Verifiable Distributed Aggregation Functions
draft-irtf-cfrg-vdaf-05
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| Authors | Richard Barnes , David Cook , Christopher Patton , Phillipp Schoppmann | ||
| Last updated | 2023-03-13 | ||
| Replaces | draft-patton-cfrg-vdaf | ||
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draft-irtf-cfrg-vdaf-05
CFRG R. L. Barnes
Internet-Draft Cisco
Intended status: Informational D. Cook
Expires: 14 September 2023 ISRG
C. Patton
Cloudflare
P. Schoppmann
Google
13 March 2023
Verifiable Distributed Aggregation Functions
draft-irtf-cfrg-vdaf-05
Abstract
This document describes Verifiable Distributed Aggregation Functions
(VDAFs), a family of multi-party protocols for computing aggregate
statistics over user measurements. These protocols are designed to
ensure that, as long as at least one aggregation server executes the
protocol honestly, individual measurements are never seen by any
server in the clear. At the same time, VDAFs allow the servers to
detect if a malicious or misconfigured client submitted an 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 (cfrg@ietf.org), which is archived at
https://mailarchive.ietf.org/arch/search/?email_list=cfrg.
Source for this draft and an issue tracker can be found at
https://github.com/cjpatton/vdaf.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at https://datatracker.ietf.org/drafts/current/.
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Internet-Drafts are draft documents valid for a maximum of six months
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This Internet-Draft will expire on 14 September 2023.
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Copyright (c) 2023 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents (https://trustee.ietf.org/
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Please review these documents carefully, as they describe your rights
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4
1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 8
2. Conventions and Definitions . . . . . . . . . . . . . . . . . 11
3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4. Definition of DAFs . . . . . . . . . . . . . . . . . . . . . 14
4.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 16
4.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 17
4.3. Validity of Aggregation Parameters . . . . . . . . . . . 17
4.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 17
4.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 18
4.6. Execution of a DAF . . . . . . . . . . . . . . . . . . . 19
5. Definition of VDAFs . . . . . . . . . . . . . . . . . . . . . 21
5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 23
5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 23
5.3. Validity of Aggregation Parameters . . . . . . . . . . . 26
5.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 26
5.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 27
5.6. Execution of a VDAF . . . . . . . . . . . . . . . . . . . 27
6. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 28
6.1. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 29
6.1.1. Auxiliary Functions . . . . . . . . . . . . . . . . . 30
6.1.2. FFT-Friendly Fields . . . . . . . . . . . . . . . . . 31
6.1.3. Parameters . . . . . . . . . . . . . . . . . . . . . 31
6.2. Pseudorandom Generators . . . . . . . . . . . . . . . . . 32
6.2.1. PrgSha3 . . . . . . . . . . . . . . . . . . . . . . . 33
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6.2.2. PrgFixedKeyAes128 . . . . . . . . . . . . . . . . . . 34
6.2.3. The Customization and Binder Strings . . . . . . . . 35
7. Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
7.1. Fully Linear Proof (FLP) Systems . . . . . . . . . . . . 37
7.1.1. Encoding the Input . . . . . . . . . . . . . . . . . 40
7.2. Construction . . . . . . . . . . . . . . . . . . . . . . 41
7.2.1. Sharding . . . . . . . . . . . . . . . . . . . . . . 43
7.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 46
7.2.3. Validity of Aggregation Parameters . . . . . . . . . 48
7.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 48
7.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 48
7.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 49
7.3. A General-Purpose FLP . . . . . . . . . . . . . . . . . . 52
7.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . 52
7.3.2. Validity Circuits . . . . . . . . . . . . . . . . . . 55
7.3.3. Construction . . . . . . . . . . . . . . . . . . . . 57
7.4. Instantiations . . . . . . . . . . . . . . . . . . . . . 60
7.4.1. Prio3Count . . . . . . . . . . . . . . . . . . . . . 60
7.4.2. Prio3Sum . . . . . . . . . . . . . . . . . . . . . . 61
7.4.3. Prio3Histogram . . . . . . . . . . . . . . . . . . . 63
8. Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
8.1. Incremental Distributed Point Functions (IDPFs) . . . . . 66
8.2. Construction . . . . . . . . . . . . . . . . . . . . . . 68
8.2.1. Client . . . . . . . . . . . . . . . . . . . . . . . 69
8.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 72
8.2.3. Validity of Aggregation Parameters . . . . . . . . . 75
8.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 75
8.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 75
8.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 75
8.3. The IDPF scheme of BBCGGI21 . . . . . . . . . . . . . . . 77
8.3.1. Key Generation . . . . . . . . . . . . . . . . . . . 78
8.3.2. Key Evaluation . . . . . . . . . . . . . . . . . . . 79
8.3.3. Auxiliary Functions . . . . . . . . . . . . . . . . . 81
8.4. Instantiation . . . . . . . . . . . . . . . . . . . . . . 83
9. Security Considerations . . . . . . . . . . . . . . . . . . . 83
9.1. Requirements for the Verification Key . . . . . . . . . . 84
9.2. Requirements for the Nonce . . . . . . . . . . . . . . . 85
9.3. Requirements for the Aggregation Parameters . . . . . . . 85
9.4. Additional Privacy Considerations for Aggregation
Parameters . . . . . . . . . . . . . . . . . . . . . . . 86
9.5. Pseudorandom Generators and random oracles . . . . . . . 86
10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 86
11. References . . . . . . . . . . . . . . . . . . . . . . . . . 87
11.1. Normative References . . . . . . . . . . . . . . . . . . 87
11.2. Informative References . . . . . . . . . . . . . . . . . 87
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 89
Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Prio3Count . . . . . . . . . . . . . . . . . . . . . . . . . . 89
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Prio3Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
Prio3Histogram . . . . . . . . . . . . . . . . . . . . . . . . 91
Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Sharding . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Preparation, Aggregation, and Unsharding . . . . . . . . . . 93
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 97
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.
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
precision.
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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 (MPC): 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 input measurements.
In addition to these MPC-style security goals, VDAFs can be composed
with various mechanisms for differential privacy, thereby providing
the added assurance that the aggregate result itself does not leak
too much information about any one measurement.
TODO(issue #94) Provide guidance for local and central DP and
point to it here.
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].
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The VDAF abstraction laid out in Section 5 represents a class of
multi-party protocols for privacy-preserving measurement proposed in
the literature. These protocols vary in their operational and
security considerations, sometimes in subtle but consequential ways.
This document therefore has two important goals:
1. Providing higher-level protocols like [DAP] with a simple,
uniform interface for accessing privacy-preserving measurement
schemes, 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 the
collector of the aggregate result);
2. Capabilities of a malicious coalition of servers attempting
to divulge information about client measurements; and
3. Conditions that are necessary to ensure that malicious
clients cannot corrupt the computation.
2. Providing cryptographers with design criteria that provide a
clear deployment roadmap for new constructions.
This document also specifies two concrete VDAF schemes, each based on
a protocol from the literature.
