Skip to main content

Verifiable Distributed Aggregation Functions

The information below is for an old version of the document.
Document Type
This is an older version of an Internet-Draft whose latest revision state is "Active".
Authors Richard Barnes , Christopher Patton , Phillipp Schoppmann
Last updated 2022-04-27
Replaces draft-patton-cfrg-vdaf
RFC stream Internet Research Task Force (IRTF)
Additional resources Mailing list discussion
Stream IRTF state (None)
Consensus boilerplate Unknown
Document shepherd (None)
IESG IESG state I-D Exists
Telechat date (None)
Responsible AD (None)
Send notices to (None)
CFRG                                                        R. L. Barnes
Internet-Draft                                                     Cisco
Intended status: Informational                                 C. Patton
Expires: 29 October 2022                                Cloudflare, Inc.
                                                           P. Schoppmann
                                                           27 April 2022

              Verifiable Distributed Aggregation Functions


   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 input that
   would result in an incorrect 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 (, which is archived at

   Source for this draft and an issue tracker can be found at

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

   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."

Barnes, et al.           Expires 29 October 2022                [Page 1]
Internet-Draft                    VDAF                        April 2022

   This Internet-Draft will expire on 29 October 2022.

Copyright Notice

   Copyright (c) 2022 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 (
   license-info) in effect on the date of publication of this document.
   Please review these documents carefully, as they describe your rights
   and restrictions with respect to this document.  Code Components
   extracted from this document must include Revised BSD License text as
   described in Section 4.e of the Trust Legal Provisions and are
   provided without warranty as described in the Revised BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   3
   2.  Conventions and Definitions . . . . . . . . . . . . . . . . .   6
   3.  Overview  . . . . . . . . . . . . . . . . . . . . . . . . . .   7
   4.  Definition of VDAFs . . . . . . . . . . . . . . . . . . . . .   9
     4.1.  Setup . . . . . . . . . . . . . . . . . . . . . . . . . .  10
     4.2.  Sharding  . . . . . . . . . . . . . . . . . . . . . . . .  11
     4.3.  Preparation . . . . . . . . . . . . . . . . . . . . . . .  11
     4.4.  Aggregation . . . . . . . . . . . . . . . . . . . . . . .  14
     4.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . . . .  15
     4.6.  Execution of a VDAF . . . . . . . . . . . . . . . . . . .  15
   5.  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . .  17
     5.1.  Finite Fields . . . . . . . . . . . . . . . . . . . . . .  17
       5.1.1.  Auxiliary Functions . . . . . . . . . . . . . . . . .  18
       5.1.2.  FFT-Friendly Fields . . . . . . . . . . . . . . . . .  19
       5.1.3.  Parameters  . . . . . . . . . . . . . . . . . . . . .  19
     5.2.  Pseudorandom Generators . . . . . . . . . . . . . . . . .  20
       5.2.1.  PrgAes128 . . . . . . . . . . . . . . . . . . . . . .  21
   6.  Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . .  22
     6.1.  Fully Linear Proof (FLP) Systems  . . . . . . . . . . . .  23
       6.1.1.  Encoding the Input  . . . . . . . . . . . . . . . . .  26
     6.2.  Construction  . . . . . . . . . . . . . . . . . . . . . .  26
       6.2.1.  Setup . . . . . . . . . . . . . . . . . . . . . . . .  27
       6.2.2.  Sharding  . . . . . . . . . . . . . . . . . . . . . .  28
       6.2.3.  Preparation . . . . . . . . . . . . . . . . . . . . .  30
       6.2.4.  Aggregation . . . . . . . . . . . . . . . . . . . . .  32
       6.2.5.  Unsharding  . . . . . . . . . . . . . . . . . . . . .  33
       6.2.6.  Auxiliary Functions . . . . . . . . . . . . . . . . .  33
     6.3.  A General-Purpose FLP . . . . . . . . . . . . . . . . . .  35
       6.3.1.  Overview  . . . . . . . . . . . . . . . . . . . . . .  35
       6.3.2.  Validity Circuits . . . . . . . . . . . . . . . . . .  38

Barnes, et al.           Expires 29 October 2022                [Page 2]
Internet-Draft                    VDAF                        April 2022

       6.3.3.  Construction  . . . . . . . . . . . . . . . . . . . .  40
     6.4.  Instantiations  . . . . . . . . . . . . . . . . . . . . .  43
       6.4.1.  Prio3Aes128Count  . . . . . . . . . . . . . . . . . .  44
       6.4.2.  Prio3Aes128Sum  . . . . . . . . . . . . . . . . . . .  45
       6.4.3.  Prio3Aes128Histogram  . . . . . . . . . . . . . . . .  46
   7.  Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . .  48
     7.1.  Incremental Distributed Point Functions (IDPFs) . . . . .  49
     7.2.  Construction  . . . . . . . . . . . . . . . . . . . . . .  50
       7.2.1.  Setup . . . . . . . . . . . . . . . . . . . . . . . .  50
       7.2.2.  Preparation . . . . . . . . . . . . . . . . . . . . .  52
       7.2.3.  Aggregation . . . . . . . . . . . . . . . . . . . . .  54
       7.2.4.  Unsharding  . . . . . . . . . . . . . . . . . . . . .  54
       7.2.5.  Helper Functions  . . . . . . . . . . . . . . . . . .  55
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  55
   9.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  57
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  57
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  57
     10.2.  Informative References . . . . . . . . . . . . . . . . .  57
   Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .  58
   Test Vectors  . . . . . . . . . . . . . . . . . . . . . . . . . .  59
     Prio3Aes128Count  . . . . . . . . . . . . . . . . . . . . . . .  59
     Prio3Aes128Sum  . . . . . . . . . . . . . . . . . . . . . . . .  60
     Prio3Aes128Histogram  . . . . . . . . . . . . . . . . . . . . .  61
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  62

1.  Introduction

   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.

Barnes, et al.           Expires 29 October 2022                [Page 3]
Internet-Draft                    VDAF                        April 2022

   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 input before submitting to
   the aggregation server.  The aggregation server then adds up the
   noisy inputs, and because it knows the distribution from whence the
   noise was sampled, it can estimate the true sum with reasonable

   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 input, it may
   be possible for a curious aggregator to make a reasonable guess of
   the input'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: No participant in the protocol should
   learn anything about an individual input 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 input is ever accessible
   to any party besides the client that submitted it.  At the same time,
   VDAFs are "verifiable" in the sense that malformed inputs that would
   otherwise garble the output of the computation can be detected and
   removed from the set of inputs.

   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].

Barnes, et al.           Expires 29 October 2022                [Page 4]
Internet-Draft                    VDAF                        April 2022

   The VDAF abstraction laid out in Section 4 represents a class of
   multi-party protocols for privacy-preserving measurement proposed in
   the literature.  These protocols vary in their operational and
   security considerations, sometimes in subtle but consequential ways.
   This document therefore has two important goals:

   1.  Providing applications like [I-D.draft-gpew-priv-ppm] with a
       simple, uniform interface for accessing privacy-preserving
       measurement schemes, and documenting relevant operational and
       security bounds for that interface:

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

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

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

   2.  Providing cryptographers with design criteria that allow new
       constructions to be easily used by applications.

   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 input into a sequence of additive
          shares and distributes the shares among the aggregation

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

      3.  Finally, the aggregation servers combine their additive shares
          to obtain the final aggregate.

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

Barnes, et al.           Expires 29 October 2022                [Page 5]
Internet-Draft                    VDAF                        April 2022

      In Section 6 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 inputs that occur at least t times.  The core primitive
      used in their protocol is a generalization of a 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
      input.  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 input.

      In Section 7 we describe a VDAF called Poplar1 that implements
      this functionality.

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

2.  Conventions and Definitions

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "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.

   Some common functionalities:

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

Barnes, et al.           Expires 29 October 2022                [Page 6]
Internet-Draft                    VDAF                        April 2022

   *  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).

   *  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.

   *  I2OSP and OS2IP from [RFC8017], which are used, respectively, to
      convert a non-negative integer to a byte string and convert a byte
      string to a non-negative integer.

