
   malicious prover to produce a gadget polynomial that would result in
   C(x) being computed incorrectly, potentially resulting in an invalid
   measurement being accepted.  To prevent this, the verifier performs a
   probabilistic test to check that the gadget polynomial is well-
   formed.  This test, and the procedure for constructing the gadget
   polynomial, are described in detail in Section 7.3.3.

7.3.1.1.  Extensions

   The FLP described in the next section extends the proof system of
   [BBCGGI19], Section 4.2 in three ways.

   First, the validity circuit in our construction includes an
   additional, random input (this is the "joint randomness" derived from
   the measurement shares in Prio3; see Section 7.2).  This allows for
   circuit optimizations that trade a small soundness error for a
   shorter proof.  For example, consider a circuit that recognizes the
   set of length-N vectors for which each element is either one or zero.
   A deterministic circuit could be constructed for this language, but
   it would involve a large number of multiplications that would result
   in a large proof.  (See the discussion in [BBCGGI19], Section 5.2 for
   details).  A much shorter proof can be constructed for the following
   randomized circuit:

   C(meas, r) = r * Range2(meas[0]) + ... + r^N * Range2(meas[N-1])

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   (Note that this is a special case of [BBCGGI19], Theorem 5.2.)  Here
   meas is the length-N input and r is a random field element.  The
   gadget circuit Range2 is the "range-check" polynomial described
   above, i.e., Range2(x) = x^2 - x.  The idea is that, if meas is valid
   (i.e., each meas[j] is in [0,2)), then the circuit will evaluate to 0
   regardless of the value of r; but if meas[j] is not in [0,2) for some
   j, the output will be non-zero with high probability.

   The second extension implemented by our FLP allows the validity
   circuit to contain multiple gadget types.  (This generalization was
   suggested in [BBCGGI19], Remark 4.5.)  This provides additional
   flexibility for designing circuits by allowing multiple, non-affine
   sub-components.  For example, the following circuit is allowed:

   C(meas, r) = r * Range2(meas[0]) + ... + r^L * Range2(meas[L-1]) + \
               r^(L+1) * Range3(meas[L]) + ... + r^N * Range3(meas[N-1])

   where Range3(x) = x^3 - 3x^2 + 2x.  This circuit checks that the
   first L inputs are in range [0,2) and the last N-L inputs are in
   range [0,3).  Of course, the same circuit can be expressed using a
   sub-component that the gadgets have in common, namely Mul, but the
   resulting proof would be longer.

   Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice
   of the fixed points alpha[0], ..., alpha[M-1], other than to require
   that the points are distinct.  In this document, the fixed points are
   chosen so that the gadget polynomial can be constructed efficiently
   using the Cooley-Tukey FFT ("Fast Fourier Transform") algorithm.
   Note that this requires the field to be "FFT-friendly" as defined in
   Section 6.1.2.

7.3.2.  Validity Circuits

   The FLP described in Section 7.3.3 is defined in terms of a validity
   circuit Valid that implements the interface described here.

   A concrete Valid defines the following parameters:

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        +================+=======================================+
        | Parameter      | Description                           |
        +================+=======================================+
        | GADGETS        | A list of gadgets                     |
        +----------------+---------------------------------------+
        | GADGET_CALLS   | Number of times each gadget is called |
        +----------------+---------------------------------------+
        | MEAS_LEN       | Length of the measurement             |
        +----------------+---------------------------------------+
        | OUTPUT_LEN     | Length of the aggregatable output     |
        +----------------+---------------------------------------+
        | JOINT_RAND_LEN | Length of the random input            |
        +----------------+---------------------------------------+
        | Measurement    | The type of measurement               |
        +----------------+---------------------------------------+
        | AggResult      | Type of the aggregate result          |
        +----------------+---------------------------------------+
        | Field          | An FFT-friendly finite field as       |
        |                | defined in Section 6.1.2              |
        +----------------+---------------------------------------+

                  Table 7: Validity circuit parameters.

   Each gadget G in GADGETS defines a constant DEGREE that specifies the
   circuit's "arithmetic degree".  This is defined to be the degree of
   the polynomial that computes it.  For example, the Mul circuit in
   Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has
   degree 2.  Hence, the arithmetic degree of this gadget is 2.

   Each gadget also defines a parameter ARITY that specifies the
   circuit's arity (i.e., the number of input wires).

   Gadgets provide a method to evaluate their circuit on a list of
   inputs, eval().  The inputs can either belong to the validity
   circuit's field, or the polynomial ring over that field.

   A concrete Valid provides the following methods for encoding a
   measurement as an input vector, truncating an input vector to the
   length of an aggregatable output, and converting an aggregated output
   to an aggregate result:

   *  Valid.encode(measurement: Measurement) -> Vec[Field] returns a
      vector of length MEAS_LEN representing a measurement.

   *  Valid.truncate(meas: Vec[Field]) -> Vec[Field] returns a vector of
      length OUTPUT_LEN representing an aggregatable output.

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   *  Valid.decode(output: Vec[Field], num_measurements: Unsigned) ->
      AggResult returns an aggregate result.

   Finally, the following methods are derived for each concrete Valid:

   def prove_rand_len(self):
       """Length of the prover randomness."""
       return sum(g.ARITY for g in Valid.GADGETS)

   def query_rand_len(self):
       """Length of the query randomness."""
       return len(Valid.GADGETS)

   def proof_len(self):
       """Length of the proof."""
       length = 0
       for (g, g_calls) in zip(self.GADGETS, self.GADGET_CALLS):
           P = next_power_of_2(1 + g_calls)
           length += g.ARITY + g.DEGREE * (P - 1) + 1
       return length

   def verifier_len(self):
       """Length of the verifier message."""
       length = 1
       for g in self.GADGETS:
           length += g.ARITY + 1
       return length

             Figure 22: Derived methods for validity circuits.

7.3.3.  Construction

   This section specifies FlpGeneric, an implementation of the Flp
   interface (Section 7.1).  It has as a generic parameter a validity
   circuit Valid implementing the interface defined in Section 7.3.2.

      NOTE A reference implementation can be found in
      https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/
      flp_generic.py.

   The FLP parameters for FlpGeneric are defined in Table 8.  The
   required methods for generating the proof, generating the verifier,
   and deciding validity are specified in the remaining subsections.

   In the remainder, we let [n] denote the set {1, ..., n} for positive
   integer n.  We also define the following constants:

   *  Let H = len(Valid.GADGETS)

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   *  For each i in [H]:

      -  Let G_i = Valid.GADGETS[i]

      -  Let L_i = Valid.GADGETS[i].ARITY

      -  Let M_i = Valid.GADGET_CALLS[i]

      -  Let P_i = next_power_of_2(M_i+1)

      -  Let alpha_i = Field.gen()^(Field.GEN_ORDER / P_i)

      +================+============================================+
      | Parameter      | Value                                      |
      +================+============================================+
      | PROVE_RAND_LEN | Valid.prove_rand_len() (see Section 7.3.2) |
      +----------------+--------------------------------------------+
      | QUERY_RAND_LEN | Valid.query_rand_len() (see Section 7.3.2) |
      +----------------+--------------------------------------------+
      | JOINT_RAND_LEN | Valid.JOINT_RAND_LEN                       |
      +----------------+--------------------------------------------+
      | MEAS_LEN       | Valid.MEAS_LEN                             |
      +----------------+--------------------------------------------+
      | OUTPUT_LEN     | Valid.OUTPUT_LEN                           |
      +----------------+--------------------------------------------+
      | PROOF_LEN      | Valid.proof_len() (see Section 7.3.2)      |
      +----------------+--------------------------------------------+
      | VERIFIER_LEN   | Valid.verifier_len() (see Section 7.3.2)   |
      +----------------+--------------------------------------------+
      | Measurement    | Valid.Measurement                          |
      +----------------+--------------------------------------------+
      | Field          | Valid.Field                                |
      +----------------+--------------------------------------------+

                   Table 8: FLP Parameters of FlpGeneric.

7.3.3.1.  Proof Generation

   On input of meas, prove_rand, and joint_rand, the proof is computed
   as follows:

   1.  For each i in [H] create an empty table wire_i.

   2.  Partition the prover randomness prove_rand into sub-vectors
       seed_1, ..., seed_H where len(seed_i) == L_i for all i in [H].
       Let us call these the "wire seeds" of each gadget.

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   3.  Evaluate Valid on input of meas and joint_rand, recording the
       inputs of each gadget in the corresponding table.  Specifically,
       for every i in [H], set wire_i[j-1,k-1] to the value on the jth
       wire into the kth call to gadget G_i.

   4.  Compute the "wire polynomials".  That is, for every i in [H] and
       j in [L_i], construct poly_wire_i[j-1], the jth wire polynomial
       for the ith gadget, as follows:

       *  Let w = [seed_i[j-1], wire_i[j-1,0], ..., wire_i[j-1,M_i-1]].

       *  Let padded_w = w + Field.zeros(P_i - len(w)).

          NOTE We pad w to the nearest power of 2 so that we can use FFT
          for interpolating the wire polynomials.  Perhaps there is some
          clever math for picking wire_inp in a way that avoids having
          to pad.

       *  Let poly_wire_i[j-1] be the lowest degree polynomial for which
          poly_wire_i[j-1](alpha_i^k) == padded_w[k] for all k in [P_i].

   5.  Compute the "gadget polynomials".  That is, for every i in [H]:

       *  Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-
          1]).  That is, evaluate the circuit G_i on the wire
          polynomials for the ith gadget.  (Arithmetic is in the ring of
          polynomials over Field.)

   The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H +
   coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i
   for each i in [H].

7.3.3.2.  Query Generation

   On input of meas, proof, query_rand, and joint_rand, the verifier
   message is generated as follows:

   1.  For every i in [H] create an empty table wire_i.

   2.  Partition proof into the sub-vectors seed_1, coeff_1, ...,
       seed_H, coeff_H defined in Section 7.3.3.1.

   3.  Evaluate Valid on input of meas and joint_rand, recording the
       inputs of each gadget in the corresponding table.  This step is
       similar to the prover's step (3.) except the verifier does not
       evaluate the gadgets.  Instead, it computes the output of the kth
       call to G_i by evaluating poly_gadget_i(alpha_i^k).  Let v denote
       the output of the circuit evaluation.