* The aforementioned Prio system [CGB17] allows for the privacy-
preserving computation of a variety aggregate statistics. The
basic idea underlying Prio is fairly simple:
1. Each client shards its measurement into a sequence of additive
shares and distributes the shares among the aggregation
servers.
2. Next, each server adds up its shares locally, resulting in an
additive share of the aggregate.
3. Finally, the aggregation servers send their aggregate shares
to the data collector, who combines them to obtain the
aggregate result.
The difficult part of this system is ensuring that the servers
hold shares of a valid input, e.g., the input is an integer in a
specific range. Thus Prio specifies a multi-party protocol for
accomplishing this task.
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In Section 7 we describe Prio3, a VDAF that follows the same
overall framework as the original Prio protocol, but incorporates
techniques introduced in [BBCGGI19] that result in significant
performance gains.
* More recently, Boneh et al. [BBCGGI21] described a protocol
called Poplar for solving the t-heavy-hitters problem in a
privacy-preserving manner. Here each client holds a bit-string of
length n, and the goal of the aggregation servers is to compute
the set of inputs that occur at least t times. The core primitive
used in their protocol is a specialized Distributed Point Function
(DPF) [GI14] that allows the servers to "query" their DPF shares
on any bit-string of length shorter than or equal to n. As a
result of this query, each of the servers has an additive share of
a bit indicating whether the string is a prefix of the client's
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 8 we describe a VDAF called Poplar1 that implements
this functionality.
Finally, perhaps the most complex aspect of schemes like Prio3 and
Poplar1 is the process by which the client-generated measurements are
prepared for aggregation. Because these constructions are based on
secret sharing, the servers will be required to exchange some amount
of information in order to verify the measurement is valid and can be
aggregated. Depending on the construction, this process may require
multiple round trips over the network.
There are applications in which this verification step may not be
necessary, e.g., when the client's software is run a trusted
execution environment. To support these applications, this document
also defines Distributed Aggregation Functions (DAFs) as a simpler
class of protocols that aim to provide the same privacy guarantee as
VDAFs but fall short of being verifiable.
OPEN ISSUE Decide if we should give one or two example DAFs.
There are natural variants of Prio3 and Poplar1 that might be
worth describing.
The remainder of this document is organized as follows: Section 3
gives a brief overview of DAFs and VDAFs; Section 4 defines the
syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6
defines various functionalities that are common to our constructions;
Section 7 describes the Prio3 construction; Section 8 describes the
Poplar1 construction; and Section 9 enumerates the security
considerations for VDAFs.
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1.1. Change Log
(*) Indicates a change that breaks wire compatibility with the
previous draft.
05:
* IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixed-key
mode for AES-128 based on a construction from [GKWWY20]. This
change is intended to improve performance of IDPF evaluation.
Note that the new PRG is not suitable for all applications. (*)
* Idpf: Add a binder string to the key-generation and evaluation
algorithms. This is used to plumb the nonce generated by the
Client to the PRG.
* Plumb random coins through the interface of randomized algorithms.
Specifically, add a random input to (V)DAF sharding algorithm and
IDPF key-generation algorithm and require implementations to
specify the length of the random input. Accordingly, update
Prio3, Poplar1, and IdpfPoplar to match the new interface. This
change is intended to improve coverage of test vectors.
* Use little-endian byte-order for field element encoding. (*)
* Poplar1: Move the last step of sketch evaluation from prep_next()
to prep_shares_to_prep().
04:
* Align security considerations with the security analysis of
[DPRS23].
* Vdaf: Pass the nonce to the sharding algorithm.
* Vdaf: Rather than allow the application to choose the nonce
length, have each implementation of the Vdaf interface specify the
expected nonce length. (*)
* Prg: Split "info string" into two components: the "customization
string", intended for domain separation; and the "binder string",
used to bind the output to ephemeral values, like the nonce,
associated with execution of a (V)DAF.
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* Replace PrgAes128 with PrgSha3, an implementation of the Prg
interface based on SHA-3, and use the new scheme as the default.
Accordingly, replace Prio3Aes128Count with Prio3Count,
Poplar1Aes128 with Poplar1, and so on. SHA-3 is a safer choice
for instantiating a random oracle, which is used in the analysis
of Prio3 of [DPRS23]. (*)
* Prio3, Poplar1: Ensure each invocation of the Prg uses a distinct
customization string, as suggested by [DPRS23]. This is intended
to make domain separation clearer, thereby simplifying security
analysis. (*)
* Prio3: Replace "joint randomness hints" sent in each input share
with "joint randomness parts" sent in the public share. This
reduces communication overhead when the number of shares exceeds
two. (*)
* Prio3: Bind nonce to joint randomness parts. This is intended to
address birthday attacks on robustness pointed out by [DPRS23].
(*)
* Poplar1: Use different Prg invocations for producing the
correlated randomness for inner and leaf nodes of the IDPF tree.
This is intended to simplify implementations. (*)
* Poplar1: Don't bind the candidate prefixes to the verifier
randomness. This is intended to improve performance, while not
impacting security. According to the analysis of [DPRS23], it is
necessary to restrict Poplar1 usage such that no report is
aggregated more than once at a given level of the IDPF tree;
otherwise, attacks on privacy may be possible. In light of this
restriction, there is no added benefit of binding to the prefixes
themselves. (*)
* Poplar1: During preparation, assert that all candidate prefixes
are unique and appear in order. Uniqueness is required to avoid
erroneously rejecting a valid report; the ordering constraint
ensures the uniqueness check can be performed efficiently. (*)
* Poplar1: Increase the maximum candidate prefix count in the
encoding of the aggregation parameter. (*)
* Poplar1: Bind the nonce to the correlated randomness derivation.
This is intended to provide defense-in-depth by ensuring the
Aggregators reject the report if the nonce does not match what the
Client used for sharding. (*)
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* Poplar1: Clarify that the aggregation parameter encoding is
OPTIONAL. Accordingly, update implementation considerations
around cross-aggregation state.
* IdpfPoplar: Add implementation considerations around branching on
the values of control bits.
* IdpfPoplar: When decoding the the control bits in the public
share, assert that the trailing bits of the final byte are all
zero. (*)
03:
* Define codepoints for (V)DAFs and use them for domain separation
in Prio3 and Poplar1. (*)
* Prio3: Align joint randomness computation with revised paper
[BBCGGI19]. This change mitigates an attack on robustness. (*)
* Prio3: Remove an intermediate PRG evaluation from query randomness
generation. (*)
* Add additional guidance for choosing FFT-friendly fields.
02:
* Complete the initial specification of Poplar1.
* Extend (V)DAF syntax to include a "public share" output by the
Client and distributed to all of the Aggregators. This is to
accommodate "extractable" IDPFs as required for Poplar1. (See
[BBCGGI21], Section 4.3 for details.)
* Extend (V)DAF syntax to allow the unsharding step to take into
account the number of measurements aggregated.