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

3.  Overview

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

         Input shares           Aggregate shares

                   Figure 1: Overall data flow of a VDAF

   In a 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:

Barnes, et al.           Expires 29 October 2022                [Page 7]
Internet-Draft                    VDAF                        April 2022

   *  Clients are configured with public parameters for a set of

   *  To submit an individual measurement, the Client shards the
      measurement into "input shares" and sends one input share to each

   *  The Aggregators verify the validity of the input shares, producing
      a set of "output shares".

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

      -  Just as each Aggregator receives one input share of each input,
         at the end of the validation process, each aggregator holds one
         output share.

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

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

   *  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 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 inputs.  Thus clients trust Aggregators not to collude
   (typically it is required that at least one Aggregator is honest),
   and Collectors trust Aggregators to properly verify Client inputs.

   Within the bounds of the non-collusion requirements of a given VDAF
   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 Aggregators to augment a basic client-
   server protocol.

Barnes, et al.           Expires 29 October 2022                [Page 8]
Internet-Draft                    VDAF                        April 2022

   In this document, we describe the computations performed by the
   actors in this system.  It is up to applications 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 VDAFs

   A concrete VDAF specifies the algorithms involved in evaluating an
   aggregation function across a batch of inputs.  This section
   specifies the interfaces of these algorithms as they would be exposed
   to applications.

   The overall execution of a VDAF comprises the following steps:

   *  Setup - Generating shared parameters for the Aggregators

   *  Sharding - Computing input shares from an individual measurement

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

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

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

   The setup algorithm is performed once for a given collection of
   Aggregators.  Sharding and preparation are done once per measurement
   input.  Aggregation and unsharding are done over a batch of inputs
   (more precisely, over the output shares recovered from those inputs).

   Note that the preparation step performs two functions: Verification
   and conversion.  Conversion translates input shares into output
   shares that are compatible with the aggregation function.
   Verification ensures that aggregating the recovered output shares
   will not lead to a garbled aggregate result.

   The remainder of this section defines the VDAF interface in terms of
   an abstract base class Vdaf.  This class defines the set of methods
   and attributes a concrete VDAF must provide.  The attributes are
   listed in Table 1; the methods are defined in the subsections that

Barnes, et al.           Expires 29 October 2022                [Page 9]
Internet-Draft                    VDAF                        April 2022

       | Parameter   | Description                                 |
       | ROUNDS      | Number of rounds of communication during    |
       |             | the preparation phase (Section 4.3)         |
       | SHARES      | Number of input shares into which each      |
       |             | measurement is sharded (Section 4.2)        |
       | Measurement | Type of each measurement                    |
       | PublicParam | Type of public parameter used by the Client |
       |             | during the sharding phase (Section 4.2)     |
       | VerifyParam | Type of verification parameter used by each |
       |             | Aggregator during the preparation phase     |
       |             | (Section 4.3)                               |
       | AggParam    | Type of aggregation parameter               |
       | Prep        | State of each Aggregator during the         |
       |             | preparation phase (Section 4.3)             |
       | OutShare    | Type of each output share                   |
       | AggShare    | Type of each aggregate share                |
       | AggResult   | Type of the aggregate result                |

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

4.1.  Setup

   Before execution of the VDAF can begin, it is necessary to distribute
   long-lived parameters to the Client and Aggregators.  The long-lived
   parameters are generated by the following algorithm:

   *  Vdaf.setup() -> (PublicParam, Vec[VerifyParam]) is the randomized
      setup algorithm used to generate the public parameter used by the
      Clients and the verification parameters used by the Aggregators.
      The length of the latter MUST be equal to SHARES.  In general, an
      Aggregator's verification parameter is considered secret and MUST
      NOT be revealed to the Clients, Collector or other Aggregators.
      The parameters MAY be reused across multiple VDAF evaluations.
      See Section 8 for a discussion of the security implications this
      has depending on the threat model.

Barnes, et al.           Expires 29 October 2022               [Page 10]
Internet-Draft                    VDAF                        April 2022

4.2.  Sharding

   In order to protect the privacy of its measurements, a VDAF Client
   splits its measurements into "input shares".  The
   measurement_to_input_shares method is used for this purpose.

   *  Vdaf.measurement_to_input_shares(public_param: PublicParam, input:
      Measurement) -> Vec[Bytes] is the randomized input-distribution
      algorithm run by each Client.  It consumes the public parameter
      and input measurement and produces a sequence of input shares, one
      for each Aggregator.  The length of the output MUST be SHARES.


       | measurement_to_input_shares                  |
         |              |              ...  |
         V              V                   V
        input_share_0  input_share_1       input_share_[SHARES-1]
         |              |              ...  |
         V              V                   V
       Aggregator 0   Aggregator 1        Aggregator SHARES-1

       Figure 2: The Client divides its measurement input into input
              shares and distributes them to the Aggregators.

      CP The public_param is intended to allow for protocols that
      require the Client to use a public key for sharding its
      measurement.  When rotating the verify_param for such a scheme, it
      would be necessary to also update the public_param with which the
      clients are configured.  For PPM it would be nice if we could
      rotate the verify_param without also having to update the clients.
      We should consider dropping this at some point.  See

4.3.  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".  (Each of the outbound messages are called "preparation-
   message shares".)  Finally, the preparation message is distributed to

Barnes, et al.           Expires 29 October 2022               [Page 11]
Internet-Draft                    VDAF                        April 2022

   the Aggregators to begin the next round.

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

       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_next |  | prep_next |       | prep_next |   |
       +-----------+  +-----------+       +-----------+   |
         |              |              ...  |             |
         V              V                   V             | x ROUNDS
       +----------------------------------------------+   |
       | prep_shares_to_prep                          |   |
       +----------------------------------------------+   |
                        |                                 |
         +--------------+-------------------+             |
         |              |              ...  |             |
         V              V                   V             /
        ...            ...                 ...
         |              |                   |
         V              V                   V
       +-----------+  +-----------+       +-----------+
       | prep_next |  | prep_next |       | prep_next |
       +-----------+  +-----------+       +-----------+
         |              |              ...  |
         V              V                   V
       out_share_0    out_share_1         out_share_[SHARES-1]

    Figure 3: 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.

Barnes, et al.           Expires 29 October 2022               [Page 12]
Internet-Draft                    VDAF                        April 2022

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

   *  Vdaf.prep_init(verify_param: VerifyParam, agg_param: AggParam,
      nonce: Bytes, input_share: Bytes) -> Prep is the deterministic
      preparation-state initialization algorithm run by each Aggregator
      to begin processing its input share into an output share.  Its
      inputs are its verification parameter (verify_param), the
      aggregation parameter (agg_param), the nonce provided by the
      environment (nonce, see Figure 6), and one of the input shares
      generated by the client (input_share).  Its output is the
      Aggregator's initial preparation state.

   *  Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) ->
      Union[Tuple[Prep, Bytes], OutShare] is the deterministic
      preparation-state update algorithm run by each Aggregator.  It
      updates the Aggregator's preparation state (prep) 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 or, if this is the first round, None.

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

   In effect, each Aggregator moves through a linear state machine with
   ROUNDS+1 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.

   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

Barnes, et al.           Expires 29 October 2022               [Page 13]
Internet-Draft                    VDAF                        April 2022

   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 Depending on what we do for issue#20, we may end up
      needing to revise the above paragraph.

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.  This algorithm is performed
   locally at each Aggregator, without communication with the other

   *  Vdaf.out_shares_to_agg_share(agg_param: AggParam, output_shares:
      Vec[OutShare]) -> agg_share: AggShare is the deterministic
      aggregation algorithm.  It is run by each Aggregator over the
      output shares it has computed over a batch of measurement inputs.

       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
       +-----------+   +-----------+       +-----------+
       | out2agg   |   | out2agg   |   ... | out2agg   |
       +-----------+   +-----------+       +-----------+
         |               |                   |
         V               V                   V
       agg_share_0     agg_share_1         agg_share_[SHARES-1]

    Figure 4: Aggregation of output shares. `B` indicates the number of
                         measurements in the batch.

Barnes, et al.           Expires 29 October 2022               [Page 14]
Internet-Draft                    VDAF                        April 2022

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

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:

   *  Vdaf.agg_shares_to_result(agg_param: AggParam, agg_shares:
      Vec[AggShare]) -> 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.  This algorithm is

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

       agg_share_0     agg_share_1         agg_share_[SHARES-1]
         |               |                   |
         V               V                   V
       | agg_shares_to_result                          |


          Figure 5: Computation of the final aggregate result from
                             aggregate shares.