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   4.  Compute the wire polynomials just as in the prover's step (4.).

   5.  Compute the tests for well-formedness of the gadget polynomials.
       That is, for every i in [H]:

       *  Let t = query_rand[i].  Check if t^(P_i) == 1: If so, then
          raise ERR_ABORT and halt.  (This prevents the verifier from
          inadvertently leaking a gadget output in the verifier
          message.)

       *  Let y_i = poly_gadget_i(t).

       *  For each j in [0,L_i) let x_i[j-1] = poly_wire_i[j-1](t).

   The verifier message is the vector verifier = [v] + x_1 + [y_1] + ...
   + x_H + [y_H].

7.3.3.3.  Decision

   On input of vector verifier, the verifier decides if the measurement
   is valid as follows:

   1.  Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in
       Section 7.3.3.2.

   2.  Check for well-formedness of the gadget polynomials.  For every i
       in [H]:

       *  Let z = G_i(x_i).  That is, evaluate the circuit G_i on x_i
          and set z to the output.

       *  If z != y_i, then return False and halt.

   3.  Return True if v == 0 and False otherwise.

7.3.3.4.  Encoding

   The FLP encoding and truncation methods invoke Valid.encode,
   Valid.truncate, and Valid.decode in the natural way.

7.4.  Instantiations

   This section specifies instantiations of Prio3 for various
   measurement types.  Each uses FlpGeneric as the FLP (Section 7.3) and
   is determined by a validity circuit (Section 7.3.2) and a XOF
   (Section 6.2).  Test vectors for each can be found in Appendix "Test
   Vectors".

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      NOTE Reference implementations of each of these VDAFs can be found
      in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/
      vdaf_prio3.sage.

7.4.1.  Prio3Count

   Our first instance of Prio3 is for a simple counter: Each measurement
   is either one or zero and the aggregate result is the sum of the
   measurements.

   This instance uses XofTurboShake128 (Section 6.2.1) as its XOF.  Its
   validity circuit, denoted Count, uses Field64 (Table 3) as its finite
   field.  Its gadget, denoted Mul, is the degree-2, arity-2 gadget
   defined as

   def eval(self, Field, inp):
       self.check_gadget_eval(inp)
       return inp[0] * inp[1]

   The call to check_gadget_eval() raises an error if the length of the
   input is not equal to the gadget's ARITY parameter.

   The Count validity circuit is defined as

   def eval(self, meas, joint_rand, _num_shares):
       return self.GADGETS[0].eval(self.Field, [meas[0], meas[0]]) \
           - meas[0]

   The measurement is encoded and decoded as a singleton vector in the
   natural way.  The parameters for this circuit are summarized below.

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               +================+==========================+
               | Parameter      | Value                    |
               +================+==========================+
               | GADGETS        | [Mul]                    |
               +----------------+--------------------------+
               | GADGET_CALLS   | [1]                      |
               +----------------+--------------------------+
               | MEAS_LEN       | 1                        |
               +----------------+--------------------------+
               | OUTPUT_LEN     | 1                        |
               +----------------+--------------------------+
               | JOINT_RAND_LEN | 0                        |
               +----------------+--------------------------+
               | Measurement    | Unsigned, in range [0,2) |
               +----------------+--------------------------+
               | AggResult      | Unsigned                 |
               +----------------+--------------------------+
               | Field          | Field64 (Table 3)        |
               +----------------+--------------------------+

                  Table 9: Parameters of validity circuit
                                   Count.

7.4.2.  Prio3Sum

   The next instance of Prio3 supports summing of integers in a pre-
   determined range.  Each measurement is an integer in range [0,
   2^bits), where bits is an associated parameter.

   This instance of Prio3 uses XofTurboShake128 (Section 6.2.1) as its
   XOF.  Its validity circuit, denoted Sum, uses Field128 (Table 3) as
   its finite field.  The measurement is encoded as a length-bits vector
   of field elements, where the lth element of the vector represents the
   lth bit of the summand:

   def encode(self, measurement):
       if 0 > measurement or measurement >= 2 ** self.MEAS_LEN:
           raise ERR_INPUT

       return self.Field.encode_into_bit_vector(measurement,
                                                self.MEAS_LEN)

   def truncate(self, meas):
       return [self.Field.decode_from_bit_vector(meas)]

   def decode(self, output, _num_measurements):
       return output[0].as_unsigned()

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   The validity circuit checks that the input consists of ones and
   zeros.  Its gadget, denoted Range2, is the degree-2, arity-1 gadget
   defined as

   def eval(self, Field, inp):
       self.check_gadget_eval(inp)
       return inp[0] * inp[0] - inp[0]

   The Sum validity circuit is defined as

   def eval(self, meas, joint_rand, _num_shares):
       self.check_valid_eval(meas, joint_rand)
       out = self.Field(0)
       r = joint_rand[0]
       for b in meas:
           out += r * self.GADGETS[0].eval(self.Field, [b])
           r *= joint_rand[0]
       return out

            +================+================================+
            | Parameter      | Value                          |
            +================+================================+
            | GADGETS        | [Range2]                       |
            +----------------+--------------------------------+
            | GADGET_CALLS   | [bits]                         |
            +----------------+--------------------------------+
            | MEAS_LEN       | bits                           |
            +----------------+--------------------------------+
            | OUTPUT_LEN     | 1                              |
            +----------------+--------------------------------+
            | JOINT_RAND_LEN | 1                              |
            +----------------+--------------------------------+
            | Measurement    | Unsigned, in range [0, 2^bits) |
            +----------------+--------------------------------+
            | AggResult      | Unsigned                       |
            +----------------+--------------------------------+
            | Field          | Field128 (Table 3)             |
            +----------------+--------------------------------+

               Table 10: Parameters of validity circuit Sum.

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7.4.3.  Prio3SumVec

   This instance of Prio3 supports summing a vector of integers.  It has
   three parameters, length, bits, and chunk_length.  Each measurement
   is a vector of positive integers with length equal to the length
   parameter.  Each element of the measurement is an integer in the
   range [0, 2^bits).  It is RECOMMENDED to set chunk_length to an
   integer near the square root of length * bits (see Section 7.4.3.1).

   This instance uses XofTurboShake128 (Section 6.2.1) as its XOF.  Its
   validity circuit, denoted SumVec, uses Field128 (Table 3) as its
   finite field.

   Measurements are encoded as a vector of field elements with length
   length * bits.  The field elements in the encoded vector represent
   all the bits of the measurement vector's elements, consecutively, in
   LSB to MSB order:

   def encode(self, measurement: Vec[Unsigned]):
       if len(measurement) != self.length:
           raise ERR_INPUT

       encoded = []
       for val in measurement:
           if 0 > val or val >= 2 ** self.bits:
               raise ERR_INPUT

           encoded += self.Field.encode_into_bit_vector(val, self.bits)
       return encoded

   def truncate(self, meas):
       truncated = []
       for i in range(self.length):
           truncated.append(self.Field.decode_from_bit_vector(
               meas[i * self.bits: (i + 1) * self.bits]
           ))
       return truncated

   def decode(self, output, _num_measurements):
       return [x.as_unsigned() for x in output]

   This validity circuit uses a ParallelSum gadget to achieve a smaller
   proof size.  This optimization for "parallel-sum circuits" is
   described in [BBCGGI19], section 4.4.  Briefly, for circuits that add
   up the output of multiple identical subcircuits, it is possible to
   achieve smaller proof sizes (on the order of O(sqrt(MEAS_LEN))
   instead of O(MEAS_LEN)) by packaging more than one such subcircuit
   into a gadget.

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   The ParallelSum gadget is parameterized with an arithmetic
   subcircuit, and a count of how many times it evaluates that
   subcircuit.  It takes in a list of inputs and passes them through to
   instances of the subcircuit in the same order.  It returns the sum of
   the subcircuit outputs.  Note that only the ParallelSum gadget
   itself, and not its subcircuit, participates in FlpGeneric's wire
   recording during evaluation, gadget consistency proofs, and proof
   validation, even though the subcircuit is provided to ParallelSum as
   an implementation of the Gadget interface.

  def eval(self, Field, inp):
      self.check_gadget_eval(inp)
      out = Field(0)
      for i in range(self.count):
          start_index = i * self.subcircuit.ARITY
          end_index = (i + 1) * self.subcircuit.ARITY
          out += self.subcircuit.eval(Field, inp[start_index:end_index])
      return out

   The SumVec validity circuit checks that the encoded measurement
   consists of ones and zeros.  Rather than use the Range2 gadget on
   each element, as in the Sum validity circuit, it instead uses Mul
   subcircuits and "free" constant multiplication and addition gates to
   simultaneously evaluate the same range check polynomial on each
   element, and multiply by a constant.  One of the two Mul subcircuit
   inputs is equal to a measurement element multiplied by a power of the
   joint randomness value, and the other is equal to the same
   measurement element minus one.  These Mul subcircuits are evaluated
   by a ParallelSum gadget, and the results are added up both within the
   ParallelSum gadget and after it.

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   def eval(self, meas, joint_rand, num_shares):
       self.check_valid_eval(meas, joint_rand)

       out = Field128(0)
       r = joint_rand[0]
       r_power = r
       shares_inv = self.Field(num_shares).inv()

       for i in range(self.GADGET_CALLS[0]):
           inputs = [None] * (2 * self.chunk_length)
           for j in range(self.chunk_length):
               index = i * self.chunk_length + j
               if index < len(meas):
                   meas_elem = meas[index]
               else:
                   meas_elem = self.Field(0)

               inputs[j * 2] = r_power * meas_elem
               inputs[j * 2 + 1] = meas_elem - shares_inv

               r_power *= r

           out += self.GADGETS[0].eval(self.Field, inputs)

       return out

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          +================+====================================+
          | Parameter      | Value                              |
          +================+====================================+
          | GADGETS        | [ParallelSum(Mul(), chunk_length)] |
          +----------------+------------------------------------+
          | GADGET_CALLS   | [(length * bits + chunk_length -   |
          |                | 1) // chunk_length]                |
          +----------------+------------------------------------+
          | MEAS_LEN       | length * bits                      |
          +----------------+------------------------------------+
          | OUTPUT_LEN     | length                             |
          +----------------+------------------------------------+
          | JOINT_RAND_LEN | 1                                  |
          +----------------+------------------------------------+
          | Measurement    | Vec[Unsigned], each element in     |
          |                | range [0, 2^bits)                  |
          +----------------+------------------------------------+
          | AggResult      | Vec[Unsigned]                      |
          +----------------+------------------------------------+
          | Field          | Field128 (Table 3)                 |
          +----------------+------------------------------------+

              Table 11: Parameters of validity circuit SumVec.