* Extend FLP syntax by adding a method for decoding the aggregate
result from a vector of field elements. The new method takes into
account the number of measurements.
* Prio3: Align aggregate result computation with updated FLP syntax.
* Prg: Add a method for statefully generating a vector of field
elements.
* Field: Require that field elements are fully reduced before
decoding. (*)
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* Define new field Field255.
01:
* Require that VDAFs specify serialization of aggregate shares.
* Define Distributed Aggregation Functions (DAFs).
* Prio3: Move proof verifier check from prep_next() to
prep_shares_to_prep(). (*)
* Remove public parameter and replace verification parameter with a
"verification key" and "Aggregator ID".
2. Conventions and Definitions
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in
BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
Algorithms in this document are written in Python 3. Type hints are
used to define input and output types. A fatal error in a program
(e.g., failure to parse one of the function parameters) is usually
handled by raising an exception.
A variable with type Bytes is a byte string. This document defines
several byte-string constants. When comprised of printable ASCII
characters, they are written as Python 3 byte-string literals (e.g.,
b'some constant string').
A global constant VERSION of type Unsigned is defined, which
algorithms are free to use as desired. Its value SHALL be 5.
This document describes algorithms for multi-party computations in
which the parties typically communicate over a network. Wherever a
quantity is defined that must be be transmitted from one party to
another, this document prescribes a particular encoding of that
quantity as a byte string.
OPEN ISSUE It might be better to not be prescriptive about how
quantities are encoded on the wire. See issue #58.
Some common functionalities:
* zeros(len: Unsigned) -> Bytes returns an array of zero bytes. The
length of output MUST be len.
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* gen_rand(len: Unsigned) -> Bytes returns an array of random bytes.
The length of output MUST be len.
* byte(int: Unsigned) -> Bytes returns the representation of int as
a byte string. The value of int MUST be in [0,256).
* concat(parts: Vec[Bytes]) -> Bytes returns the concatenation of
the input byte strings, i.e., parts[0] || ... || parts[len(parts)-
1].
* xor(left: Bytes, right: Bytes) -> Bytes returns the bitwise XOR of
left and right. An exception is raised if the inputs are not the
same length.
* to_be_bytes(val: Unsigned, length: Unsigned) -> Bytes converts val
to big-endian bytes; its value MUST be in range [0, 2^(8*length)).
Function from_be_bytes(encoded: Bytes) -> Unsigned computes the
inverse.
* to_le_bytes(val: Unsigned, length: Unsigned) -> Bytes converts val
to little-endian bytes; its value MUST be in range [0,
2^(8*length)). Function from_le_bytes(encoded: Bytes) -> Unsigned
computes the inverse.
* next_power_of_2(n: Unsigned) -> Unsigned returns the smallest
integer greater than or equal to n that is also a power of two.
3. Overview
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+--------------+
+---->| Aggregator 0 |----+
| +--------------+ |
| ^ |
| | |
| V |
| +--------------+ |
| +-->| Aggregator 1 |--+ |
| | +--------------+ | |
+--------+-+ | ^ | +->+-----------+
| Client |---+ | +--->| Collector |--> Aggregate
+--------+-+ +->+-----------+
| ... |
| |
| | |
| V |
| +----------------+ |
+--->| Aggregator N-1 |---+
+----------------+
Input shares Aggregate shares
Figure 1: Overall data flow of a (V)DAF
In a DAF- or VDAF-based private measurement system, we distinguish
three types of actors: Clients, Aggregators, and Collectors. The
overall flow of the measurement process is as follows:
* To submit an individual measurement, the Client shards the
measurement into "input shares" and sends one input share to each
Aggregator. We sometimes refer to this sequence of input shares
collectively as the Client's "report".
* The Aggregators convert their input shares into "output shares".
- Output shares are in one-to-one correspondence with the input
shares.
- Just as each Aggregator receives one input share of each input,
at the end of this process, each aggregator holds one output
share.
- In VDAFs, Aggregators will need to exchange information among
themselves as part of the validation process.
* Each Aggregator combines the output shares across inputs in the
batch to compute the "aggregate share" for that batch, i.e., its
share of the desired aggregate result.
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* The Aggregators submit their aggregate shares to the Collector,
who combines them to obtain the aggregate result over the batch.
Aggregators are a new class of actor relative to traditional
measurement systems where clients submit measurements to a single
server. They are critical for both the privacy properties of the
system and, in the case of VDAFs, the correctness of the measurements
obtained. The privacy properties of the system are assured by non-
collusion among Aggregators, and Aggregators are the entities that
perform validation of Client measurements. Thus clients trust
Aggregators not to collude (typically it is required that at least
one Aggregator is honest), and Collectors trust Aggregators to
correctly run the protocol.
Within the bounds of the non-collusion requirements of a given (V)DAF
instance, it is possible for the same entity to play more than one
role. For example, the Collector could also act as an Aggregator,
effectively using the other Aggregator(s) to augment a basic client-
server protocol.
In this document, we describe the computations performed by the
actors in this system. It is up to the higher-level protocol making
use of the (V)DAF to arrange for the required information to be
delivered to the proper actors in the proper sequence. In general,
we assume that all communications are confidential and mutually
authenticated, with the exception that Clients submitting
measurements may be anonymous.
4. Definition of DAFs
By way of a gentle introduction to VDAFs, this section describes a
simpler class of schemes called Distributed Aggregation Functions
(DAFs). Unlike VDAFs, DAFs do not provide verifiability of the
computation. Clients must therefore be trusted to compute their
input shares correctly. Because of this fact, the use of a DAF is
NOT RECOMMENDED for most applications. See Section 9 for additional
discussion.
A DAF scheme is used to compute a particular "aggregation function"
over a set of measurements generated by Clients. Depending on the
aggregation function, the Collector might select an "aggregation
parameter" and disseminates it to the Aggregators. The semantics of
this parameter is specific to the aggregation function, but in
general it is used to represent the set of "queries" that can be made
on the measurement set. For example, the aggregation parameter is
used to represent the candidate prefixes in Poplar1 Section 8.
Execution of a DAF has four distinct stages:
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* Sharding - Each Client generates input shares from its measurement
and distributes them among the Aggregators.
* Preparation - Each Aggregator converts each input share into an
output share compatible with the aggregation function. This
computation involves the aggregation parameter. In general, each
aggregation parameter may result in a different an output share.
* Aggregation - Each Aggregator combines a sequence of output shares
into its aggregate share and sends the aggregate share to the
Collector.
* Unsharding - The Collector combines the aggregate shares into the
aggregate result.
Sharding and Preparation are done once per measurement. Aggregation
and Unsharding are done over a batch of measurements (more precisely,
over the recovered output shares).
A concrete DAF specifies an algorithm for the computation needed in
each of these stages. The interface of each algorithm is defined in
the remainder of this section. In addition, a concrete DAF defines
the associated constants and types enumerated in the following table.