      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
      [I-D.draft-gpew-priv-ppm].)  Or is this out-of-scope of this

4.6.  Execution of a VDAF

   Executing a VDAF involves the concurrent evaluation of the VDAF on
   individual inputs and aggregation of the recovered output shares.
   This is captured by the following example algorithm:

Barnes, et al.           Expires 29 October 2022               [Page 15]
Internet-Draft                    VDAF                        April 2022

   def run_vdaf(Vdaf,
                agg_param: Vdaf.AggParam,
                nonces: Vec[Bytes],
                measurements: Vec[Vdaf.Measurement]):
       # Distribute long-lived parameters.
       (public_param, verify_params) = Vdaf.setup()

       out_shares = []
       for (nonce, measurement) in zip(nonces, measurements):

           # Each Client shards its input into shares.
           input_shares = Vdaf.measurement_to_input_shares(public_param,

           # Each Aggregator initializes its preparation state.
           prep_states = []
           for j in range(Vdaf.SHARES):
               state = Vdaf.prep_init(verify_params[j],

           # Aggregators recover their output shares.
           inbound = None
           for i in range(Vdaf.ROUNDS+1):
               outbound = []
               for j in range(Vdaf.SHARES):
                   out = Vdaf.prep_next(prep_states[j], inbound)
                   if i < Vdaf.ROUNDS:
                       (prep_states[j], out) = out
               # This is where we would send messages over the
               # network in a distributed VDAF computation.
               if i < Vdaf.ROUNDS:
                   inbound = Vdaf.prep_shares_to_prep(agg_param,

           # The final outputs of prepare phasre are the output shares.

       # Each Aggregator aggregates its output shares into an
       # aggregate share.
       agg_shares = []
       for j in range(Vdaf.SHARES):
           out_shares_j = [out[j] for out in out_shares]
           agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,

Barnes, et al.           Expires 29 October 2022               [Page 16]
Internet-Draft                    VDAF                        April 2022


       # Collector unshards the aggregate.
       agg_result = Vdaf.agg_shares_to_result(agg_param, agg_shares)
       return agg_result

                       Figure 6: Execution of a VDAF.

   The inputs to this algorithm are the aggregation parameter agg_param,
   a list of nonces nonces, and a batch of Client inputs input_batch.
   The aggregation parameter is chosen by the Aggregators prior to
   executing the VDAF.  This document does not specify how the nonces
   are chosen, but security requires that the nonces be unique for each
   measurement.  See Section 8 for details.

   Another important question this document leaves out of scope is how a
   VDAF is to be executed by Aggregators distributed over a real
   network.  Algorithm run_vdaf prescribes the protocol's execution in a
   "benign" environment in which there is no adversary and messages 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 [I-D.draft-gpew-priv-ppm] that implements
   suitable cryptographic functionalities.  Moreover, some fraction of
   the Aggregators (or Clients) may be malicious and diverge from their
   prescribed behaviors.  Section 8 describes the execution of the VDAF
   in various adversarial environments and what properties the wrapper
   protocol needs to provide in each.

5.  Preliminaries

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

5.1.  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:

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

Barnes, et al.           Expires 29 October 2022               [Page 17]
Internet-Draft                    VDAF                        April 2022

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

   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.  If integer >= Field.MODULUS, then integer is first
      reduced modulo Field.MODULUS.

   Finally, 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 += I2OSP(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 = Field(OS2IP(encoded_x))
       return vec

             Figure 7: Derived class methods for finite fields.

5.1.1.  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.

Barnes, et al.           Expires 29 October 2022               [Page 18]
Internet-Draft                    VDAF                        April 2022

   # Compute the inner product of the operands.
   def inner_product(left: Vec[Field], right: Vec[Field]) -> Field:
       return sum(map(lambda x: x[0] * x[1], zip(left, right)))

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

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

               Figure 8: Common functions for finite fields.

5.1.2.  FFT-Friendly Fields

   Some VDAFs require fields that are suitable for efficient computation
   of the discrete Fourier transform.  (One example is Prio3 (Section 6)
   when instantiated with the generic FLP of Section 6.3.3.)
   Specifically, a field is said to be "FFT-friendly" if, in addition to
   satisfying the interface described in Section 5.1, it implements the
   following method:

   *  Field.gen() -> Field returns the generator of a large subgroup of
      the multiplicative group.

   FFT-friendly fields also define the following parameter:

   *  GEN_ORDER: Unsigned is the order of a multiplicative subgroup
      generated by Field.gen().  This value MUST be a power of 2.

5.1.3.  Parameters

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

Barnes, et al.           Expires 29 October 2022               [Page 19]
Internet-Draft                    VDAF                        April 2022

                 | Parameter    | Value                 |
                 | MODULUS      | 2^32 * 4294967295 + 1 |
                 | ENCODED_SIZE | 8                     |
                 | Generator    | 7^4294967295          |
                 | GEN_ORDER    | 2^32                  |

                 Table 2: Field64, an FFT-friendly field.

             | Parameter    | Value                          |
             | MODULUS      | 2^66 * 4611686018427387897 + 1 |
             | ENCODED_SIZE | 16                             |
             | Generator    | 7^4611686018427387897          |
             | GEN_ORDER    | 2^66                           |

                 Table 3: Field128, an FFT-friendly field.

5.2.  Pseudorandom Generators

   A pseudorandom generator (PRG) is used to expand a short,
   (pseudo)random seed into a long string of pseudorandom bits.  A PRG
   suitable for this document implements the interface specified in this
   section.  Concrete constructions are described in the subsections
   that folllow.

   PRGs are defined by a class Prg with the following associated

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

   A concrete Prg implements the following class method:

   *  Prg(seed: Bytes, info: Bytes) constructs an instance of Prg from
      the given seed and info string.  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 PRG).  The info
      string is used for domain separation.

Barnes, et al.           Expires 29 October 2022               [Page 20]
Internet-Draft                    VDAF                        April 2022

   * Unsigned) returns the next length bytes of output
      of PRG.  If the seed was securely generated, the output can be
      treated as pseudorandom.

   Each Prg has two derived class methods.  The first is used to derive
   a fresh seed from an existing one.  The second is used to compute a
   sequence of pseudorandom field elements.  For each method, 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

   # Derive a new seed.
   def derive_seed(Prg, seed: Bytes, info: Bytes) -> bytes:
       prg = Prg(seed, info)

   # Expand a seed into a vector of Field elements.
   def expand_into_vec(Prg,
                       seed: Bytes,
                       info: Bytes,
                       length: Unsigned):
       m = next_power_of_2(Field.MODULUS) - 1
       prg = Prg(seed, info)
       vec = []
       while len(vec) < length:
           x = OS2IP(
           x &= m
           if x < Field.MODULUS:
       return vec

                 Figure 9: Derived class methods for PRGs.

5.2.1.  PrgAes128

      OPEN ISSUE Phillipp points out that a fixed-key mode of AES may be
      more performant (  See for details.

   Our first construction, PrgAes128, converts a blockcipher, namely
   AES-128, into a PRG.  Seed expansion involves two steps.  In the
   first step, CMAC [RFC4493] is applied to the seed and info string to
   get a fresh key.  In the second step, the fresh key is used in CTR-
   mode to produce a key stream for generating the output.  A fixed
   initialization vector (IV) is used.

Barnes, et al.           Expires 29 October 2022               [Page 21]
Internet-Draft                    VDAF                        April 2022

class PrgAes128:

    SEED_SIZE: Unsigned = 16

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

        # Use CMAC as a pseudorandom function to derive a key.
        self.key = AES128-CMAC(seed, info)

    def next(self, length):
        self.length_consumed += length

        # CTR-mode encryption of the all-zero string of the desired
        # length and using a fixed, all-zero IV.
        stream = AES128-CTR(key, zeros(16), zeros(self.length_consumed))
        return stream[-length:]

               Figure 10: Definition of PRG PrgAes128.

6.  Prio3

      NOTE This construction has not undergone significant security

   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.