7.4.3.1.  Selection of ParallelSum chunk length

   The chunk_length parameter provides a trade-off between the arity of
   the ParallelSum gadget and the number of times the gadget is called.
   The proof length is asymptotically minimized when the chunk length is
   near the square root of the length of the measurement.  However, the
   relationship between VDAF parameters and proof length is complicated,
   involving two forms of rounding (the circuit pads the inputs to its
   last ParallelSum gadget call, up to the chunk length, and FlpGeneric
   rounds the degree of wire polynomials -- determined by the number of
   times a gadget is called -- up to the next power of two).  Therefore,
   the optimal choice of chunk_length for a concrete measurement size
   will vary, and must be found through trial and error.  Setting
   chunk_length equal to the square root of the appropriate measurement
   length will result in proofs up to 50% larger than the optimal proof
   size.

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7.4.4.  Prio3Histogram

   This instance of Prio3 allows for estimating the distribution of some
   quantity by computing a simple histogram.  Each measurement
   increments one histogram bucket, out of a set of fixed buckets.
   (Bucket indexing begins at 0.)  For example, the buckets might
   quantize the real numbers, and each measurement would report the
   bucket that the corresponding client's real-numbered value falls
   into.  The aggregate result counts the number of measurements in each
   bucket.

   This instance of Prio3 uses XofTurboShake128 (Section 6.2.1) as its
   XOF.  Its validity circuit, denoted Histogram, uses Field128
   (Table 3) as its finite field.  It has two parameters, length, the
   number of histogram buckets, and chunk_length, which is used by by a
   circuit optimization described below.  It is RECOMMENDED to set
   chunk_length to an integer near the square root of length (see
   Section 7.4.3.1).

   The measurement is encoded as a one-hot vector representing the
   bucket into which the measurement falls:

   def encode(self, measurement):
       encoded = [self.Field(0)] * self.length
       encoded[measurement] = self.Field(1)
       return encoded

   def truncate(self, meas):
       return meas

   def decode(self, output, _num_measurements):
       return [bucket_count.as_unsigned() for bucket_count in output]

   The Histogram validity circuit checks for one-hotness in two steps,
   by checking that the encoded measurement consists of ones and zeros,
   and by checking that the sum of all elements in the encoded
   measurement is equal to one.  All the individual checks are combined
   together in a random linear combination.

   As in the SumVec validity circuit (Section 7.4.3), the first part of
   the validity circuit uses the ParallelSum gadget to perform range
   checks while achieving a smaller proof size.  The ParallelSum gadget
   uses Mul subcircuits to evaluate a range check polynomial on each
   element, and includes an additional constant multiplication.  One of
   the two Mul subcircuit inputs is equal to a measurement element
   multiplied by a power of the first joint randomness value, and the
   other is equal to the same measurement element minus one.  The
   results are added up both within the ParallelSum gadget and after it.

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   def eval(self, meas, joint_rand, num_shares):
       self.check_valid_eval(meas, joint_rand)

       # Check that each bucket is one or zero.
       range_check = self.Field(0)
       r = joint_rand[0]
       r_power = r
       shares_inv = self.Field(num_shares).inv()
       for i in range(self.GADGET_CALLS[0]):
           inputs = [None] * (2 * self.chunk_length)
           for j in range(self.chunk_length):
               index = i * self.chunk_length + j
               if index < len(meas):
                   meas_elem = meas[index]
               else:
                   meas_elem = self.Field(0)

               inputs[j * 2] = r_power * meas_elem
               inputs[j * 2 + 1] = meas_elem - shares_inv

               r_power *= r

           range_check += r * self.GADGETS[0].eval(self.Field, inputs)

       # Check that the buckets sum to 1.
       sum_check = -shares_inv
       for b in meas:
           sum_check += b

       out = joint_rand[1] * range_check + \
           joint_rand[1] ** 2 * sum_check
       return out

   Note that this circuit depends on the number of shares into which the
   measurement is sharded.  This is provided to the FLP by Prio3.

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    +================+===============================================+
    | Parameter      | Value                                         |
    +================+===============================================+
    | GADGETS        | [ParallelSum(Mul(), chunk_length)]            |
    +----------------+-----------------------------------------------+
    | GADGET_CALLS   | [(length + chunk_length - 1) // chunk_length] |
    +----------------+-----------------------------------------------+
    | MEAS_LEN       | length                                        |
    +----------------+-----------------------------------------------+
    | OUTPUT_LEN     | length                                        |
    +----------------+-----------------------------------------------+
    | JOINT_RAND_LEN | 2                                             |
    +----------------+-----------------------------------------------+
    | Measurement    | Unsigned                                      |
    +----------------+-----------------------------------------------+
    | AggResult      | Vec[Unsigned]                                 |
    +----------------+-----------------------------------------------+
    | Field          | Field128 (Table 3)                            |
    +----------------+-----------------------------------------------+

           Table 12: Parameters of validity circuit Histogram.

8.  Poplar1

   This section specifies Poplar1, a VDAF for the following task.  Each
   Client holds a string of length BITS and the Aggregators hold a set
   of l-bit strings, where l <= BITS.  We will refer to the latter as
   the set of "candidate prefixes".  The Aggregators' goal is to count
   how many measurements are prefixed by each candidate prefix.

   This functionality is the core component of the Poplar protocol
   [BBCGGI21], which was designed to compute the heavy hitters over a
   set of input strings.  At a high level, the protocol works as
   follows.

   1.  Each Client splits its string into input shares and sends one
       share to each Aggregator.

   2.  The Aggregators agree on an initial set of candidate prefixes,
       say 0 and 1.

   3.  The Aggregators evaluate the VDAF on each set of input shares and
       aggregate the recovered output shares.  The aggregation parameter
       is the set of candidate prefixes.

   4.  The Aggregators send their aggregate shares to the Collector, who
       combines them to recover the counts of each candidate prefix.

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   5.  Let H denote the set of prefixes that occurred at least t times.
       If the prefixes all have length BITS, then H is the set of t-
       heavy-hitters.  Otherwise compute the next set of candidate
       prefixes, e.g., for each p in H, add p || 0 and p || 1 to the
       set.  Repeat step 3 with the new set of candidate prefixes.

   Poplar1 is constructed from an "Incremental Distributed Point
   Function (IDPF)", a primitive described by [BBCGGI21] that
   generalizes the notion of a Distributed Point Function (DPF) [GI14].
   Briefly, a DPF is used to distribute the computation of a "point
   function", a function that evaluates to zero on every input except at
   a programmable "point".  The computation is distributed in such a way
   that no one party knows either the point or what it evaluates to.

   An IDPF generalizes this "point" to a path on a full binary tree from
   the root to one of the leaves.  It is evaluated on an "index"
   representing a unique node of the tree.  If the node is on the
   programmed path, then the function evaluates to a non-zero value;
   otherwise it evaluates to zero.  This structure allows an IDPF to
   provide the functionality required for the above protocol: To compute
   the hit count for an index, just evaluate each set of IDPF shares at
   that index and add up the results.

   Consider the sub-tree constructed from a set of input strings and a
   target threshold t by including all indices that prefix at least t of
   the input strings.  We shall refer to this structure as the "prefix
   tree" for the batch of inputs and target threshold.  To compute the
   t-heavy hitters for a set of inputs, the Aggregators and Collector
   first compute the prefix tree, then extract the heavy hitters from
   the leaves of this tree.  (Note that the prefix tree may leak more
   information about the set than the heavy hitters themselves; see
   Section 9.3.1 for details.)

   Poplar1 composes an IDPF with the "secure sketching" protocol of
   [BBCGGI21].  This protocol ensures that evaluating a set of input
   shares on a unique set of candidate prefixes results in shares of a
   "one-hot" vector, i.e., a vector that is zero everywhere except for
   one element, which is equal to one.

   The remainder of this section is structured as follows.  IDPFs are
   defined in Section 8.1; a concrete instantiation is given
   Section 8.3.  The Poplar1 VDAF is defined in Section 8.2 in terms of
   a generic IDPF.  Finally, a concrete instantiation of Poplar1 is
   specified in Section 8.4; test vectors can be found in Appendix "Test
   Vectors".

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8.1.  Incremental Distributed Point Functions (IDPFs)

   An IDPF is defined over a domain of size 2^BITS, where BITS is
   constant defined by the IDPF.  Indexes into the IDPF tree are encoded
   as integers in range [0, 2^BITS).  The Client specifies an index
   alpha and a vector of values beta, one for each "level" L in range
   [0, BITS).  The key generation algorithm generates one IDPF "key" for
   each Aggregator.  When evaluated at level L and index 0 <= prefix <
   2^L, each IDPF key returns an additive share of beta[L] if prefix is
   the L-bit prefix of alpha and shares of zero otherwise.

   An index x is defined to be a prefix of another index y as follows.
   Let LSB(x, N) denote the least significant N bits of positive integer
   x.  By definition, a positive integer 0 <= x < 2^L is said to be the
   length-L prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is
   equal to the most significant L bits of LSB(y, BITS), For example, 6
   (110 in binary) is the length-3 prefix of 25 (11001), but 7 (111) is
   not.

   Each of the programmed points beta is a vector of elements of some
   finite field.  We distinguish two types of fields: One for inner
   nodes (denoted FieldInner), and one for leaf nodes (FieldLeaf).  (Our
   instantiation of Poplar1 (Section 8.4) will use a much larger field
   for leaf nodes than for inner nodes.  This is to ensure the IDPF is
   "extractable" as defined in [BBCGGI21], Definition 1.)