+=============+====================================+
| Parameter | Description |
+=============+====================================+
| ID | Algorithm identifier for this DAF. |
+-------------+------------------------------------+
| SHARES | Number of input shares into which |
| | each measurement is sharded |
+-------------+------------------------------------+
| RAND_SIZE | Size of the random byte string |
| | passed to sharding algorithm |
+-------------+------------------------------------+
| Measurement | Type of each measurement |
+-------------+------------------------------------+
| AggParam | Type of aggregation parameter |
+-------------+------------------------------------+
| OutShare | Type of each output share |
+-------------+------------------------------------+
| AggResult | Type of the aggregate result |
+-------------+------------------------------------+
Table 1: Constants and types defined by each
concrete DAF.
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These types define some of the inputs and outputs of DAF methods at
various stages of the computation. Observe that only the
measurements, output shares, the aggregate result, and the
aggregation parameter have an explicit type. All other values --- in
particular, the input shares and the aggregate shares --- have type
Bytes and are treated as opaque byte strings. This is because these
values must be transmitted between parties over a network.
OPEN ISSUE It might be cleaner to define a type for each value,
then have that type implement an encoding where necessary. This
way each method parameter has a meaningful type hint. See
issue#58.
Each DAF is identified by a unique, 32-bit integer ID. Identifiers
for each (V)DAF specified in this document are defined in Table 16.
4.1. Sharding
In order to protect the privacy of its measurements, a DAF Client
shards its measurements into a sequence of input shares. The
measurement_to_input_shares method is used for this purpose.
* Daf.measurement_to_input_shares(input: Measurement, rand:
Bytes[Daf.RAND_SIZE]) -> (Bytes, Vec[Bytes]) is the randomized
sharding algorithm run by each Client. (The input rand consists
of the random coins consumed by the algorithm.) It consumes the
measurement and produces a "public share", distributed to each of
the Aggregators, and a corresponding sequence of input shares, one
for each Aggregator. The length of the output vector MUST be
SHARES.
Client
======
measurement
|
V
+----------------------------------------------+
| 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
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Figure 2: The Client divides its measurement into input shares
and distributes them to the Aggregators.
4.2. Preparation
Once an Aggregator has received the public share and one of the input
shares, the next step is to prepare the input share for aggregation.
This is accomplished using the following algorithm:
* Daf.prep(agg_id: Unsigned, agg_param: AggParam, public_share:
Bytes, input_share: Bytes) -> OutShare is the deterministic
preparation algorithm. It takes as input the public share and one
of the input shares generated by a Client, the Aggregator's unique
identifier, and the aggregation parameter selected by the
Collector and returns an output share.
The protocol in which the DAF is used MUST ensure that the
Aggregator's identifier is equal to the integer in range [0, SHARES)
that matches the index of input_share in the sequence of input shares
output by the Client.
4.3. Validity of Aggregation Parameters
Concrete DAFs implementations MAY impose certain restrictions for
input shares and aggregation parameters. Protocols using a DAF MUST
ensure that for each input share and aggregation parameter agg_param,
Daf.prep is only called if Daf.is_valid(agg_param,
previous_agg_params) returns True, where previous_agg_params contains
all aggregation parameters that have previously been used with the
same input share.
DAFs MUST implement the following function:
* Daf.is_valid(agg_param: AggParam, previous_agg_params:
Vec[AggParam]) -> Bool: Checks if the agg_param is compatible with
all elements of previous_agg_params.
4.4. Aggregation
Once an Aggregator holds output shares for a batch of measurements
(where batches are defined by the application), it combines them into
a share of the desired aggregate result:
* Daf.out_shares_to_agg_share(agg_param: AggParam, out_shares:
Vec[OutShare]) -> agg_share: Bytes is the deterministic
aggregation algorithm. It is run by each Aggregator a set of
recovered output shares.
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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 3: Aggregation of output shares. `B` indicates the number of
measurements in the batch.
For simplicity, we have written this algorithm in a "one-shot" form,
where all output shares for a batch are provided at the same time.
Many DAFs may also support a "streaming" form, where shares are
processed one at a time.
Implementation note: For most natural DAFs (and VDAFs) it is not
necessary for an Aggregator to store all output shares individually
before aggregating. Typically it is possible to merge output shares
into aggregate shares as they arrive, merge these into other
aggregate shares, and so on. In particular, this is the case when
the output shares are vectors over some finite field and aggregating
them involves merely adding up the vectors element-wise. Such is the
case for Prio3 Section 7 and Poplar1 Section 8.
4.5. Unsharding
After the Aggregators have aggregated a sufficient number of output
shares, each sends its aggregate share to the Collector, who runs the
following algorithm to recover the following output:
* Daf.agg_shares_to_result(agg_param: AggParam, agg_shares:
Vec[Bytes], num_measurements: Unsigned) -> AggResult is run by the
Collector in order to compute the aggregate result from the
Aggregators' shares. The length of agg_shares MUST be SHARES.
num_measurements is the number of measurements that contributed to
each of the aggregate shares. This algorithm is deterministic.
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Aggregator 0 Aggregator 1 Aggregator SHARES-1
============ ============ ===================
agg_share_0 agg_share_1 agg_share_[SHARES-1]
| | |
V V V
+-----------------------------------------------+
| agg_shares_to_result |
+-----------------------------------------------+
|
V
agg_result
Collector
=========
Figure 4: 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 [DAP].)
Or is this out-of-scope of this document? See https://github.com/
ietf-wg-ppm/ppm-specification/issues/19.
4.6. Execution of a DAF
Securely executing a DAF involves emulating the following procedure.
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def run_daf(Daf,
agg_param: Daf.AggParam,
measurements: Vec[Daf.Measurement]):
out_shares = [ [] for j in range(Daf.SHARES) ]
for measurement in measurements:
# Each Client shards its measurement into input shares and
# distributes them among the Aggregators.
rand = gen_rand(Daf.RAND_SIZE)
(public_share, input_shares) = \
Daf.measurement_to_input_shares(measurement, rand)
# Each Aggregator prepares its input share for aggregation.
for j in range(Daf.SHARES):
out_shares[j].append(
Daf.prep(j, agg_param, public_share, input_shares[j]))
# Each Aggregator aggregates its output shares into an aggregate
# share and sends it to the Collector.
agg_shares = []
for j in range(Daf.SHARES):
agg_share_j = Daf.out_shares_to_agg_share(agg_param,
out_shares[j])
agg_shares.append(agg_share_j)
# Collector unshards the aggregate result.
num_measurements = len(measurements)
agg_result = Daf.agg_shares_to_result(agg_param, agg_shares,
num_measurements)
return agg_result
Figure 5: Execution of a DAF.
The inputs to this procedure are the same as the aggregation function
computed by the DAF: An aggregation parameter and a sequence of
measurements. The procedure prescribes how a DAF is executed in a
"benign" environment in which there is no adversary and the messages
are passed among the protocol participants over secure point-to-point
channels. In reality, these channels need to be instantiated by some
"wrapper protocol", such as [DAP], that realizes these channels using
suitable cryptographic mechanisms. Moreover, some fraction of the
Aggregators (or Clients) may be malicious and diverge from their
prescribed behaviors. Section 9 describes the execution of the DAF
in various adversarial environments and what properties the wrapper
protocol needs to provide in each.