   *  Input validity is determined by an arithmetic circuit evaluated
      over the encoded input.  (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 input is said
      to be "valid"; otherwise, if the output is non-zero, then the
      input is said to "invalid".

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

   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

Barnes, et al.           Expires 29 October 2022               [Page 22]
Internet-Draft                    VDAF                        April 2022

   out a multi-party computation to determine if their shares correspond
   to a valid input (as determined by the arithmetic circuit).  This
   computation involves a "proof" of validity generated by the Client.
   Next, each Aggregator sums up its input shares locally.  Finally, the
   Collector sums up the aggregate shares and computes the aggregate

   This VDAF does not have an aggregation parameter.  Instead, the
   output share is derived from the input share by applying a fixed map.
   See Section 7 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 inputs.  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 6.1.  The generic transformation of
   an FLP into Prio3 is specified in Section 6.2.  Next, a concrete FLP
   suitable for any validity circuit is specified in Section 6.3.
   Finally, instantiations of Prio3 for various types of measurements
   are specified in Section 6.4.  Test vectors can be found in
   Appendix "Test Vectors".

6.1.  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 input's validity and distributes shares of the proof to the
   Aggregators.  Each Aggregator then performs some a computation on its
   input share and proof share locally and sends the result to the other
   Aggregators.  Combining the exchanged messages allows each Aggregator
   to decide if it holds a share of a valid input.  (See Section 6.2 for

Barnes, et al.           Expires 29 October 2022               [Page 23]
Internet-Draft                    VDAF                        April 2022

   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          |
       | INPUT_LEN      | Length of the encoded measurement        |
       |                | (Section 6.1.1)                          |
       | OUTPUT_LEN     | Length of the aggregatable output        |
       |                | (Section 6.1.1)                          |
       | PROOF_LEN      | Length of the proof                      |
       | VERIFIER_LEN   | Length of the verifier message generated |
       |                | by querying the input and proof          |
       | Measurement    | Type of the measurement                  |
       | Field          | As defined in (Section 5.1)              |

          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 6.1.1):

Barnes, et al.           Expires 29 October 2022               [Page 24]
Internet-Draft                    VDAF                        April 2022

   *  Flp.prove(input: 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 input,
      the "prover randomness" prove_rand, and the "joint randomness"
      joint_rand.  The proof randomness is used only by the prover, but
      the joint randomness is shared by both the prover and verifier.

   *  Flp.query(input: Vec[Field], proof: Vec[Field], query_rand:
      Vec[Field], joint_rand: Vec[Field]) -> Vec[Field] is the query-
      generation algorithm run by the verifier.  This is used to "query"
      the input and proof.  The result of the query (i.e., the output of
      this function) is called the "verifier message".  In addition to
      the input 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.

   *  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 input
      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 input and proof, the
   query-generation algorithm outputs a share of the verifier message.
   Furthermore, the "zero-knowledge" property of the FLP system ensures
   that the verifier message reveals nothing about the input's validity.
   Therefore, to decide if an input 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

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

def run_flp(Flp, inp: 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(inp, prove_rand, joint_rand)

    # Verifier queries the input and proof.
    verifier = Flp.query(inp, proof, query_rand, joint_rand, num_shares)

    # Verifier decides if the input is valid.
    return Flp.decide(verifier)

Barnes, et al.           Expires 29 October 2022               [Page 25]
Internet-Draft                    VDAF                        April 2022

                   Figure 11: Execution of an FLP.

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

   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.

6.1.1.  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 INPUT_LEN.

   For some FLPs, the encoded input 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 (this type is also defined in
   Section 6.4).  After consuming this vector, all that is needed is the
   integer it represents.  Thus the FLP defines an algorithm for
   truncating the input to the length of the aggregated output:

   *  Flp.truncate(input: Vec[Field]) -> Vec[Field] maps an encoded
      input to an aggregatable output.  The length of the input MUST be
      INPUT_LEN and the length of the output MUST be OUTPUT_LEN.

   We remark that, taken together, these two functionalities correspond
   roughly to the notion of "Affine-aggregatable encodings (AFEs)" from

6.2.  Construction

   This section specifies Prio3, an implementation of the Vdaf interface
   (Section 4).  It has two generic parameters: an Flp (Section 6.1) and
   a Prg (Section 5.2).  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.

Barnes, et al.           Expires 29 October 2022               [Page 26]
Internet-Draft                    VDAF                        April 2022

      | Parameter   | Value                                         |
      | ROUNDS      | 1                                             |
      | SHARES      | in [2, 255)                                   |
      | Measurement | Flp.Measurement                               |
      | PublicParam | None                                          |
      | VerifyParam | Tuple[Unsigned, Bytes]                        |
      | AggParam    | None                                          |
      | Prep        | Tuple[Vec[Flp.Field], Optional[Bytes], Bytes] |
      | OutShare    | Vec[Flp.Field]                                |
      | AggShare    | Vec[Flp.Field]                                |
      | AggResult   | Vec[Unsigned]                                 |

             Table 5: Associated parameters for the Prio3 VDAF.

6.2.1.  Setup

   The setup algorithm generates a symmetric key shared by all of the
   Aggregators.  The key is used to derive query randomness for the FLP
   query-generation algorithm run by the Aggregators during preparation.
   An Aggregator's verification parameter also includes its "ID", a
   unique integer in [0, SHARES).

   def setup(Prio3):
       k_query_init = gen_rand(Prio3.Prg.SEED_SIZE)
       verify_param = [(j, k_query_init) for j in range(Prio3.SHARES)]
       return (None, verify_param)

                 Figure 12: The setup algorithm for Prio3.

Barnes, et al.           Expires 29 October 2022               [Page 27]
Internet-Draft                    VDAF                        April 2022

6.2.2.  Sharding

   Recall from Section 6.1 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 input in a way that
   allows the Aggregators to reconstruct it from their input shares.
   (This idea is based on the Fiat-Shamir heuristic and is described in
   Section 6.2.3 of [BBCGGI19].)

   The input-distribution algorithm involves the following steps:

   1.  Encode the Client's raw measurement as an input for the FLP

   2.  Shard the input into a sequence of input shares

   3.  Derive the joint randomness from the input shares

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

   5.  Shard the proof into a sequence of proof shares

   The algorithm is specified below.  Notice that only one set input and
   proof shares (called the "leader" shares below) are vectors of field
   elements.  The other shares (called the "helper" shares) are
   represented instead by PRG seeds, which are expanded into vectors of
   field elements.

   The code refers to a pair of auxiliary functions for encoding the
   shares.  These are called encode_leader_share and encode_helper_share
   respectively and they are described in Section 6.2.6.

   def measurement_to_input_shares(Prio3, _public_param, measurement):
       dst = b"vdaf-00 prio3"
       inp = Prio3.Flp.encode(measurement)
       k_joint_rand = zeros(Prio3.Prg.SEED_SIZE)

       # Generate input shares.
       leader_input_share = inp
       k_helper_input_shares = []
       k_helper_blinds = []
       k_helper_hints = []
       for j in range(Prio3.SHARES-1):
           k_blind = gen_rand(Prio3.Prg.SEED_SIZE)
           k_share = gen_rand(Prio3.Prg.SEED_SIZE)
           helper_input_share = Prio3.Prg.expand_into_vec(

Barnes, et al.           Expires 29 October 2022               [Page 28]
Internet-Draft                    VDAF                        April 2022

               dst + byte(j+1),
           leader_input_share = vec_sub(leader_input_share,
           encoded = Prio3.Flp.Field.encode_vec(helper_input_share)
           k_hint = Prio3.Prg.derive_seed(k_blind,
                                          byte(j+1) + encoded)
           k_joint_rand = xor(k_joint_rand, k_hint)
       k_leader_blind = gen_rand(Prio3.Prg.SEED_SIZE)
       encoded = Prio3.Flp.Field.encode_vec(leader_input_share)
       k_leader_hint = Prio3.Prg.derive_seed(k_leader_blind,
                                             byte(0) + encoded)
       k_joint_rand = xor(k_joint_rand, k_leader_hint)

       # Finish joint randomness hints.
       for j in range(Prio3.SHARES-1):
           k_helper_hints[j] = xor(k_helper_hints[j], k_joint_rand)
       k_leader_hint = xor(k_leader_hint, k_joint_rand)

       # Generate the proof shares.
       prove_rand = Prio3.Prg.expand_into_vec(
       joint_rand = Prio3.Prg.expand_into_vec(
       proof = Prio3.Flp.prove(inp, prove_rand, joint_rand)
       leader_proof_share = proof
       k_helper_proof_shares = []
       for j in range(Prio3.SHARES-1):
           k_share = gen_rand(Prio3.Prg.SEED_SIZE)
           helper_proof_share = Prio3.Prg.expand_into_vec(
               dst + byte(j+1),

Barnes, et al.           Expires 29 October 2022               [Page 29]
Internet-Draft                    VDAF                        April 2022

           leader_proof_share = vec_sub(leader_proof_share,

       input_shares = []
       for j in range(Prio3.SHARES-1):
       return input_shares

             Figure 13: Input-distribution algorithm for Prio3.