   A concrete IDPF defines the types and constants enumerated in
   Table 13.  In the remainder we write Output as shorthand for the type
   Union[list[list[FieldInner]], list[list[FieldLeaf]]].  (This type
   denotes either a vector of inner node field elements or leaf node
   field elements.)  The scheme is comprised of the following
   algorithms:

   *  Idpf.gen(alpha: Unsigned, beta_inner: list[list[FieldInner]],
      beta_leaf: list[FieldLeaf], binder: bytes, rand:
      bytes[Idpf.RAND_SIZE]) -> tuple[bytes, list[bytes]] is the
      randomized IDPF-key generation algorithm.  (Input rand consists of
      the random bytes it consumes.)  Its inputs are the index alpha the
      values beta, and a binder string.  The value of alpha MUST be in
      range [0, 2^BITS).  The output is a public part that is sent to
      all Aggregators and a vector of private IDPF keys, one for each
      aggregator.  The binder string is used to derive the key in the
      underlying XofFixedKeyAes128 XOF that is used for expanding seeds
      at each level.  It MUST be chosen uniformly at random by the
      Client (see Section 9.2).

         TODO(issue #255) Decide whether to treat the public share as an
         opaque byte string or to replace it with an explicit type.

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   *  Idpf.eval(agg_id: Unsigned, public_share: bytes, key: bytes,
      level: Unsigned, prefixes: tuple[Unsigned, ...], binder: Bytes) ->
      Output is the deterministic, stateless IDPF-key evaluation
      algorithm run by each Aggregator.  Its inputs are the Aggregator's
      unique identifier, the public share distributed to all of the
      Aggregators, the Aggregator's IDPF key, the "level" at which to
      evaluate the IDPF, the sequence of candidate prefixes, and a
      binder string.  It returns the share of the value corresponding to
      each candidate prefix.

      The output type (i.e., Output) depends on the value of level: If
      level < Idpf.BITS-1, the output is the value for an inner node,
      which has type list[list[Idpf.FieldInner]]; otherwise, if level ==
      Idpf.BITS-1, then the output is the value for a leaf node, which
      has type list[list[Idpf.FieldLeaf]].

      The value of level MUST be in range [0, BITS).  The indexes in
      prefixes MUST all be distinct and in range [0, 2^level).

      Applications MUST ensure that the Aggregator's identifier is equal
      to the integer in range [0, SHARES) that matches the index of key
      in the sequence of IDPF keys output by the Client.

   In addition, the following method is derived for each concrete Idpf:

   def current_field(Idpf, level):
       return Idpf.FieldInner if level < Idpf.BITS-1 \
                   else Idpf.FieldLeaf

   Finally, an implementation note.  The interface for IDPFs specified
   here is stateless, in the sense that there is no state carried
   between IDPF evaluations.  This is to align the IDPF syntax with the
   VDAF abstraction boundary, which does not include shared state across
   across VDAF evaluations.  In practice, of course, it will often be
   beneficial to expose a stateful API for IDPFs and carry the state
   across evaluations.  See Section 8.3 for details.

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    +============+====================================================+
    | Parameter  | Description                                        |
    +============+====================================================+
    | SHARES     | Number of IDPF keys output by IDPF-key generator   |
    +------------+----------------------------------------------------+
    | BITS       | Length in bits of each input string                |
    +------------+----------------------------------------------------+
    | VALUE_LEN  | Number of field elements of each output value      |
    +------------+----------------------------------------------------+
    | RAND_SIZE  | Size of the random string consumed by the IDPF-key |
    |            | generator.  Equal to twice the XOF's seed size.    |
    +------------+----------------------------------------------------+
    | KEY_SIZE   | Size in bytes of each IDPF key                     |
    +------------+----------------------------------------------------+
    | FieldInner | Implementation of Field (Section 6.1) used for     |
    |            | values of inner nodes                              |
    +------------+----------------------------------------------------+
    | FieldLeaf  | Implementation of Field used for values of leaf    |
    |            | nodes                                              |
    +------------+----------------------------------------------------+
    | Output     | Alias of Union[list[list[FieldInner]],             |
    |            | list[list[FieldLeaf]]]                             |
    +------------+----------------------------------------------------+
    | FieldVec   | Alias of Union[list[FieldInner], list[FieldLeaf]]  |
    +------------+----------------------------------------------------+

         Table 13: Constants and types defined by a concrete IDPF.

8.2.  Construction

   This section specifies Poplar1, an implementation of the Vdaf
   interface (Section 5).  It is defined in terms of any Idpf
   (Section 8.1) for which Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2 and
   an implementation of Xof (Section 6.2).  The associated constants and
   types required by the Vdaf interface are defined in Table 14.  The
   methods required for sharding, preparation, aggregation, and
   unsharding are described in the remaining subsections.  These methods
   make use of constants defined in Table 15.

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     +=================+============================================+
     | Parameter       | Value                                      |
     +=================+============================================+
     | VERIFY_KEY_SIZE | Xof.SEED_SIZE                              |
     +-----------------+--------------------------------------------+
     | RAND_SIZE       | Xof.SEED_SIZE * 3 + Idpf.RAND_SIZE         |
     +-----------------+--------------------------------------------+
     | NONCE_SIZE      | 16                                         |
     +-----------------+--------------------------------------------+
     | ROUNDS          | 2                                          |
     +-----------------+--------------------------------------------+
     | SHARES          | 2                                          |
     +-----------------+--------------------------------------------+
     | Measurement     | Unsigned                                   |
     +-----------------+--------------------------------------------+
     | AggParam        | Tuple[Unsigned, Tuple[Unsigned, ...]]      |
     +-----------------+--------------------------------------------+
     | PublicShare     | bytes (IDPF public share)                  |
     +-----------------+--------------------------------------------+
     | InputShare      | tuple[bytes, bytes, list[Idpf.FieldInner], |
     |                 | list[Idpf.FieldLeaf]]                      |
     +-----------------+--------------------------------------------+
     | OutShare        | Idpf.FieldVec                              |
     +-----------------+--------------------------------------------+
     | AggShare        | Idpf.FieldVec                              |
     +-----------------+--------------------------------------------+
     | AggResult       | Vec[Unsigned]                              |
     +-----------------+--------------------------------------------+
     | PrepState       | tuple[bytes, Unsigned, Idpf.FieldVec]      |
     +-----------------+--------------------------------------------+
     | PrepShare       | Idpf.FieldVec                              |
     +-----------------+--------------------------------------------+
     | PrepMessage     | Optional[Idpf.FieldVec]                    |
     +-----------------+--------------------------------------------+

                  Table 14: VDAF parameters for Poplar1.

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                  +=============================+=======+
                  | Variable                    | Value |
                  +=============================+=======+
                  | USAGE_SHARD_RAND: Unsigned  | 1     |
                  +-----------------------------+-------+
                  | USAGE_CORR_INNER: Unsigned  | 2     |
                  +-----------------------------+-------+
                  | USAGE_CORR_LEAF: Unsigned   | 3     |
                  +-----------------------------+-------+
                  | USAGE_VERIFY_RAND: Unsigned | 4     |
                  +-----------------------------+-------+

                    Table 15: Constants used by Poplar1.

8.2.1.  Client

   The Client's measurement is interpreted as an IDPF index, denoted
   alpha.  The programmed IDPF values are pairs of field elements (1, k)
   where each k is chosen at random.  This random value is used as part
   of the secure sketching protocol of [BBCGGI21], Appendix C.4.  After
   evaluating their IDPF key shares on a given sequence of candidate
   prefixes, the sketching protocol is used by the Aggregators to verify
   that they hold shares of a one-hot vector.  In addition, for each
   level of the tree, the prover generates random elements a, b, and c
   and computes

       A = -2*a + k
       B = a^2 + b - k*a + c

   and sends additive shares of a, b, c, A and B to the Aggregators.
   Putting everything together, the sharding algorithm is defined as
   follows.

   def shard(Poplar1, measurement, nonce, rand):
       l = Poplar1.Xof.SEED_SIZE

       # Split the random input into random input for IDPF key
       # generation, correlated randomness, and sharding.
       if len(rand) != Poplar1.RAND_SIZE:
           raise ERR_INPUT # unexpected length for random input
       idpf_rand, rand = front(Poplar1.Idpf.RAND_SIZE, rand)
       seeds = [rand[i:i+l] for i in range(0,3*l,l)]
       corr_seed, seeds = front(2, seeds)
       (k_shard,), seeds = front(1, seeds)

       xof = Poplar1.Xof(
           k_shard,
           Poplar1.domain_separation_tag(USAGE_SHARD_RAND),

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           nonce,
       )

       # Construct the IDPF values for each level of the IDPF tree.
       # Each "data" value is 1; in addition, the Client generates
       # a random "authenticator" value used by the Aggregators to
       # compute the sketch during preparation. This sketch is used
       # to verify the one-hotness of their output shares.
       beta_inner = [
           [Poplar1.Idpf.FieldInner(1), k]
           for k in xof.next_vec(Poplar1.Idpf.FieldInner,
                                 Poplar1.Idpf.BITS - 1)
       ]
       beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \
           xof.next_vec(Poplar1.Idpf.FieldLeaf, 1)

       # Generate the IDPF keys.
       (public_share, keys) = Poplar1.Idpf.gen(measurement,
                                               beta_inner,
                                               beta_leaf,
                                               nonce,
                                               idpf_rand)

       # Generate correlated randomness used by the Aggregators to
       # compute a sketch over their output shares. XOF seeds are
       # used to encode shares of the `(a, b, c)` triples.
       # (See [BBCGGI21, Appendix C.4].)
       corr_offsets = vec_add(
           Poplar1.Xof.expand_into_vec(
               Poplar1.Idpf.FieldInner,
               corr_seed[0],
               Poplar1.domain_separation_tag(USAGE_CORR_INNER),
               byte(0) + nonce,
               3 * (Poplar1.Idpf.BITS-1),
           ),
           Poplar1.Xof.expand_into_vec(
               Poplar1.Idpf.FieldInner,
               corr_seed[1],
               Poplar1.domain_separation_tag(USAGE_CORR_INNER),
               byte(1) + nonce,
               3 * (Poplar1.Idpf.BITS-1),
           ),
       )
       corr_offsets += vec_add(
           Poplar1.Xof.expand_into_vec(
               Poplar1.Idpf.FieldLeaf,
               corr_seed[0],
               Poplar1.domain_separation_tag(USAGE_CORR_LEAF),