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5. Definition of VDAFs
Like DAFs described in the previous section, a VDAF scheme is used to
compute a particular aggregation function over a set of Client-
generated measurements. Evaluation of a VDAF involves the same four
stages as for DAFs: Sharding, Preparation, Aggregation, and
Unsharding. However, the Preparation stage will require interaction
among the Aggregators in order to facilitate verifiability of the
computation's correctness. Accommodating this interaction will
require syntactic changes.
Overall execution of a VDAF comprises the following stages:
* Sharding - Computing input shares from an individual measurement
* Preparation - Conversion and verification of input shares to
output shares compatible with the aggregation function being
computed
* Aggregation - Combining a sequence of output shares into an
aggregate share
* Unsharding - Combining a sequence of aggregate shares into an
aggregate result
In contrast to DAFs, the Preparation stage for VDAFs now performs an
additional task: Verification of the validity of the recovered output
shares. This process ensures that aggregating the output shares will
not lead to a garbled aggregate result.
The remainder of this section defines the VDAF interface. The
attributes are listed in Table 2 are defined by each concrete VDAF.
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+=================+==========================================+
| Parameter | Description |
+=================+==========================================+
| ID | Algorithm identifier for this VDAF |
+-----------------+------------------------------------------+
| VERIFY_KEY_SIZE | Size (in bytes) of the verification key |
| | (Section 5.2) |
+-----------------+------------------------------------------+
| RAND_SIZE | Size of the random byte string passed to |
| | sharding algorithm |
+-----------------+------------------------------------------+
| NONCE_SIZE | Size (in bytes) of the nonce |
+-----------------+------------------------------------------+
| ROUNDS | Number of rounds of communication during |
| | the Preparation stage (Section 5.2) |
+-----------------+------------------------------------------+
| SHARES | Number of input shares into which each |
| | measurement is sharded (Section 5.1) |
+-----------------+------------------------------------------+
| Measurement | Type of each measurement |
+-----------------+------------------------------------------+
| AggParam | Type of aggregation parameter |
+-----------------+------------------------------------------+
| Prep | State of each Aggregator during |
| | Preparation (Section 5.2) |
+-----------------+------------------------------------------+
| OutShare | Type of each output share |
+-----------------+------------------------------------------+
| AggResult | Type of the aggregate result |
+-----------------+------------------------------------------+
Table 2: Constants and types defined by each concrete VDAF.
Similarly to DAFs (see {[sec-daf}}), any output of a VDAF method that
must be transmitted from one party to another is treated as an opaque
byte string. All other quantities are given a concrete type.
OPEN ISSUE It might be cleaner to define a type for each value,
then have that type implement an encoding where necessary. See
issue#58.
Each VDAF is identified by a unique, 32-bit integer ID. Identifiers
for each (V)DAF specified in this document are defined in Table 16.
The following method is defined for every VDAF:
def custom(Vdaf, usage: Unsigned) -> Bytes:
return format_custom(0, Vdaf.ID, usage)
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It is used to construct a customization string for an instance of Prg
used by the VDAF. (See Section 6.2.)
5.1. Sharding
Sharding transforms a measurement into input shares as it does in
DAFs (cf. Section 4.1); in addition, it takes a nonce as input and
produces a public share:
* Vdaf.measurement_to_input_shares(measurement: Measurement, nonce:
Bytes[Vdaf.NONCE_SIZE], rand: Bytes[Vdaf.RAND_SIZE]) -> (Bytes,
Vec[Bytes]) is the randomized sharding algorithm run by each
Client. (Input rand consists of the random coins consumed by the
algorithm.) It consumes the measurement and the nonce and
produces a public share, distributed to each of Aggregators, and
the corresponding sequence of input shares, one for each
Aggregator. Depending on the VDAF, the input shares may encode
additional information used to verify the recovered output shares
(e.g., the "proof shares" in Prio3 Section 7). The length of the
output vector MUST be SHARES.
In order to ensure privacy of the measurement, the Client MUST
generate the nonce using a cryptographically secure pseudorandom
number generator (CSPRNG). (See Section 9 for details.)
5.2. Preparation
To recover and verify output shares, the Aggregators interact with
one another over ROUNDS rounds. Prior to each round, each Aggregator
constructs an outbound message. Next, the sequence of outbound
messages is combined into a single message, called a "preparation
message". (Each of the outbound messages are called "preparation-
message shares".) Finally, the preparation message is distributed to
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
aggregation.
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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 6: VDAF preparation process on the input shares for a single
measurement. At the end of the computation, each Aggregator holds an
output share or an error.
To facilitate the preparation process, a concrete VDAF implements the
following class methods:
* Vdaf.prep_init(verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE], agg_id:
Unsigned, agg_param: AggParam, nonce: Bytes[Vdaf.NONCE_SIZE],
public_share: 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 the shared verification key (verify_key),
the Aggregator's unique identifier (agg_id), the aggregation
parameter (agg_param), the nonce provided by the environment
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(nonce, see Figure 7), the public share (public_share), and one of
the input shares generated by the client (input_share). Its
output is the Aggregator's initial preparation state.
It is up to the high level protocol in which the VDAF is used to
arrange for the distribution of the verification key prior to
generating and processing reports. (See Section 9 for details.)
Protocols using the VDAF MUST ensure that the Aggregator's
identifier is equal to the integer in range [0, SHARES) that
matches the index of input_share in the sequence of input shares
output by the Client.
Protocols MUST ensure that public share consumed by each of the
Aggregators is identical. This is security critical for VDAFs
such as Poplar1 that require an extractable distributed point
function. (See Section 8 for details.)
* 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
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an Aggregator only recovers an output share if it is deemed valid (at
least, based on the Aggregator's view of the protocol). Another way
to draw this boundary would be to have the Aggregators recover output
shares first, then verify that they are valid. However, this would
allow the possibility of misusing the API by, say, aggregating an
invalid output share. Moreover, in protocols like Prio+ [AGJOP21]
based on oblivious transfer, it is necessary for the Aggregators to
interact in order to recover aggregatable output shares at all.
Note that it is possible for a VDAF to specify ROUNDS == 0, in which
case each Aggregator runs the preparation-state update algorithm once
and immediately recovers its output share without interacting with
the other Aggregators. However, most, if not all, constructions will
require some amount of interaction in order to ensure validity of the
output shares (while also maintaining privacy).
OPEN ISSUE accommodating 0-round VDAFs may require syntax changes
if, for example, public keys are required. On the other hand, we
could consider defining this class of schemes as a different
primitive. See issue#77.