6.2.3.  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 input and proof share locally, then exchanges its
   verifier share with the other Aggregators.  The verifier shares are
   then combined into the verifier message, which is used to decide
   whether to accept.

   In addition, the Aggregators must ensure that they have all used the
   same joint randomness for the query-generation algorithm.  The joint
   randomness is generated by a PRG seed.  Each Aggregator derives an
   XOR secret share of this seed from its input share and the "blind"
   generated by the client.  Thus, before running the query-generation
   algorithm, it must first gather the XOR secret shares derived by the
   other Aggregators.

   In order to avoid extra round of communication, the Client sends each
   Aggregator a "hint" equal to the XOR of the other Aggregators' shares
   of the joint randomness seed.  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 XOR shares of the PRG
   seed along with their verifier shares.

Barnes, et al.           Expires 29 October 2022               [Page 30]
Internet-Draft                    VDAF                        April 2022

      NOTE This optimization somewhat diverges from Section 6.2.3 of
      [BBCGGI19].  Security analysis is needed.

   The algorithms required for preparation are defined as follows.
   These algorithms make use of encoding and decoding methods defined in
   Section 6.2.6.

   def prep_init(Prio3, verify_param, _agg_param, nonce, input_share):
       dst = b"vdaf-00 prio3"
       (j, k_query_init) = verify_param

       (input_share, proof_share, k_blind, k_hint) = \
           Prio3.decode_leader_share(input_share) if j == 0 else \
           Prio3.decode_helper_share(dst, j, input_share)

       out_share = Prio3.Flp.truncate(input_share)

       k_query_rand = Prio3.Prg.derive_seed(k_query_init,
                                            byte(255) + nonce)
       query_rand = Prio3.Prg.expand_into_vec(
       joint_rand, k_joint_rand, k_joint_rand_share = [], None, None
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           encoded = Prio3.Flp.Field.encode_vec(input_share)
           k_joint_rand_share = Prio3.Prg.derive_seed(k_blind,
                                                      byte(j) + encoded)
           k_joint_rand = xor(k_hint, k_joint_rand_share)
           joint_rand = Prio3.Prg.expand_into_vec(
       verifier_share = Prio3.Flp.query(input_share,

       prep_msg = Prio3.encode_prepare_message(verifier_share,
       return (out_share, k_joint_rand, prep_msg)

   def prep_next(Prio3, prep, inbound):

Barnes, et al.           Expires 29 October 2022               [Page 31]
Internet-Draft                    VDAF                        April 2022

       (out_share, k_joint_rand, prep_msg) = prep

       if inbound is None:
           return (prep, prep_msg)

       (verifier, k_joint_rand_check) = \

       if k_joint_rand_check != k_joint_rand or \
               not Prio3.Flp.decide(verifier):
           raise ERR_VERIFY

       return out_share

   def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
       verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN)
       k_joint_rand_check = zeros(Prio3.Prg.SEED_SIZE)
       for encoded in prep_shares:
           (verifier_share, k_joint_rand_share) = \

           verifier = vec_add(verifier, verifier_share)

           if Prio3.Flp.JOINT_RAND_LEN > 0:
               k_joint_rand_check = xor(k_joint_rand_check,

       return Prio3.encode_prepare_message(verifier,

                  Figure 14: Preparation state for Prio3.

6.2.4.  Aggregation

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

   def out_shares_to_agg_share(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 15: Aggregation algorithm for Prio3.

Barnes, et al.           Expires 29 October 2022               [Page 32]
Internet-Draft                    VDAF                        April 2022

6.2.5.  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.

   def agg_shares_to_result(Prio3, _agg_param, agg_shares):
       agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
       for agg_share in agg_shares:
           agg = vec_add(agg, agg_share)
       return list(map(lambda x: x.as_unsigned(), agg))

         Figure 16: Computation of the aggregate result for Prio3.

6.2.6.  Auxiliary Functions

   def encode_leader_share(Prio3,
       encoded = Bytes()
       encoded += Prio3.Flp.Field.encode_vec(input_share)
       encoded += Prio3.Flp.Field.encode_vec(proof_share)
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           encoded += k_blind
           encoded += k_hint
       return encoded

   def decode_leader_share(Prio3, encoded):
       l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN
       encoded_input_share, encoded = encoded[:l], encoded[l:]
       input_share = Prio3.Flp.Field.decode_vec(encoded_input_share)
       l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN
       encoded_proof_share, encoded = encoded[:l], encoded[l:]
       proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share)
       l = Prio3.Prg.SEED_SIZE
       k_blind, k_hint = None, None
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           k_blind, encoded = encoded[:l], encoded[l:]
           k_hint, encoded = encoded[:l], encoded[l:]
       if len(encoded) != 0:
           raise ERR_DECODE
       return (input_share, proof_share, k_blind, k_hint)

   def encode_helper_share(Prio3,

Barnes, et al.           Expires 29 October 2022               [Page 33]
Internet-Draft                    VDAF                        April 2022

       encoded = Bytes()
       encoded += k_input_share
       encoded += k_proof_share
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           encoded += k_blind
           encoded += k_hint
       return encoded

   def decode_helper_share(Prio3, dst, j, encoded):
       l = Prio3.Prg.SEED_SIZE
       k_input_share, encoded = encoded[:l], encoded[l:]
       input_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
                                               dst + byte(j),
       k_proof_share, encoded = encoded[:l], encoded[l:]
       proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
                                               dst + byte(j),
       k_blind, k_hint = None, None
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           k_blind, encoded = encoded[:l], encoded[l:]
           k_hint, encoded = encoded[:l], encoded[l:]
       if len(encoded) != 0:
           raise ERR_DECODE
       return (input_share, proof_share, k_blind, k_hint)

   def encode_prepare_message(Prio3, verifier, k_joint_rand):
       encoded = Bytes()
       encoded += Prio3.Flp.Field.encode_vec(verifier)
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           encoded += k_joint_rand
       return encoded

   def decode_prepare_message(Prio3, encoded):
       l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.VERIFIER_LEN
       encoded_verifier, encoded = encoded[:l], encoded[l:]
       verifier = Prio3.Flp.Field.decode_vec(encoded_verifier)
       k_joint_rand = None
       if Prio3.Flp.JOINT_RAND_LEN > 0:
           l = Prio3.Prg.SEED_SIZE
           k_joint_rand, encoded = encoded[:l], encoded[l:]
       if len(encoded) != 0:
           raise ERR_DECODE
       return (verifier, k_joint_rand)

Barnes, et al.           Expires 29 October 2022               [Page 34]
Internet-Draft                    VDAF                        April 2022

              Figure 17: Helper functions required for Prio3.

6.3.  A General-Purpose FLP

   This section describes an FLP based on the construction from in
   [BBCGGI19], Section 4.2.  We begin in Section 6.3.1 with an overview
   of their proof system and the extensions to their proof system made
   here.  The construction is specified in Section 6.3.3.

      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

      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.

6.3.1.  Overview

   In the proof system of [BBCGGI19], validity is defined via an
   arithmetic circuit evaluated over the input: If the circuit output is
   zero, then the input is deemed valid; otherwise, if the circuit
   output is non-zero, then the input is deemed invalid.  Thus the goal
   of the proof system is merely to allow the verifier to evaluate the
   validity circuit over the input.  For our application (Section 6),
   this computation is distributed among multiple Aggregators, each of
   which has only a share of the input.