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               byte(0) + nonce,
               3,
           ),
           Poplar1.Xof.expand_into_vec(
               Poplar1.Idpf.FieldLeaf,
               corr_seed[1],
               Poplar1.domain_separation_tag(USAGE_CORR_LEAF),
               byte(1) + nonce,
               3,
           ),
       )

       # For each level of the IDPF tree, shares of the `(A, B)`
       # pairs are computed from the corresponding `(a, b, c)`
       # triple and authenticator value `k`.
       corr_inner = [[], []]
       for level in range(Poplar1.Idpf.BITS):
           Field = Poplar1.Idpf.current_field(level)
           k = beta_inner[level][1] if level < Poplar1.Idpf.BITS - 1 \
               else beta_leaf[1]
           (a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
           A = -Field(2) * a + k
           B = a ** 2 + b - a * k + c
           corr1 = xof.next_vec(Field, 2)
           corr0 = vec_sub([A, B], corr1)
           if level < Poplar1.Idpf.BITS - 1:
               corr_inner[0] += corr0
               corr_inner[1] += corr1
           else:
               corr_leaf = [corr0, corr1]

       # Each input share consists of the Aggregator's IDPF key
       # and a share of the correlated randomness.
       input_shares = list(zip(keys, corr_seed, corr_inner, corr_leaf))
       return (public_share, input_shares)

               Figure 23: The sharding algorithm for Poplar1.

8.2.2.  Preparation

   The aggregation parameter encodes a sequence of candidate prefixes.
   When an Aggregator receives an input share from the Client, it begins
   by evaluating its IDPF share on each candidate prefix, recovering a
   data_share and auth_share for each.  The Aggregators use these and
   the correlation shares provided by the Client to verify that the
   sequence of data_share values are additive shares of a one-hot
   vector.

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   Aggregators MUST ensure the candidate prefixes are all unique and
   appear in lexicographic order.  (This is enforced in the definition
   of prep_init() below.)  Uniqueness is necessary to ensure the refined
   measurement (i.e., the sum of the output shares) is in fact a one-hot
   vector.  Otherwise, sketch verification might fail, causing the
   Aggregators to erroneously reject a report that is actually valid.
   Note that enforcing the order is not strictly necessary, but this
   does allow uniqueness to be determined more efficiently.

  def prep_init(Poplar1, verify_key, agg_id, agg_param,
                nonce, public_share, input_share):
      (level, prefixes) = agg_param
      (key, corr_seed, corr_inner, corr_leaf) = input_share
      Field = Poplar1.Idpf.current_field(level)

      # Ensure that candidate prefixes are all unique and appear in
      # lexicographic order.
      for i in range(1,len(prefixes)):
          if prefixes[i-1] >= prefixes[i]:
              raise ERR_INPUT # out-of-order prefix

      # Evaluate the IDPF key at the given set of prefixes.
      value = Poplar1.Idpf.eval(
          agg_id, public_share, key, level, prefixes, nonce)

      # Get shares of the correlated randomness for computing the
      # Aggregator's share of the sketch for the given level of the IDPF
      # tree.
      if level < Poplar1.Idpf.BITS - 1:
          corr_xof = Poplar1.Xof(
              corr_seed,
              Poplar1.domain_separation_tag(USAGE_CORR_INNER),
              byte(agg_id) + nonce,
          )
          # Fast-forward the XOF state to the current level.
          corr_xof.next_vec(Field, 3 * level)
      else:
          corr_xof = Poplar1.Xof(
              corr_seed,
              Poplar1.domain_separation_tag(USAGE_CORR_LEAF),
              byte(agg_id) + nonce,
          )
      (a_share, b_share, c_share) = corr_xof.next_vec(Field, 3)
      (A_share, B_share) = corr_inner[2*level:2*(level+1)] \
          if level < Poplar1.Idpf.BITS - 1 else corr_leaf

      # Compute the Aggregator's first round of the sketch. These are
      # called the "masked input values" [BBCGGI21, Appendix C.4].

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      verify_rand_xof = Poplar1.Xof(
          verify_key,
          Poplar1.domain_separation_tag(USAGE_VERIFY_RAND),
          nonce + to_be_bytes(level, 2),
      )
      verify_rand = verify_rand_xof.next_vec(Field, len(prefixes))
      sketch_share = [a_share, b_share, c_share]
      out_share = []
      for (i, r) in enumerate(verify_rand):
          [data_share, auth_share] = value[i]
          sketch_share[0] += data_share * r
          sketch_share[1] += data_share * r ** 2
          sketch_share[2] += auth_share * r
          out_share.append(data_share)

      prep_mem = [A_share, B_share, Field(agg_id)] + out_share
      return ((b'sketch round 1', level, prep_mem),
              sketch_share)

  def prep_next(Poplar1, prep_state, prep_msg):
      prev_sketch = prep_msg
      (step, level, prep_mem) = prep_state
      Field = Poplar1.Idpf.current_field(level)

      if step == b'sketch round 1':
          if prev_sketch == None:
              prev_sketch = Field.zeros(3)
          elif len(prev_sketch) != 3:
              raise ERR_INPUT  # prep message malformed
          (A_share, B_share, agg_id), prep_mem = \
              prep_mem[:3], prep_mem[3:]
          sketch_share = [
              agg_id * (prev_sketch[0] ** 2
                        - prev_sketch[1]
                        - prev_sketch[2])
              + A_share * prev_sketch[0]
              + B_share
          ]
          return ((b'sketch round 2', level, prep_mem),
                  sketch_share)

      elif step == b'sketch round 2':
          if prev_sketch == None:
              return prep_mem  # Output shares
          else:
              raise ERR_INPUT  # prep message malformed

      raise ERR_INPUT  # unexpected input

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  def prep_shares_to_prep(Poplar1, agg_param, prep_shares):
      if len(prep_shares) != 2:
          raise ERR_INPUT  # unexpected number of prep shares
      (level, _) = agg_param
      Field = Poplar1.Idpf.current_field(level)
      sketch = vec_add(prep_shares[0], prep_shares[1])
      if len(sketch) == 3:
          return sketch
      elif len(sketch) == 1:
          if sketch == Field.zeros(1):
              # In order to reduce communication overhead, let `None`
              # denote a successful sketch verification.
              return None
          else:
              raise ERR_VERIFY  # sketch verification failed
      else:
          raise ERR_INPUT  # unexpected input length

                Figure 24: Preparation state for Poplar1.

8.2.3.  Validity of Aggregation Parameters

   Aggregation parameters are valid for a given input share if no
   aggregation parameter with the same level has been used with the same
   input share before.  The whole preparation phase MUST NOT be run more
   than once for a given combination of input share and level.

   def is_valid(agg_param, previous_agg_params):
       (level, _) = agg_param
       return all(
           level != other_level
           for (other_level, _) in previous_agg_params
       )

         Figure 25: Validity of aggregation parameters for Poplar1.

8.2.4.  Aggregation

   Aggregation involves simply adding up the output shares.

   def aggregate(Poplar1, agg_param, out_shares):
       (level, prefixes) = agg_param
       Field = Poplar1.Idpf.current_field(level)
       agg_share = Field.zeros(len(prefixes))
       for out_share in out_shares:
           agg_share = vec_add(agg_share, out_share)
       return agg_share

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               Figure 26: Aggregation algorithm for Poplar1.

8.2.5.  Unsharding

   Finally, the Collector unshards the aggregate result by adding up the
   aggregate shares.

   def unshard(Poplar1, agg_param,
               agg_shares, _num_measurements):
       (level, prefixes) = agg_param
       Field = Poplar1.Idpf.current_field(level)
       agg = Field.zeros(len(prefixes))
       for agg_share in agg_shares:
           agg = vec_add(agg, agg_share)
       return list(map(lambda x: x.as_unsigned(), agg))

        Figure 27: Computation of the aggregate result for Poplar1.

8.2.6.  Message Serialization

   This section defines serialization formats for messages exchanged
   over the network while executing Poplar1.  It is RECOMMENDED that
   implementations provide serialization methods for them.

   Message structures are defined following Section 3 of [RFC8446]).  In
   the remainder we use S as an alias for Poplar1.Xof.SEED_SIZE, Fi as
   an alias for Poplar1.Idpf.FieldInner and Fl as an alias for
   Poplar1.Idpf.FieldLeaf.  XOF seeds are represented as follows:

   opaque Poplar1Seed[S];

   Elements of the inner field are encoded in little-endian byte order
   (as defined in Section 6.1) and are represented as follows:

   opaque Poplar1FieldInner[Fi];

   Likewise, elements of the leaf field are encoded in little-endian
   byte order (as defined in Section 6.1) and are represented as
   follows:

   opaque Poplar1FieldLeaf[Fl];

8.2.6.1.  Public Share

   The public share is equal to the IDPF public share, which is a byte
   string.  (See Section 8.1.)

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8.2.6.2.  Input Share

   Each input share is structured as follows:

   struct {
       opaque idpf_key[Poplar1.Idpf.KEY_SIZE];
       Poplar1Seed corr_seed;
       Poplar1FieldInner corr_inner[Fi * 2 * (Poplar1.Idpf.BITS - 1)];
       Poplar1FieldLeaf corr_leaf[Fl * 2];
   } Poplar1InputShare;

8.2.6.3.  Prep Share

   Encoding of the prep share depends on the round of sketching: if the
   first round, then each sketch share has three field elements; if the
   second round, then each sketch share has one field element.  The
   field that is used depends on the level of the IDPF tree specified by
   the aggregation parameter, either the inner field or the leaf field.