5.3. Validity of Aggregation Parameters
Similar to DAFs (see Section 4.3), VDAFs MAY impose restrictions for
input shares and aggregation parameters. Protocols using a VDAF MUST
ensure that for each input share and aggregation parameter agg_param,
the preparation phase (including Vdaf.prep_init, Vdaf.prep_next, and
Vdaf.prep_shares_to_prep; see Section 5.2) is only called if
Vdaf.is_valid(agg_param, previous_agg_params) returns True, where
previous_agg_params contains all aggregation parameters that have
previously been used with the same input share.
VDAFs MUST implement the following function:
* Vdaf.is_valid(agg_param: AggParam, previous_agg_params:
Vec[AggParam]) -> Bool: Checks if the agg_param is compatible with
all elements of previous_agg_params.
5.4. Aggregation
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):
* Vdaf.out_shares_to_agg_share(agg_param: AggParam, out_shares:
Vec[OutShare]) -> agg_share: Bytes is the deterministic
aggregation algorithm. It is run by each Aggregator over the
output shares it has computed over a batch of measurement inputs.
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The data flow for this stage is illustrated in Figure 3. Here again,
we have the aggregation algorithm in a "one-shot" form, where all
shares for a batch are provided at the same time. VDAFs typically
also support a "streaming" form, where shares are processed one at a
time.
5.5. Unsharding
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):
* Vdaf.agg_shares_to_result(agg_param: AggParam, agg_shares:
Vec[Bytes], num_measurements: Unsigned) -> AggResult is run by the
Collector in order to compute the aggregate result from the
Aggregators' shares. The length of agg_shares MUST be SHARES.
num_measurements is the number of measurements that contributed to
each of the aggregate shares. This algorithm is deterministic.
The data flow for this stage is illustrated in Figure 4.
5.6. Execution of a VDAF
Secure execution of a VDAF involves simulating the following
procedure.
def run_vdaf(Vdaf,
verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE],
agg_param: Vdaf.AggParam,
nonces: Vec[Bytes[Vdaf.NONCE_SIZE]],
measurements: Vec[Vdaf.Measurement]):
out_shares = []
for (nonce, measurement) in zip(nonces, measurements):
# Each Client shards its measurement into input shares.
rand = gen_rand(Vdaf.RAND_SIZE)
(public_share, input_shares) = \
Vdaf.measurement_to_input_shares(measurement, nonce, rand)
# Each Aggregator initializes its preparation state.
prep_states = []
for j in range(Vdaf.SHARES):
state = Vdaf.prep_init(verify_key, j,
agg_param,
nonce,
public_share,
input_shares[j])
prep_states.append(state)
# Aggregators recover their output shares.
inbound = None
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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
outbound.append(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,
outbound)
# The final outputs of prepare phase are the output shares.
out_shares.append(outbound)
# Each Aggregator aggregates its output shares into an
# aggregate share. In a distributed VDAF computation, the
# aggregate shares are sent over the network.
agg_shares = []
for j in range(Vdaf.SHARES):
out_shares_j = [out[j] for out in out_shares]
agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
out_shares_j)
agg_shares.append(agg_share_j)
# Collector unshards the aggregate.
num_measurements = len(measurements)
agg_result = Vdaf.agg_shares_to_result(agg_param, agg_shares,
num_measurements)
return agg_result
Figure 7: Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter, a list of
measurements, and a nonce for each measurement. This document does
not specify how the nonces are chosen, but security requires that the
nonces be unique. See Section 9 for details. As explained in
Section 4.6, the secure execution of a VDAF requires the application
to instantiate secure channels between each of the protocol
participants.
6. Preliminaries
This section describes the primitives that are common to the VDAFs
specified in this document.
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6.1. Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite
field elements are represented by a class Field with the following
associated parameters:
* MODULUS: Unsigned is the prime modulus that defines the field.
* ENCODED_SIZE: Unsigned is the number of bytes used to encode a
field element as a byte string.
A concrete Field also implements the following class methods:
* Field.zeros(length: Unsigned) -> output: Vec[Field] returns a
vector of zeros. The length of output MUST be length.
* Field.rand_vec(length: Unsigned) -> output: Vec[Field] returns a
vector of random field elements. The length of output MUST be
length.
A field element is an instance of a concrete Field. The concrete
class defines the usual arithmetic operations on field elements. In
addition, it defines the following instance method for converting a
field element to an unsigned integer:
* elem.as_unsigned() -> Unsigned returns the integer representation
of field element elem.
Likewise, each concrete Field implements a constructor for converting
an unsigned integer into a field element:
* Field(integer: Unsigned) returns integer represented as a field
element. The value of integer MUST be less than Field.MODULUS.
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.
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def encode_vec(Field, data: Vec[Field]) -> Bytes:
encoded = Bytes()
for x in data:
encoded += to_le_bytes(x.as_unsigned(), Field.ENCODED_SIZE)
return encoded
def decode_vec(Field, encoded: Bytes) -> Vec[Field]:
L = Field.ENCODED_SIZE
if len(encoded) % L != 0:
raise ERR_DECODE
vec = []
for i in range(0, len(encoded), L):
encoded_x = encoded[i:i+L]
x = from_le_bytes(encoded_x)
if x >= Field.MODULUS:
raise ERR_DECODE # Integer is larger than modulus
vec.append(Field(x))
return vec
Figure 8: Derived class methods for finite fields.
6.1.1. Auxiliary Functions
The following auxiliary functions on vectors of field elements are
used in the remainder of this document. Note that an exception is
raised by each function if the operands are not the same length.
# 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 9: Common functions for finite fields.
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6.1.2. FFT-Friendly Fields
Some VDAFs require fields that are suitable for efficient computation
of the discrete Fourier transform, as this allows for fast polynomial
interpolation. (One example is Prio3 (Section 7) when instantiated
with the generic FLP of Section 7.3.3.) Specifically, a field is
said to be "FFT-friendly" if, in addition to satisfying the interface
described in Section 6.1, it implements the following method:
* Field.gen() -> Field returns the generator of a large subgroup of
the multiplicative group. To be FFT-friendly, the order of this
subgroup MUST be a power of 2. In addition, the size of the
subgroup dictates how large interpolated polynomials can be. It
is RECOMMENDED that a generator is chosen with order at least
2^20.
FFT-friendly fields also define the following parameter:
* GEN_ORDER: Unsigned is the order of a multiplicative subgroup
generated by Field.gen().
6.1.3. Parameters
The tables below define finite fields used in the remainder of this
document.
+==============+================+=======================+==========+
| Parameter | Field64 | Field128 | Field255 |
+==============+================+=======================+==========+
| MODULUS | 2^32 * | 2^66 * | 2^255 - |
| | 4294967295 + 1 | 4611686018427387897 + | 19 |
| | | 1 | |
+--------------+----------------+-----------------------+----------+
| ENCODED_SIZE | 8 | 16 | 32 |
+--------------+----------------+-----------------------+----------+
| Generator | 7^4294967295 | 7^4611686018427387897 | n/a |
+--------------+----------------+-----------------------+----------+
| GEN_ORDER | 2^32 | 2^66 | n/a |
+--------------+----------------+-----------------------+----------+
Table 3: Parameters for the finite fields used in this document.