   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 an
   input 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

Barnes, et al.           Expires 29 October 2022               [Page 35]
Internet-Draft                    VDAF                        April 2022

   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 input 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

   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]

Barnes, et al.           Expires 29 October 2022               [Page 36]
Internet-Draft                    VDAF                        April 2022

   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 input 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; 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)

   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
   input 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 6.3.3.  Extensions

   The FLP described in the next section extends the proof system
   [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 input shares in Prio3; see Section 6.2).  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(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1])

Barnes, et al.           Expires 29 October 2022               [Page 37]
Internet-Draft                    VDAF                        April 2022

   (Note that this is a special case of [BBCGGI19], Theorem 5.2.)  Here
   inp 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 inp is valid
   (i.e., each inp[j] is in [0,2)), then the circuit will evaluate to 0
   regardless of the value of r; but if inp[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.)  For example, the following
   circuit is allowed, where Mul and Range2 are the gadgets defined
   above (the input has length N+1):

   C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1]) + \
               2^0 * inp[0]       + ... + 2^(N-1) * inp[N-1]     - \
               Mul(inp[N], inp[N])

   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 5.1.2.

6.3.2.  Validity Circuits

   The FLP described in Section 6.3.3 is defined in terms of a validity
   circuit Valid that implements the interface described here.

   A concrete Valid defines the following parameters:

Barnes, et al.           Expires 29 October 2022               [Page 38]
Internet-Draft                    VDAF                        April 2022

        | Parameter      | Description                           |
        | GADGETS        | A list of gadgets                     |
        | GADGET_CALLS   | Number of times each gadget is called |
        | INPUT_LEN      | Length of the input                   |
        | OUTPUT_LEN     | Length of the aggregatable output     |
        | JOINT_RAND_LEN | Length of the random input            |
        | Measurement    | The type of measurement               |
        | Field          | An FFT-friendly finite field as       |
        |                | defined in Section 5.1.2              |

                  Table 6: 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 6.3.1 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).

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

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

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

   Finally, the following class methods are derived for each concrete

Barnes, et al.           Expires 29 October 2022               [Page 39]
Internet-Draft                    VDAF                        April 2022

   # Length of the prover randomness.
   def prove_rand_len(Valid):
       return sum(map(lambda g: g.ARITY, Valid.GADGETS))

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

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

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

             Figure 18: Derived methods for validity circuits.

6.3.3.  Construction

   This section specifies FlpGeneric, an implementation of the Flp
   interface (Section 6.1).  It has as a generic parameter a validity
   circuit Valid implementing the interface defined in Section 6.3.2.

      NOTE A reference implementation can be found in

   The FLP parameters for FlpGeneric are defined in Table 7.  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)

   *  For each i in [H]:

      -  Let G_i = Valid.GADGETS[i]

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

Barnes, et al.           Expires 29 October 2022               [Page 40]
Internet-Draft                    VDAF                        April 2022

      -  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 6.3.2) |
      | QUERY_RAND_LEN | Valid.query_rand_len() (see Section 6.3.2) |
      | JOINT_RAND_LEN | Valid.JOINT_RAND_LEN                       |
      | INPUT_LEN      | Valid.INPUT_LEN                            |
      | OUTPUT_LEN     | Valid.OUTPUT_LEN                           |
      | PROOF_LEN      | Valid.proof_len() (see Section 6.3.2)      |
      | VERIFIER_LEN   | Valid.verifier_len() (see Section 6.3.2)   |
      | Measurement    | Valid.Measurement                          |
      | Field          | Valid.Field                                |

                   Table 7: FLP Parameters of FlpGeneric.  Proof Generation

   On input inp, prove_rand, and joint_rand, the proof is computed as

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

   2.  Partition the prover randomness prove_rand into subvectors
       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.

   3.  Evaluate Valid on input of inp 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.

Barnes, et al.           Expires 29 October 2022               [Page 41]
Internet-Draft                    VDAF                        April 2022

   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].  Query Generation

   On input of inp, 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 subvectors seed_1, coeff_1, ..., seed_H,
       coeff_H defined in Section

   3.  Evaluate Valid on input of inp 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.

   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]:

Barnes, et al.           Expires 29 October 2022               [Page 42]
Internet-Draft                    VDAF                        April 2022

       *  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

       *  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].  Decision

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

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

   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.  Encoding

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

6.4.  Instantiations

   This section specifies instantiations of Prio3 for various
   measurement types.  Each uses FlpGeneric as the FLP (Section 6.3) and
   is determined by a validity circuit (Section 6.3.2) and a PRG
   (Section 5.2).  Test vectors for each can be found in Appendix "Test

      NOTE Reference implementations of each of these VDAFs can be found

Barnes, et al.           Expires 29 October 2022               [Page 43]
Internet-Draft                    VDAF                        April 2022

6.4.1.  Prio3Aes128Count

   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

   This instance uses PrgAes128 (Section 5.2.1) as its PRG.  Its
   validity circuit, denoted Count, uses Field64 (Table 2) as its finite
   field.  Its gadget, denoted Mul, is the degree-2, arity-2 gadget
   defined as

   def Mul(x, y):
       return x * y

   The validity circuit is defined as

   def Count(inp: Vec[Field64]):
       return Mul(inp[0], inp[0]) - inp[0]

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

               | Parameter      | Value                    |
               | GADGETS        | [Mul]                    |
               | GADGET_CALLS   | [1]                      |
               | INPUT_LEN      | 1                        |
               | OUTPUT_LEN     | 1                        |
               | JOINT_RAND_LEN | 0                        |
               | Measurement    | Unsigned, in range [0,2) |
               | Field          | Field64 (Table 2)        |

                  Table 8: Parameters of validity circuit

Barnes, et al.           Expires 29 October 2022               [Page 44]
Internet-Draft                    VDAF                        April 2022

6.4.2.  Prio3Aes128Sum

   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 PrgAes128 (Section 5.2.1) as its PRG.
   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(Sum, measurement: Integer):
       if 0 > measurement or measurement >= 2^Sum.INPUT_LEN:
           raise ERR_INPUT

       encoded = []
       for l in range(Sum.INPUT_LEN):
           encoded.append(Sum.Field((measurement >> l) & 1))
       return encoded

   def truncate(Sum, inp):
       decoded = Sum.Field(0)
       for (l, b) in enumerate(inp):
           w = Sum.Field(1 << l)
           decoded += w * b
       return [decoded]

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

   def Range2(x):
       return x^2 - x

   The validity circuit is defined as

   def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]):
       out = Field128(0)
       r = joint_rand[0]
       for x in inp:
           out += r * Range2(x)
           r *= joint_rand[0]
       return out

Barnes, et al.           Expires 29 October 2022               [Page 45]
Internet-Draft                    VDAF                        April 2022

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

                Table 9: Parameters of validity circuit Sum.

6.4.3.  Prio3Aes128Histogram

   This instance of Prio3 allows for estimating the distribution of the
   measurements by computing a simple histogram.  Each measurement is an
   arbitrary integer and the aggregate result counts the number of
   measurements that fall in a set of fixed buckets.

   This instance of Prio3 uses PrgAes128 (Section 5.2.1) as its PRG.
   Its validity circuit, denoted Histogram, uses Field128 (Table 3) as
   its finite field.  The measurement is encoded as a one-hot vector
   representing the bucket into which the measurement falls (let bucket
   denote a sequence of monotonically increasing integers):

   def encode(Histogram, measurement: Integer):
       boundaries = buckets + [Infinity]
       encoded = [Field128(0) for _ in range(len(boundaries))]
       for i in range(len(boundaries)):
           if measurement <= boundaries[i]:
               encoded[i] = Field128(1)
               return encoded

   def truncate(Histogram, inp: Vec[Field128]):
       return inp

   The validity circuit uses Range2 (see Section 6.4.2) as its single
   gadget.  It checks for one-hotness in two steps, as follows:

Barnes, et al.           Expires 29 October 2022               [Page 46]
Internet-Draft                    VDAF                        April 2022

   def Histogram(inp: Vec[Field128],
                 joint_rand: Vec[Field128],
                 num_shares: Unsigned):
       # Check that each bucket is one or zero.
       range_check = Field128(0)
       r = joint_rand[0]
       for x in inp:
           range_check += r * Range2(x)
           r *= joint_rand[0]

       # Check that the buckets sum to 1.
       sum_check = -Field128(1) * Field128(num_shares).inv()
       for b in inp:
           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
   input is sharded.  This is provided to the FLP by Prio3.

                  | Parameter      | Value              |
                  | GADGETS        | [Range2]           |
                  | GADGET_CALLS   | [buckets + 1]      |
                  | INPUT_LEN      | buckets + 1        |
                  | OUTPUT_LEN     | buckets + 1        |
                  | JOINT_RAND_LEN | 2                  |
                  | Measurement    | Integer            |
                  | Field          | Field128 (Table 3) |

                      Table 10: Parameters of validity
                             circuit Histogram.