   For the first round and inner field:

   struct {
       Poplar1FieldInner sketch_share[Fi * 3];
   } Poplar1PrepShareRoundOneInner;

   For the first round and leaf field:

   struct {
       Poplar1FieldLeaf sketch_share[Fl * 3];
   } Poplar1PrepShareRoundOneLeaf;

   For the second round and inner field:

   struct {
       Poplar1FieldInner sketch_share;
   } Poplar1PrepShareRoundTwoInner;

   For the second round and leaf field:

   struct {
       Poplar1FieldLeaf sketch_share;
   } Poplar1PrepShareRoundTwoLeaf;

8.2.6.4.  Prep Message

   Likewise, the structure of the prep message for Poplar1 depends on
   the sketching round and field.  For the first round and inner field:

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   struct {
       Poplar1FieldInner[Fi * 3];
   } Poplar1PrepMessageRoundOneInner;

   For the first round and leaf field:

   struct {
       Poplar1FieldLeaf sketch[Fl * 3];
   } Poplar1PrepMessageRoundOneLeaf;

   Note that these messages have the same structures as the prep shares
   for the first round.

   The second-round prep message is the empty string.  This is because
   the sketch shares are expected to sum to a particular value if the
   output shares are valid; we represent a successful preparation with
   the empty string and otherwise return an error.

8.2.6.5.  Aggregate Share

   The encoding of the aggregate share depends on whether the inner or
   leaf field is used, and the number of candidate prefixes.  Both of
   these are determined by the aggregation parameter.

   Let prefix_count denote the number of candidate prefixes.  For the
   inner field:

   struct {
       Poplar1FieldInner agg_share[Fi * prefix_count];
   } Poplar1AggShareInner;

   For the leaf field:

   struct {
       Poplar1FieldLeaf agg_share[Fl * prefix_count];
   } Poplar1AggShareLeaf;

8.2.6.6.  Aggregation Parameter

   The aggregation parameter is encoded as follows:

      TODO(issue #255) Express the aggregation parameter encoding in TLS
      syntax.  Decide whether to RECOMMEND this encoding, and if so, add
      it to test vectors.

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   def encode_agg_param(Poplar1, (level, prefixes)):
       if level > 2 ** 16 - 1:
           raise ERR_INPUT # level too deep
       if len(prefixes) > 2 ** 32 - 1:
           raise ERR_INPUT # too many prefixes
       encoded = Bytes()
       encoded += to_be_bytes(level, 2)
       encoded += to_be_bytes(len(prefixes), 4)
       packed = 0
       for (i, prefix) in enumerate(prefixes):
           packed |= prefix << ((level+1) * i)
       l = ((level+1) * len(prefixes) + 7) // 8
       encoded += to_be_bytes(packed, l)
       return encoded

   def decode_agg_param(Poplar1, encoded):
       encoded_level, encoded = encoded[:2], encoded[2:]
       level = from_be_bytes(encoded_level)
       encoded_prefix_count, encoded = encoded[:4], encoded[4:]
       prefix_count = from_be_bytes(encoded_prefix_count)
       l = ((level+1) * prefix_count + 7) // 8
       encoded_packed, encoded = encoded[:l], encoded[l:]
       packed = from_be_bytes(encoded_packed)
       prefixes = []
       m = 2 ** (level+1) - 1
       for i in range(prefix_count):
           prefixes.append(packed >> ((level+1) * i) & m)
       if len(encoded) != 0:
           raise ERR_INPUT
       return (level, tuple(prefixes))

   Implementation note: The aggregation parameter includes the level of
   the IDPF tree and the sequence of indices to evaluate.  For
   implementations that perform per-report caching across executions of
   the VDAF, this may be more information than is strictly needed.  In
   particular, it may be sufficient to convey which indices from the
   previous execution will have their children included in the next.
   This would help reduce communication overhead.

8.3.  The IDPF scheme of [BBCGGI21]

   In this section we specify a concrete IDPF, called IdpfPoplar,
   suitable for instantiating Poplar1.  The scheme gets its name from
   the name of the protocol of [BBCGGI21].

      TODO We should consider giving IdpfPoplar a more distinctive name.

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   The constant and type definitions required by the Idpf interface are
   given in Table 16.

   IdpfPoplar requires a XOF for deriving the output shares, as well as
   a variety of other artifacts used internally.  For performance
   reasons, we instantiate this object using XofFixedKeyAes128
   (Section 6.2.2).  See Section 9.4 for justification of this choice.

                   +============+======================+
                   | Parameter  | Value                |
                   +============+======================+
                   | SHARES     | 2                    |
                   +------------+----------------------+
                   | BITS       | any positive integer |
                   +------------+----------------------+
                   | VALUE_LEN  | any positive integer |
                   +------------+----------------------+
                   | KEY_SIZE   | Xof.SEED_SIZE        |
                   +------------+----------------------+
                   | FieldInner | Field64 (Table 3)    |
                   +------------+----------------------+
                   | FieldLeaf  | Field255 (Table 3)   |
                   +------------+----------------------+

                        Table 16: Constants and type
                        definitions for IdpfPoplar.

8.3.1.  Key Generation

      TODO Describe the construction in prose, beginning with a gentle
      introduction to the high level idea.

   The description of the IDPF-key generation algorithm makes use of
   auxiliary functions extend(), convert(), and encode_public_share()
   defined in Section 8.3.3.  In the following, we let Field2 denote the
   field GF(2).

   def gen(IdpfPoplar, alpha, beta_inner, beta_leaf, binder, rand):
       if alpha >= 2 ** IdpfPoplar.BITS:
           raise ERR_INPUT # alpha too long
       if len(beta_inner) != IdpfPoplar.BITS - 1:
           raise ERR_INPUT # beta_inner vector is the wrong size
       if len(rand) != IdpfPoplar.RAND_SIZE:
           raise ERR_INPUT # unexpected length for random input

       init_seed = [
           rand[:XofFixedKeyAes128.SEED_SIZE],
           rand[XofFixedKeyAes128.SEED_SIZE:],

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       ]

       seed = init_seed.copy()
       ctrl = [Field2(0), Field2(1)]
       correction_words = []
       for level in range(IdpfPoplar.BITS):
           Field = IdpfPoplar.current_field(level)
           keep = (alpha >> (IdpfPoplar.BITS - level - 1)) & 1
           lose = 1 - keep
           bit = Field2(keep)

           (s0, t0) = IdpfPoplar.extend(seed[0], binder)
           (s1, t1) = IdpfPoplar.extend(seed[1], binder)
           seed_cw = xor(s0[lose], s1[lose])
           ctrl_cw = (
               t0[0] + t1[0] + bit + Field2(1),
               t0[1] + t1[1] + bit,
           )

           x0 = xor(s0[keep], ctrl[0].conditional_select(seed_cw))
           x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw))
           (seed[0], w0) = IdpfPoplar.convert(level, x0, binder)
           (seed[1], w1) = IdpfPoplar.convert(level, x1, binder)
           ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep]
           ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep]

           b = beta_inner[level] if level < IdpfPoplar.BITS-1 \
                   else beta_leaf
           if len(b) != IdpfPoplar.VALUE_LEN:
               raise ERR_INPUT # beta too long or too short

           w_cw = vec_add(vec_sub(b, w0), w1)
           # Implementation note: Here we negate the correction word if
           # the control bit `ctrl[1]` is set. We avoid branching on the
           # value in order to reduce leakage via timing side channels.
           mask = Field(1) - Field(2) * Field(ctrl[1].as_unsigned())
           for i in range(len(w_cw)):
               w_cw[i] *= mask

           correction_words.append((seed_cw, ctrl_cw, w_cw))

       public_share = IdpfPoplar.encode_public_share(correction_words)
       return (public_share, init_seed)

          Figure 28: IDPF-key generation algorithm of IdpfPoplar.

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8.3.2.  Key Evaluation

      TODO Describe in prose how IDPF-key evaluation algorithm works.

   The description of the IDPF-evaluation algorithm makes use of
   auxiliary functions extend(), convert(), and decode_public_share()
   defined in Section 8.3.3.

def eval(IdpfPoplar, agg_id, public_share, init_seed,
         level, prefixes, binder):
    if agg_id >= IdpfPoplar.SHARES:
        raise ERR_INPUT # invalid aggregator ID
    if level >= IdpfPoplar.BITS:
        raise ERR_INPUT # level too deep
    if len(set(prefixes)) != len(prefixes):
        raise ERR_INPUT # candidate prefixes are non-unique

    correction_words = IdpfPoplar.decode_public_share(public_share)
    out_share = []
    for prefix in prefixes:
        if prefix >= 2 ** (level+1):
            raise ERR_INPUT # prefix too long

        # The Aggregator's output share is the value of a node of
        # the IDPF tree at the given `level`. The node's value is
        # computed by traversing the path defined by the candidate
        # `prefix`. Each node in the tree is represented by a seed
        # (`seed`) and a set of control bits (`ctrl`).
        seed = init_seed
        ctrl = Field2(agg_id)
        for current_level in range(level+1):
            bit = (prefix >> (level - current_level)) & 1

            # Implementation note: Typically the current round of
            # candidate prefixes would have been derived from
            # aggregate results computed during previous rounds. For
            # example, when using `IdpfPoplar` to compute heavy
            # hitters, a string whose hit count exceeded the given
            # threshold in the last round would be the prefix of each
            # `prefix` in the current round. (See [BBCGGI21,
            # Section 5.1].) In this case, part of the path would
            # have already been traversed.
            #
            # Re-computing nodes along previously traversed paths is
            # wasteful. Implementations can eliminate this added
            # complexity by caching nodes (i.e., `(seed, ctrl)`
            # pairs) output by previous calls to `eval_next()`.
            (seed, ctrl, y) = IdpfPoplar.eval_next(

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                seed,
                ctrl,
                correction_words[current_level],
                current_level,
                bit,
                binder,
            )
        out_share.append(y if agg_id == 0 else vec_neg(y))
    return out_share

def eval_next(IdpfPoplar, prev_seed, prev_ctrl,
              correction_word, level, bit, binder):
    """
    Compute the next node in the IDPF tree along the path determined by
    a candidate prefix. The next node is determined by `bit`, the bit of
    the prefix corresponding to the next level of the tree.