OPEN ISSUE We currently use big-endian for encoding field
elements. However, for implementations of GF(2^255-19), little
endian is more common. See issue#90.
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6.2. 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.
PRGs are defined by a class Prg with the following associated
parameter:
* SEED_SIZE: Unsigned is the size (in bytes) of a seed.
A concrete Prg implements the following class method:
* Prg(seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes)
constructs an instance of Prg from the given seed and
customization and binder strings. (See below for definitions of
these.) The seed MUST be of length SEED_SIZE and MUST be
generated securely (i.e., it is either the output of gen_rand or a
previous invocation of the PRG).
* prg.next(length: 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
PRG).
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# Derive a new seed.
def derive_seed(Prg, seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes):
prg = Prg(seed, custom, binder)
return prg.next(Prg.SEED_SIZE)
# Output the next `length` pseudorandom elements of `Field`.
def next_vec(self, Field, length: Unsigned):
m = next_power_of_2(Field.MODULUS) - 1
vec = []
while len(vec) < length:
x = from_le_bytes(self.next(Field.ENCODED_SIZE))
x &= m
if x < Field.MODULUS:
vec.append(Field(x))
return vec
# Expand the input `seed` into vector of `length` field elements.
def expand_into_vec(Prg,
Field,
seed: Bytes[Prg.SEED_SIZE],
custom: Bytes,
binder: Bytes,
length: Unsigned):
prg = Prg(seed, custom, binder)
return prg.next_vec(Field, length)
Figure 10: Derived class methods for PRGs.
6.2.1. PrgSha3
This section describes PrgSha3, a PRG based on the Keccak permutation
of SHA-3 [FIPS202]. Keccak is used in the cSHAKE128 mode of
operation [SP800-185]. This Prg is RECOMMENDED for all use cases
within VDAFs.
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class PrgSha3(Prg):
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed, custom, binder):
self.l = 0
self.x = seed + binder
self.s = custom
def next(self, length: Unsigned) -> Bytes:
self.l += length
# Function `cSHAKE128(x, l, n, s)` is as defined in
# [SP800-185, Section 3.3].
#
# Implementation note: Rather than re-generate the output
# stream each time `next()` is invoked, most implementations
# of SHA-3 will expose an "absorb-then-squeeze" API that
# allows stateful handling of the stream.
stream = cSHAKE128(self.x, self.l, b'', self.s)
return stream[-length:]
Figure 11: Definition of PRG PrgSha3.
6.2.2. PrgFixedKeyAes128
While PrgSha3 as described above can be securely used in all cases
where a Prg is needed in the VDAFs described in this document, there
are some cases where a more efficient instantiation based on fixed-
key AES is possible. For now, this is limited to the Prg used inside
the Idpf Section 8.1 implementation in Poplar1 Section 8.3. It is
NOT RECOMMENDED to use this Prg anywhere else. See Security
Considerations Section 9 for a more detailed discussion.
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class PrgFixedKeyAes128(Prg):
# Associated parameters
SEED_SIZE = 16
def __init__(self, seed, custom, binder):
self.length_consumed = 0
# Use SHA-3 to derive a key from the binder and customization
# strings. Note that the AES key does not need to be kept
# secret from any party. However, when used with IdpfPoplar,
# we require the binder to be a random nonce.
#
# Implementation note: This step can be cached across PRG
# evaluations with many different seeds.
self.fixed_key = cSHAKE128(binder, 16, b'', custom)
self.seed = seed
def next(self, length: Unsigned) -> Bytes:
offset = self.length_consumed % 16
new_length = self.length_consumed + length
block_range = range(
int(self.length_consumed / 16),
int(new_length / 16) + 1)
self.length_consumed = new_length
hashed_blocks = [
self.hash_block(xor(self.seed, to_le_bytes(i, 16))) \
for i in block_range
]
return concat(hashed_blocks)[offset:offset+length]
# The multi-instance tweakable circular correlation-robust hash function of
# [GKWWY20] (Section 4.2).
#
# Function `AES128(key, block)` is the AES-128 blockcipher.
def hash_block(self, block):
lo, hi = block[:8], block[8:]
sigma = hi + xor(hi, lo)
return xor(AES128(self.fixed_key, sigma), sigma)
6.2.3. The Customization and Binder Strings
PRGs are used to map a seed to a finite domain, e.g., a fresh seed or
a vector of field elements. To ensure domain separation, the
derivation is needs to be bound to some distinguished "customization
string". The customization string encodes the following values:
1. The document version (i.e.,VERSION);
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2. The "class" of the algorithm using the output (e.g., VDAF);
3. A unique identifier for the algorithm; and
4. Some indication of how the output is used (e.g., for deriving the
measurement shares in Prio3 Section 7).
The following algorithm is used in the remainder of this document in
order to format the customization string:
def format_custom(algo_class: Unsigned,
algo: Unsigned,
usage: Unsigned) -> Bytes:
return to_be_bytes(VERSION, 1) + \
to_be_bytes(algo_class, 1) + \
to_be_bytes(algo, 4) + \
to_be_bytes(usage, 2)
It is also sometimes necessary to bind the output to some ephemeral
value that multiple parties need to agree on. We call this input the
"binder string".
7. Prio3
This section describes Prio3, a VDAF for Prio [CGB17]. Prio is
suitable for a wide variety of aggregation functions, including (but
not limited to) sum, mean, standard deviation, estimation of
quantiles (e.g., median), and linear regression. In fact, the scheme
described in this section is compatible with any aggregation function
that has the following structure:
* Each measurement is encoded as a vector over some finite field.
* 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 be "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
out a multi-party computation to determine if their shares correspond
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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
result.
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 8 for an example of a VDAF that makes meaningful use of
the aggregation parameter.
As the name implies, Prio3 is a descendant of the original Prio
construction. A second iteration was deployed in the [ENPA] system,
and like the VDAF described here, the ENPA system was built from
techniques introduced in [BBCGGI19] that significantly improve
communication cost. That system was specialized for a particular
aggregation function; the goal of Prio3 is to provide the same level
of generality as the original construction.
The core component of Prio3 is a "Fully Linear Proof (FLP)" system.
Introduced by [BBCGGI19], the FLP encapsulates the functionality
required for encoding and validating 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 7.1. The generic transformation of
an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP
suitable for any validity circuit is specified in Section 7.3.
Finally, instantiations of Prio3 for various types of measurements
are specified in Section 7.4. Test vectors can be found in
Appendix "Test Vectors".
7.1. Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a two-party protocol executed by a prover and
a verifier. In actual use, however, the prover's computation is
carried out by the Client, and the verifier's computation is
distributed among the Aggregators. The Client generates a "proof" of
its 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 7.2 for
details.)
As usual, we will describe the interface implemented by a concrete
FLP in terms of an abstract base class Flp that specifies the set of
methods and parameters a concrete FLP must provide.