Barnes, et al.           Expires 29 October 2022               [Page 47]
Internet-Draft                    VDAF                        April 2022

7.  Poplar1

      NOTE The spec for Poplar1 is still a work-in-progress.  A partial
      implementation can be found at
      libprio-rs/blob/main/src/vdaf/  The verification logic
      is nearly complete, however as of this draft the code is missing
      the IDPF.  An implementation of the IDPF can be found at

   This section specifies Poplar1, a VDAF for the following task.  Each
   Client holds a BITS-bit string 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 inputs are prefixed by each candidate prefix.

   This functionality is the core component of Poplar [BBCGGI21].  At a
   high level, the protocol works as follows.

   1.  Each Clients runs the input-distribution algorithm on its n-bit
       string and sends an input 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.

   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 as follows.  For each p in H, add add p || 0 and p || 1
       to the set.  Repeat step 3 with the new set of candidate

   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.

Barnes, et al.           Expires 29 October 2022               [Page 48]
Internet-Draft                    VDAF                        April 2022

   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 path,
   then function evaluates to to a non-zero value; otherwise it
   evaluates to zero.  This structure allows an IDPF to provide the
   functionality required for the above protocol, while at the same time
   ensuring the same degree of privacy as a DPF.

   Our VDAF 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.

7.1.  Incremental Distributed Point Functions (IDPFs)

   An IDPF is defined over a domain of size 2^BITS, where BITS is
   constant defined by the IDPF.  The Client specifies an index alpha
   and a pair of values beta, one for each "level" 1 <= l <= BITS.  The
   key generation generates two IDPF keys, one for each Aggregator.
   When evaluated at index 0 <= x < 2^l, each IDPF share returns an
   additive share of beta[l] if x is the l-bit prefix of alpha and
   shares of zero otherwise.

      CP What does it mean for x to be the l-bit prefix of alpha?  We
      need to be a bit more precise here.

      CP Why isn't the domain size actually 2^(BITS+1), i.e., the number
      of nodes in a binary tree of height BITS (excluding the root)?

   Each beta[l] is a pair of elements of a finite field.  Each level MAY
   have different field parameters.  Thus a concrete IDPF specifies
   associated types Field[1], Field[2], ..., and Field[BITS] defining,
   respectively, the field parameters at level 1, level 2, ..., and
   level BITS.

   An IDPF is comprised of the following algorithms (let type Value[l]
   denote (Field[l], Field[l]) for each level l):

   *  idpf_gen(alpha: Unsigned, beta: (Value[1], ..., Value[BITS])) ->
      key: (IDPFKey, IDPFKey) is the randomized key-generation algorithm
      run by the client.  Its inputs are the index alpha and the values
      beta.  The value of alpha MUST be in range [0, 2^BITS).

Barnes, et al.           Expires 29 October 2022               [Page 49]
Internet-Draft                    VDAF                        April 2022

   *  IDPFKey.eval(l: Unsigned, x: Unsigned) -> value: Value[l]) is
      deterministic, stateless key-evaluation algorithm run by each
      Aggregator.  It returns the value corresponding to index x.  The
      value of l MUST be in [1, BITS] and the value of x MUST be in
      range [2^(l-1), 2^l).

   A concrete IDPF specifies a single associated constant:

   *  BITS: Unsigned is the length of each Client input.

   A concrete IDPF also specifies the following associated types:

   *  Field[l] for each level 1 <= l <= BITS.  Each defines the same
      methods and associated constants as Field in Section 6.

   Note that IDPF construction of [BBCGGI21] uses one field for the
   inner nodes of the tree and a different, larger field for the leaf
   nodes.  See [BBCGGI21], Section 4.3.

   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.

7.2.  Construction

   The VDAF involves two rounds of communication (ROUNDS == 2) and is
   defined for two Aggregators (SHARES == 2).

7.2.1.  Setup

   The verification parameter is a symmetric key shared by both
   Aggregators.  This VDAF has no public parameter.

   def vdaf_setup():
     k_verify_init = gen_rand(SEED_SIZE)
     return (None, [(0, k_verify_init), (1, k_verify_init)])

                Figure 19: The setup algorithm for poplar1.

Barnes, et al.           Expires 29 October 2022               [Page 50]
Internet-Draft                    VDAF                        April 2022  Client

   The client's input is an IDPF index, denoted alpha.  The 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].  After evaluating their IDPF key shares on the set 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*a + b - k*a + c

   and sends additive shares of a, b, c, A and B to the Aggregators.
   Putting everything together, the input-distribution algorithm is
   defined as follows.  Function encode_input_share is defined in
   Section 7.2.5.

Barnes, et al.           Expires 29 October 2022               [Page 51]
Internet-Draft                    VDAF                        April 2022

   def measurement_to_input_shares(_, alpha):
     if alpha < 2**BITS: raise ERR_INVALID_INPUT

     # Prepare IDPF values.
     beta = []
     correlation_shares_0, correlation_shares_1 = [], []
     for l in range(1,BITS+1):
       (k, a, b, c) = Field[l].rand_vec(4)

       # Construct values of the form (1, k), where k
       # is a random field element.
       beta += [(1, k)]

       # Create secret shares of correlations to aid
       # the Aggregators' computation.
       A = -2*a+k
       B = a*a + b - a * k + c
       correlation_share = Field[l].rand_vec(5)
         [a, b, c, A, B] - correlation_share)

     # Generate IDPF shares.
     (key_0, key_1) = idpf_gen(alpha, beta)

     input_shares = [
       encode_input_share(key_0, correlation_shares_0),
       encode_input_share(key_1, correlation_shares_1),

     return input_shares

          Figure 20: The input-distribution algorithm for poplar1.

      TODO It would be more efficient to represent the shares of a, b,
      and c using PRG seeds as suggested in [BBCGGI21].

7.2.2.  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
   pair of vectors of field elements data_share and auth_share, The
   Aggregators use auth_share and the correlation shares provided by the
   Client to verify that their data_share vectors are additive shares of
   a one-hot vector.

Barnes, et al.           Expires 29 October 2022               [Page 52]
Internet-Draft                    VDAF                        April 2022

      CP Consider adding aggregation parameter as input to k_verify_rand

class PrepState:
  def __init__(verify_param, agg_param, nonce, input_share):
    (self.l, self.candidate_prefixes) = decode_indexes(agg_param)
     self.correlation_shares) = decode_input_share(input_share)
    (self.party_id, k_verify_init) = verify_param
    self.k_verify_rand = get_key(k_verify_init, nonce)
    self.step = "ready"

  def next(self, inbound: Optional[Bytes]):
    l = self.l
    (a_share, b_share, c_share,
     A_share, B_share) = correlation_shares[l-1]

    if self.step == "ready" and inbound == None:
      # Evaluate IDPF on candidate prefixes.
      data_share, auth_share = [], []
      for x in self.candidate_prefixes:
        value = self.idpf_key.eval(l, x)

      # Prepare first sketch verification message.
      r = Prg.expand_into_vec(Field[l], self.k_verify_rand, len(data_share))
      verifier_share_1 = [
         a_share + inner_product(data_share, r),
         b_share + inner_product(data_share, r * r),
         c_share + inner_product(auth_share, r),

      self.output_share = data_share
      self.step = "sketch round 1"
      return verifier_share_1

    elif self.step == "sketch round 1" and inbound != None:
      verifier_1 = Field[l].decode_vec(inbound)
      verifier_share_2 = [
        (verifier_1[0] * verifier_1[0] \
         - verifier_1[1] \
         - verifier_1[2]) * self.party_id \
        + A_share * verifier_1[0] \
        + B_share

      self.step = "sketch round 2"

Barnes, et al.           Expires 29 October 2022               [Page 53]
Internet-Draft                    VDAF                        April 2022

      return Field[l].encode_vec(verifier_share_2)

    elif self.step == "sketch round 2" and inbound != None:
      verifier_2 = Field[l].decode_vec(inbound)
      if verifier_2 != 0: raise ERR_INVALID
      return Field[l].encode_vec(self.output_share)

    else: raise ERR_INVALID_STATE

def prep_shares_to_prep(agg_param, inbound: Vec[Bytes]):
  if len(inbound) != 2:

  (l, _) = decode_indexes(agg_param)
  verifier = Field[l].decode_vec(inbound[0]) + \

  return Field[l].encode_vec(verifier)

              Figure 21: Preparation state for poplar1.