    TODO Consider implementing some version of the optimization
    discussed at the end of [BBCGGI21, Appendix C.2]. This could on
    average reduce the number of AES calls by a constant factor.
    """

    Field = IdpfPoplar.current_field(level)
    (seed_cw, ctrl_cw, w_cw) = correction_word
    (s, t) = IdpfPoplar.extend(prev_seed, binder)
    s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw))
    s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw))
    t[0] += ctrl_cw[0] * prev_ctrl
    t[1] += ctrl_cw[1] * prev_ctrl

    next_ctrl = t[bit]
    (next_seed, y) = IdpfPoplar.convert(level, s[bit], binder)
    # Implementation note: Here we add the correction word to the
    # output if `next_ctrl` is set. We avoid branching on the value of
    # the control bit in order to reduce side channel leakage.
    mask = Field(next_ctrl.as_unsigned())
    for i in range(len(y)):
        y[i] += w_cw[i] * mask

    return (next_seed, next_ctrl, y)

    Figure 29: IDPF-evaluation generation algorithm of IdpfPoplar.

8.3.3.  Auxiliary Functions

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   def extend(IdpfPoplar, seed, binder):
       xof = XofFixedKeyAes128(seed, format_dst(1, 0, 0), binder)
       s = [
           bytearray(xof.next(XofFixedKeyAes128.SEED_SIZE)),
           bytearray(xof.next(XofFixedKeyAes128.SEED_SIZE)),
       ]
       # Use the least significant bits as the control bit correction,
       # and then zero it out. This gives effectively 127 bits of
       # security, but reduces the number of AES calls needed by 1/3.
       t = [Field2(s[0][0] & 1), Field2(s[1][0] & 1)]
       s[0][0] &= 0xFE
       s[1][0] &= 0xFE
       return (s, t)

   def convert(IdpfPoplar, level, seed, binder):
       xof = XofFixedKeyAes128(seed, format_dst(1, 0, 1), binder)
       next_seed = xof.next(XofFixedKeyAes128.SEED_SIZE)
       Field = IdpfPoplar.current_field(level)
       w = xof.next_vec(Field, IdpfPoplar.VALUE_LEN)
       return (next_seed, w)

   def encode_public_share(IdpfPoplar, correction_words):
       encoded = Bytes()
       control_bits = list(itertools.chain.from_iterable(
           cw[1] for cw in correction_words
       ))
       encoded += pack_bits(control_bits)
       for (level, (seed_cw, _, w_cw)) \
           in enumerate(correction_words):
           Field = IdpfPoplar.current_field(level)
           encoded += seed_cw
           encoded += Field.encode_vec(w_cw)
       return encoded

   def decode_public_share(IdpfPoplar, encoded):
       l = (2*IdpfPoplar.BITS + 7) // 8
       encoded_ctrl, encoded = encoded[:l], encoded[l:]
       control_bits = unpack_bits(encoded_ctrl, 2 * IdpfPoplar.BITS)
       correction_words = []
       for level in range(IdpfPoplar.BITS):
           Field = IdpfPoplar.current_field(level)
           ctrl_cw = (
               control_bits[level * 2],
               control_bits[level * 2 + 1],
           )
           l = XofFixedKeyAes128.SEED_SIZE
           seed_cw, encoded = encoded[:l], encoded[l:]
           l = Field.ENCODED_SIZE * IdpfPoplar.VALUE_LEN

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           encoded_w_cw, encoded = encoded[:l], encoded[l:]
           w_cw = Field.decode_vec(encoded_w_cw)
           correction_words.append((seed_cw, ctrl_cw, w_cw))
       if len(encoded) != 0:
           raise ERR_DECODE
       return correction_words

                Figure 30: Helper functions for IdpfPoplar.

   Here, pack_bits() takes a list of bits, packs each group of eight
   bits into a byte, in LSB to MSB order, padding the most significant
   bits of the last byte with zeros as necessary, and returns the byte
   array. unpack_bits() performs the reverse operation: it takes in a
   byte array and a number of bits, and returns a list of bits,
   extracting eight bits from each byte in turn, in LSB to MSB order,
   and stopping after the requested number of bits.  If the byte array
   has an incorrect length, or if unused bits in the last bytes are not
   zero, it throws an error.

8.4.  Instantiation

   By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2)
   and XofTurboShake128 (Section 6.2.1).  This VDAF is suitable for any
   positive value of BITS.  Test vectors can be found in Appendix "Test
   Vectors".

9.  Security Considerations

   VDAFs have two essential security goals:

   1.  Privacy: An attacker that controls the network, the Collector,
       and a subset of Clients and Aggregators learns nothing about the
       measurements of honest Clients beyond what it can deduce from the
       aggregate result.

   2.  Robustness: An attacker that controls the network and a subset of
       Clients cannot cause the Collector to compute anything other than
       the aggregate of the measurements of honest Clients.

   Formal definitions of privacy and robustness can be found in
   [DPRS23].  A VDAF is the core cryptographic primitive of a protocol
   that achieves the above privacy and robustness goals.  It is not
   sufficient on its own, however.  The application will need to assure
   a few security properties, for example:

   *  Securely distributing the long-lived parameters, in particular the
      verification key.

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   *  Establishing secure channels:

      -  Confidential and authentic channels among Aggregators, and
         between the Aggregators and the Collector; and

      -  Confidential and Aggregator-authenticated channels between
         Clients and Aggregators.

   *  Enforcing the non-collusion properties required of the specific
      VDAF in use.

   In such an environment, a VDAF provides the high-level privacy
   property described above: The Collector learns only the aggregate
   measurement, and nothing about individual measurements aside from
   what can be inferred from the aggregate result.  The Aggregators
   learn neither individual measurements nor the aggregate result.  The
   Collector is assured that the aggregate statistic accurately reflects
   the inputs as long as the Aggregators correctly executed their role
   in the VDAF.

   On their own, VDAFs do not mitigate Sybil attacks [Dou02].  In this
   attack, the adversary observes a subset of input shares transmitted
   by a Client it is interested in.  It allows the input shares to be
   processed, but corrupts and picks bogus measurements for the
   remaining Clients.  Applications can guard against these risks by
   adding additional controls on report submission, such as Client
   authentication and rate limits.

   VDAFs do not inherently provide differential privacy [Dwo06].  The
   VDAF approach to private measurement can be viewed as complementary
   to differential privacy, relying on non-collusion instead of
   statistical noise to protect the privacy of the inputs.  It is
   possible that a future VDAF could incorporate differential privacy
   features, e.g., by injecting noise before the sharding stage and
   removing it after unsharding.

9.1.  Requirements for the Verification Key

   The Aggregators are responsible for exchanging the verification key
   in advance of executing the VDAF.  Any procedure is acceptable as
   long as the following conditions are met:

   1.  To ensure robustness of the computation, the Aggregators MUST NOT
       reveal the verification key to the Clients.  Otherwise, a
       malicious Client might be able to exploit knowledge of this key
       to craft an invalid report that would be accepted by the
       Aggregators.

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   2.  To ensure privacy of the measurements, the Aggregators MUST
       commit to the verification key prior to processing reports
       generated by Clients.  Otherwise, a malicious Aggregator may be
       able to craft a verification key that, for a given report, causes
       an honest Aggregator to leak information about the measurement
       during preparation.

   Meeting these conditions is required in order to leverage security
   analysis in the framework of [DPRS23].  Their definition of
   robustness allows the attacker, playing the role of a cohort of
   malicious Clients, to submit arbitrary reports to the Aggregators and
   eavesdrop on their communications as they process them.  Security in
   this model is achievable as long as the verification key is kept
   secret from the attacker.

   The privacy definition of [DPRS23] considers an active attacker that
   controls the network and a subset of Aggregators; in addition, the
   attacker is allowed to choose the verification key used by each
   honest Aggregator over the course of the experiment.  Security is
   achievable in this model as long as the key is picked at the start of
   the experiment, prior to any reports being generated.  (The model
   also requires nonces to be generated at random; see Section 9.2
   below.)

   Meeting these requirements is relatively straightforward.  For
   example, the Aggregators may designate one of their peers to generate
   the verification key and distribute it to the others.  To assure
   Clients of key commitment, the Clients and (honest) Aggregators could
   bind reports to a shared context string derived from the key.  For
   instance, the "task ID" of DAP [DAP] could be set to the hash of the
   verification key; then as long as honest Aggregators only consume
   reports for the task indicated by the Client, forging a new key after
   the fact would reduce to finding collisions in the underlying hash
   function.  (Keeping the key secret from the Clients would require the
   hash function to be one-way.)  However, since rotating the key
   implies rotating the task ID, this scheme would not allow key
   rotation over the lifetime of a task.

9.2.  Requirements for the Nonce

   The sharding and preparation steps of VDAF execution depend on a
   nonce associated with the Client's report.  To ensure privacy of the
   underlying measurement, the Client MUST generate this nonce using a
   CSPRNG.  This is required in order to leverage security analysis for
   the privacy definition of [DPRS23], which assumes the nonce is chosen
   at random prior to generating the report.

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   Other security considerations may require the nonce to be non-
   repeating.  For example, to achieve differential privacy it is
   necessary to avoid "over exposing" a measurement by including it too
   many times in a single batch or across multiple batches.  It is
   RECOMMENDED that the nonce generated by the Client be used by the
   Aggregators for replay protection.

9.3.  Requirements for the Aggregation Parameters

   As described in Section 4.3 and Section 5.3 respectively, DAFs and
   VDAFs may impose restrictions on the re-use of input shares.  This is
   to ensure that correlated randomness provided by the Client through
   the input share is not used more than once, which might compromise
   confidentiality of the Client's measurements.

   Protocols that make use of VDAFs therefore MUST call Vdaf.is_valid on
   the set of all aggregation parameters used for a Client's input
   share, and only proceed with the preparation and aggregation phases
   if that function call returns True.

9.3.1.  Additional Privacy Considerations

   Aggregating a batch of reports multiple times, each time with a
   different aggregation parameter, could result in information leakage
   beyond what is used by the application.

   For example, when Poplar1 is used for heavy hitters, the Aggregators
   learn not only the heavy hitters themselves, but also the prefix tree
   (as defined in Section 8) computed along the way.  Indeed, this
   leakage is inherent to any construction that uses an IDPF
   (Section 8.1) in the same way.  Depending on the distribution of the
   measurements, the prefix tree can leak a significant amount of
   information about unpopular inputs.  For instance, it is possible
   (though perhaps unlikely) for a large set of non-heavy-hitter values
   to share a common prefix, which would be leaked by a prefix tree with
   a sufficiently small threshold.