7.2.3.  Aggregation

   def out_shares_to_agg_share(agg_param, output_shares: Vec[Bytes]):
     (l, candidate_prefixes) = decode_indexes(agg_param)
     if len(output_shares) != len(candidate_prefixes):
       raise ERR_INVALID_INPUT

     agg_share = Field[l].zeros(len(candidate_prefixes))
     for output_share in output_shares:
       agg_share += Field[l].decode_vec(output_share)

     return Field[l].encode_vec(agg_share)

               Figure 22: Aggregation algorithm for poplar1.

7.2.4.  Unsharding

   def agg_shares_to_result(agg_param, agg_shares: Vec[Bytes]):
     (l, _) = decode_indexes(agg_param)
     if len(agg_shares) != 2:
       raise ERR_INVALID_INPUT

     agg = Field[l].decode_vec(agg_shares[0]) + \

     return Field[l].encode_vec(agg)

Barnes, et al.           Expires 29 October 2022               [Page 54]
Internet-Draft                    VDAF                        April 2022

        Figure 23: Computation of the aggregate result for poplar1.

7.2.5.  Helper Functions

      TODO Specify the following functionalities:

   *  encode_input_share is used to encode an input share, consisting of
      an IDPF key share and correlation shares.

   *  decode_input_share is used to decode an input share.

   *  decode_indexes(encoded: Bytes) -> (l: Unsigned, indexes:
      Vec[Unsigned]) decodes a sequence of indexes, i.e., candidate
      indexes for IDPF evaluation.  The value of l MUST be in range [1,
      BITS] and indexes[i] MUST be in range [2^(l-1), 2^l) for all i.
      An error is raised if encoded cannot be decoded.

8.  Security Considerations

      NOTE: This is a brief outline of the security considerations.
      This section will be filled out more as the draft matures and
      security analyses are completed.

   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.

   Note that, to achieve robustness, it is important to ensure that the
   verification parameters distributed to the Aggregators
   (verify_params, see Section 6.2.1) is never revealed to the Clients.

   It is also possible to consider a stronger form of robustness, where
   the attacker also controls a subset of Aggregators (see [BBCGGI19],
   Section 6.3).  To satisfy this stronger notion of robustness, it is
   necessary to prevent the attacker from sharing the verification
   parameters with the Clients.  It is therefore RECOMMENDED that the
   Aggregators generate verify_params only after a set of Client inputs
   has been collected for verification, and re-generate them for each
   such set of inputs.

Barnes, et al.           Expires 29 October 2022               [Page 55]
Internet-Draft                    VDAF                        April 2022

   In order to achieve robustness, the Aggregators MUST ensure that the
   nonces used to process the measurements in a batch are all unique.

   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.

   *  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 inputs for the remaining
   Clients.  Applications can guard against these risks by adding
   additional controls on measurement 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.

Barnes, et al.           Expires 29 October 2022               [Page 56]
Internet-Draft                    VDAF                        April 2022

9.  IANA Considerations

   This document makes no request of IANA.

10.  References

10.1.  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,

   [RFC4493]  Song, JH., Poovendran, R., Lee, J., and T. Iwata, "The
              AES-CMAC Algorithm", RFC 4493, DOI 10.17487/RFC4493, June
              2006, <>.

   [RFC8017]  Moriarty, K., Ed., Kaliski, B., Jonsson, J., and A. Rusch,
              "PKCS #1: RSA Cryptography Specifications Version 2.2",
              RFC 8017, DOI 10.17487/RFC8017, November 2016,

   [RFC8174]  Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
              2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
              May 2017, <>.

10.2.  Informative References

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

   [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,

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

   [CGB17]    Corrigan-Gibbs, H. and D. Boneh, "Prio: Private, Robust,
              and Scalable Computation of Aggregate Statistics", NSDI
              2017 , 2017,

Barnes, et al.           Expires 29 October 2022               [Page 57]
Internet-Draft                    VDAF                        April 2022

   [Dou02]    Douceur, J., "The Sybil Attack", IPTPS 2002 , 2002,

   [Dwo06]    Dwork, C., "Differential Privacy", ICALP 2006 , 2006,

   [ENPA]     "Exposure Notification Privacy-preserving Analytics (ENPA)
              White Paper", 2021, <https://covid19-static.cdn-

   [EPK14]    Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR:
              Randomized Aggregatable Privacy-Preserving Ordinal
              Response", CCS 2014 , 2014,

   [GI14]     Gilboa, N. and Y. Ishai, "Distributed Point Functions and
              Their Applications", EUROCRYPT 2014 , 2014,

              Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood,
              "Privacy Preserving Measurement", Work in Progress,
              Internet-Draft, draft-gpew-priv-ppm-01, 7 March 2022,

              "Origin Telemetry", 2020, <https://firefox-source-

   [Vad16]    Vadhan, S., "The Complexity of Differential Privacy",
              2016, <


   Thanks to David Cook, Henry Corrigan-Gibbs, Armando Faz-Hernandez,
   Mariana Raykova, and Christopher Wood for useful feedback on and
   contributions to the spec.

Barnes, et al.           Expires 29 October 2022               [Page 58]
Internet-Draft                    VDAF                        April 2022

Test Vectors

   Test vectors cover the generation of input shares and the conversion
   of input shares into output shares.  Vectors specify the public and
   verification parameters, the measurement, the aggregation parameter,
   the expected input shares, the prepare messages, and the expected
   output shares.

   Test vectors are encoded in JSON.  Input shares and prepare messages
   are represented as hexadecimal streams.  To make the tests
   deterministic, gen_rand() was replaced with a function that returns
   the requested number of 0x01 octets.


   For this test, the value of SHARES is 2.

Barnes, et al.           Expires 29 October 2022               [Page 59]
Internet-Draft                    VDAF                        April 2022

    "public_param": null,
    "verify_params": [
    "agg_param": null,
    "prep": [
            "measurement": 1,
            "nonce": "01010101010101010101010101010101",
            "input_shares": [
            "prep_shares": [
            "out_shares": [


   For this test:

   *  The value of SHARES is 2.

   *  The value of bits is 8.

Barnes, et al.           Expires 29 October 2022               [Page 60]
Internet-Draft                    VDAF                        April 2022

    "public_param": null,
    "verify_params": [
    "agg_param": null,
    "prep": [
            "measurement": 100,
            "nonce": "01010101010101010101010101010101",
            "input_shares": [
            "prep_shares": [
            "out_shares": [


   For this test:

   *  The value of SHARES is 2.

   *  The value of buckets is [1, 10, 100].

Barnes, et al.           Expires 29 October 2022               [Page 61]
Internet-Draft                    VDAF                        April 2022

    "public_param": null,
    "verify_params": [
    "agg_param": null,
    "prep": [
            "measurement": 50,
            "nonce": "01010101010101010101010101010101",
            "input_shares": [
            "prep_shares": [
            "out_shares": [

Authors' Addresses

Barnes, et al.           Expires 29 October 2022               [Page 62]
Internet-Draft                    VDAF                        April 2022

   Richard L. Barnes

   Christopher Patton
   Cloudflare, Inc.

   Phillipp Schoppmann

Barnes, et al.           Expires 29 October 2022               [Page 63]