   The only known, general-purpose approach to mitigating this leakage
   is via differential privacy.

      TODO(issue #94) Describe (or point to some description of) the
      central DP mechanism for Poplar described in [BBCGGI21].

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9.4.  Requirements for XOFs

   As described in Section 6.2, our constructions rely on eXtendable
   Output Functions (XOFs).  In the security analyses of our protocols,
   these are usually modeled as random oracles.  XofTurboShake128 is
   designed to be indifferentiable from a random oracle [MRH04], making
   it a suitable choice for most situations.

   The one exception is the Idpf implementation IdpfPoplar Section 8.3.
   Here, a random oracle is not needed to prove privacy, since the
   analysis of [BBCGGI21], Proposition 1, only requires a Pseudorandom
   Generator (PRG).  As observed in [GKWY20], a PRG can be instantiated
   from a correlation-robust hash function H.  Informally, correlation
   robustness requires that for a random r, H(xor(r, x)) is
   computationally indistinguishable from a random function of x.  A PRG
   can therefore be constructed as

   PRG(r) = H(xor(r, 1)) || H(xor(r, 2)) || ...`

   since each individual hash function evaluation is indistinguishable
   from a random function.

   Our construction at Section 6.2.2 implements a correlation-robust
   hash function using fixed-key AES.  For security, it assumes that AES
   with a fixed key can be modeled as a random permutation [GKWY20].
   Additionally, we use a different AES key for every client, which in
   the ideal cipher model leads to better concrete security [GKWWY20].

   We note that for robustness, the analysis of [BBCGGI21] still assumes
   a random oracle to make the Idpf extractable.  While
   XofFixedKeyAes128 has been shown to be differentiable from a random
   oracle [GKWWY20], there are no known attacks exploiting this
   difference.  We also stress that even if the Idpf is not extractable,
   Poplar1 guarantees that every client can contribute to at most one
   prefix among the ones being evaluated by the helpers.

9.5.  Choosing the Number of Proofs to Use for Prio3

      TODO Add guidance for choosing PROOFS (Section 7.1.2) for Prio3.
      In particular when we go for a smaller field for a given circuit.
      See this (https://github.com/cfrg/draft-irtf-cfrg-vdaf/issues/177)
      for details.

10.  IANA Considerations

   A codepoint for each (V)DAF in this document is defined in the table
   below.  Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for
   private use.

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   +==========================+================+======+===============+
   | Value                    | Scheme         | Type | Reference     |
   +==========================+================+======+===============+
   | 0x00000000               | Prio3Count     | VDAF | Section 7.4.1 |
   +--------------------------+----------------+------+---------------+
   | 0x00000001               | Prio3Sum       | VDAF | Section 7.4.2 |
   +--------------------------+----------------+------+---------------+
   | 0x00000002               | Prio3SumVec    | VDAF | Section 7.4.3 |
   +--------------------------+----------------+------+---------------+
   | 0x00000003               | Prio3Histogram | VDAF | Section 7.4.4 |
   +--------------------------+----------------+------+---------------+
   | 0x00000004 to 0x00000FFF | reserved for   | VDAF | n/a           |
   |                          | Prio3          |      |               |
   +--------------------------+----------------+------+---------------+
   | 0x00001000               | Poplar1        | VDAF | Section 8.4   |
   +--------------------------+----------------+------+---------------+
   | 0xFFFF0000 to 0xFFFFFFFF | reserved       | n/a  | n/a           |
   +--------------------------+----------------+------+---------------+

                Table 17: Unique identifiers for (V)DAFs.

      TODO Add IANA considerations for the codepoints summarized in
      Table 17.

11.  References

11.1.  Normative References

   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119,
              DOI 10.17487/RFC2119, March 1997,
              <https://www.rfc-editor.org/rfc/rfc2119>.

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.

   [RFC8446]  Rescorla, E., "The Transport Layer Security (TLS) Protocol
              Version 1.3", RFC 8446, DOI 10.17487/RFC8446, August 2018,
              <https://www.rfc-editor.org/rfc/rfc8446>.

   [TurboSHAKE]
              Viguier, B., Wong, D., Van Assche, G., Dang, Q., and J.
              Daemen, "KangarooTwelve and TurboSHAKE", Work in Progress,
              Internet-Draft, draft-irtf-cfrg-kangarootwelve-11, 20 June
              2023, <https://datatracker.ietf.org/doc/html/draft-irtf-
              cfrg-kangarootwelve-11>.

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11.2.  Informative References

   [AGJOP21]  Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A.
              Polychroniadou, "Prio+: Privacy Preserving Aggregate
              Statistics via Boolean Shares", 2021,
              <https://ia.cr/2021/576>.

   [BBCGGI19] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and
              Y. Ishai, "Zero-Knowledge Proofs on Secret-Shared Data via
              Fully Linear PCPs", CRYPTO 2019 , 2019,
              <https://ia.cr/2019/188>.

   [BBCGGI21] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and
              Y. Ishai, "Lightweight Techniques for Private Heavy
              Hitters", IEEE S&P 2021 , 2021, <https://ia.cr/2021/017>.

   [CGB17]    Corrigan-Gibbs, H. and D. Boneh, "Prio: Private, Robust,
              and Scalable Computation of Aggregate Statistics", NSDI
              2017 , 2017,
              <https://dl.acm.org/doi/10.5555/3154630.3154652>.

   [DAP]      Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood,
              "Distributed Aggregation Protocol for Privacy Preserving
              Measurement", Work in Progress, Internet-Draft, draft-
              ietf-ppm-dap-08, 23 October 2023,
              <https://datatracker.ietf.org/doc/html/draft-ietf-ppm-dap-
              08>.

   [Dou02]    Douceur, J., "The Sybil Attack", IPTPS 2002 , 2002,
              <https://doi.org/10.1007/3-540-45748-8_24>.

   [DPRS23]   Davis, H., Patton, C., Rosulek, M., and P. Schoppmann,
              "Verifiable Distributed Aggregation Functions", n.d.,
              <https://ia.cr/2023/130>.

   [Dwo06]    Dwork, C., "Differential Privacy", ICALP 2006 , 2006,
              <https://link.springer.com/chapter/10.1007/11787006_1>.

   [ENPA]     "Exposure Notification Privacy-preserving Analytics (ENPA)
              White Paper", 2021, <https://covid19-static.cdn-
              apple.com/applications/covid19/current/static/contact-
              tracing/pdf/ENPA_White_Paper.pdf>.

   [EPK14]    Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR:
              Randomized Aggregatable Privacy-Preserving Ordinal
              Response", CCS 2014 , 2014,
              <https://dl.acm.org/doi/10.1145/2660267.2660348>.

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   [GI14]     Gilboa, N. and Y. Ishai, "Distributed Point Functions and
              Their Applications", EUROCRYPT 2014 , 2014,
              <https://link.springer.com/
              chapter/10.1007/978-3-642-55220-5_35>.

   [GKWWY20]  Guo, C., Katz, J., Wang, X., Weng, C., and Y. Yu, "Better
              concrete security for half-gates garbling (in the multi-
              instance setting)", CRYPTO 2020 , 2020,
              <https://link.springer.com/
              chapter/10.1007/978-3-030-56880-1_28>.

   [GKWY20]   Guo, C., Katz, J., Wang, X., and Y. Yu, "Efficient and
              Secure Multiparty Computation from Fixed-Key Block
              Ciphers", S&P 2020 , 2020,
              <https://eprint.iacr.org/2019/074>.

   [MRH04]    Maurer, U., Renner, R., and C. Holenstein,
              "Indifferentiability, impossibility results on reductions,
              and applications to the random oracle methodology", In TCC
              2004: Theory of Cryptography, pages 21-39,
              DOI 10.1007/978-3-540-24638-1_2, February 2004,
              <https://doi.org/10.1007/978-3-540-24638-1_2>.

   [OriginTelemetry]
              "Origin Telemetry", 2020, <https://firefox-source-
              docs.mozilla.org/toolkit/components/telemetry/collection/
              origin.html>.

Acknowledgments

   The security considerations in Section 9 are based largely on the
   security analysis of [DPRS23].  Thanks to Hannah Davis and Mike
   Rosulek, who lent their time to developing definitions and security
   proofs.

   Thanks to Junye Chen, Henry Corrigan-Gibbs, Armando Faz-Hernández,
   Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana
   Raykova, Jacob Rothstein, Shan Wang, Xiao Wang, and Christopher Wood
   for useful feedback on and contributions to the spec.

Test Vectors

   [TO BE REMOVED BY RFC EDITOR: Machine-readable test vectors can be
   found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
   test_vec.]

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   Test vectors cover the generation of input shares and the conversion
   of input shares into output shares.  Vectors specify the verification
   key, measurements, aggregation parameter, and any parameters needed
   to construct the VDAF.  (For example, for Prio3Sum, the user
   specifies the number of bits for representing each summand.)

   Byte strings are encoded in hexadecimal.  To make the tests
   deterministic, the random inputs of randomized algorithms were fixed
   to the byte sequence starting with 0, incrementing by 1, and wrapping
   at 256:

   0, 1, 2, ..., 255, 0, 1, 2, ...

Prio3Count

      TODO Copy the machine readable vectors from the source repository
      (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
      test_vec) and format them for humans.

Prio3Sum

      TODO Copy the machine readable vectors from the source repository
      (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
      test_vec) and format them for humans.

Prio3SumVec

      TODO Copy the machine readable vectors from the source repository
      (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
      test_vec) and format them for humans.

Prio3Histogram

      TODO Copy the machine readable vectors from the source repository
      (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
      test_vec) and format them for humans.

Poplar1

      TODO Copy the machine readable vectors from the source repository
      (https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
      test_vec) and format them for humans.

Authors' Addresses

   Richard L. Barnes
   Cisco
   Email: rlb@ipv.sx

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   David Cook
   ISRG
   Email: divergentdave@gmail.com

   Christopher Patton
   Cloudflare
   Email: chrispatton+ietf@gmail.com

   Phillipp Schoppmann
   Google
   Email: schoppmann@google.com

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