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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 7.1.1) |
+----------------+------------------------------------------+
| OUTPUT_LEN | Length of the aggregatable output |
| | (Section 7.1.1) |
+----------------+------------------------------------------+
| PROOF_LEN | Length of the proof |
+----------------+------------------------------------------+
| VERIFIER_LEN | Length of the verifier message generated |
| | by querying the input and proof |
+----------------+------------------------------------------+
| Measurement | Type of the measurement |
+----------------+------------------------------------------+
| AggResult | Type of the aggregate result |
+----------------+------------------------------------------+
| Field | As defined in (Section 6.1) |
+----------------+------------------------------------------+
Table 4: Constants and types defined by a concrete FLP.
An FLP specifies the following algorithms for generating and
verifying proofs of validity (encoding is described below in
Section 7.1.1):
* 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 prover randomness is used only by the prover, but
the joint randomness is shared by both the prover and verifier.
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* Flp.query(input: Vec[Field], proof: Vec[Field], query_rand:
Vec[Field], joint_rand: Vec[Field], num_shares: Unsigned) ->
Vec[Field] is the query-generation algorithm run by the verifier.
This is used to "query" the 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.
num_shares specifies how many input and proof shares were
generated.
* Flp.decide(verifier: Vec[Field]) -> Bool is the deterministic
decision algorithm run by the verifier. It takes as input the
verifier message and outputs a boolean indicating if the 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 "strong 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 algorithm.
The query-generation algorithm includes a parameter num_shares that
specifies the number of shares of the input and proof that were
generated. If these data are not secret shared, then num_shares ==
1. This parameter is useful for certain FLP constructions. For
example, the FLP in Section 7.3 is defined in terms of an arithmetic
circuit; when the circuit contains constants, it is sometimes
necessary to normalize those constants to ensure that the circuit's
output, when run on a valid input, is the same regardless of the
number of shares.
An FLP is executed by the prover and verifier as follows:
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def run_flp(Flp, 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)
Figure 12: Execution of an FLP.
The proof system is constructed so that, if input is a valid input,
then run_flp(Flp, input, 1) 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
probability.
We remark that [BBCGGI19] defines a much larger class of fully linear
proof systems than we consider here. In particular, what is called
an "FLP" here is called a 1.5-round, public-coin, interactive oracle
proof system in their paper.
7.1.1. Encoding the Input
The type of measurement being aggregated is defined by the FLP.
Hence, the FLP also specifies a method of encoding raw measurements
as a vector of field elements:
* Flp.encode(measurement: Measurement) -> Vec[Field] encodes a raw
measurement as a vector of field elements. The return value MUST
be of length 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 7.4.2). After consuming this vector, all that is needed is
the integer it represents. Thus the FLP defines an algorithm for
truncating the input to the length of the aggregated output:
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* 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.
Once the aggregate shares have been computed and combined together,
their sum can be converted into the aggregate result. This could be
a projection from the FLP's field to the integers, or it could
include additional post-processing.
* Flp.decode(output: Vec[Field], num_measurements: Unsigned) ->
AggResult maps a sum of aggregate shares to an aggregate result.
The length of the input MUST be OUTPUT_LEN. num_measurements is
the number of measurements that contributed to the aggregated
output.
We remark that, taken together, these three functionalities
correspond roughly to the notion of "Affine-aggregatable encodings
(AFEs)" from [CGB17].
7.2. Construction
This section specifies Prio3, an implementation of the Vdaf interface
(Section 5). It has two generic parameters: an Flp (Section 7.1) and
a Prg (Section 6.2). 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. These methods refer to constants
enumerated in Table 6.
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+=================+===============================================+
| Parameter | Value |
+=================+===============================================+
| VERIFY_KEY_SIZE | Prg.SEED_SIZE |
+-----------------+-----------------------------------------------+
| NONCE_SIZE | 16 |
+-----------------+-----------------------------------------------+
| ROUNDS | 1 |
+-----------------+-----------------------------------------------+
| SHARES | in [2, 256) |
+-----------------+-----------------------------------------------+
| Measurement | Flp.Measurement |
+-----------------+-----------------------------------------------+
| AggParam | None |
+-----------------+-----------------------------------------------+
| Prep | Tuple[Vec[Flp.Field], Optional[Bytes], Bytes] |
+-----------------+-----------------------------------------------+
| OutShare | Vec[Flp.Field] |
+-----------------+-----------------------------------------------+
| AggResult | Flp.AggResult |
+-----------------+-----------------------------------------------+
Table 5: VDAF parameters for Prio3.
+=================================+=======+
| Variable | Value |
+=================================+=======+
| DST_MEASUREMENT_SHARE: Unsigned | 1 |
+---------------------------------+-------+
| DST_PROOF_SHARE: Unsigned | 2 |
+---------------------------------+-------+
| DST_JOINT_RANDOMNESS: Unsigned | 3 |
+---------------------------------+-------+
| DST_PROVE_RANDOMNESS: Unsigned | 4 |
+---------------------------------+-------+
| DST_QUERY_RANDOMNESS: Unsigned | 5 |
+---------------------------------+-------+
| DST_JOINT_RAND_SEED: Unsigned | 6 |
+---------------------------------+-------+
| DST_JOINT_RAND_PART: Unsigned | 7 |
+---------------------------------+-------+
Table 6: Constants used by Prio3.
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7.2.1. Sharding
Recall from Section 7.1 that the FLP syntax calls for "joint
randomness" shared by the prover (i.e., the Client) and the verifier
(i.e., the Aggregators). VDAFs have no such notion. Instead, the
Client derives the joint randomness from its 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 sharding 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 and nonce
4. Run the FLP proof-generation algorithm using the derived joint
randomness
5. Shard the proof into a sequence of proof shares
The algorithm is specified below. Notice that only one set of 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 definitions of constants and a few auxiliary functions are
defined in Section 7.2.6.
def measurement_to_input_shares(Prio3, measurement, nonce, rand):
l = Prio3.Prg.SEED_SIZE
use_joint_rand = Prio3.Flp.JOINT_RAND_LEN > 0
# Split the coins into the various seeds we'll need.
if len(rand) != Prio3.RAND_SIZE:
raise ERR_INPUT # unexpected length for random coins
seeds = [rand[i:i+l] for i in range(0,Prio3.RAND_SIZE,l)]
if use_joint_rand:
k_helper_seeds, seeds = front((Prio3.SHARES-1) * 3, seeds)
k_helper_meas_shares = [
k_helper_seeds[i]
for i in range(0, (Prio3.SHARES-1) * 3, 3)
]
k_helper_proof_shares = [
k_helper_seeds[i]
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for i in range(1, (Prio3.SHARES-1) * 3, 3)
]
k_helper_blinds = [
k_helper_seeds[i]
for i in range(2, (Prio3.SHARES-1) * 3, 3)
]
(k_leader_blind,), seeds = front(1, seeds)
else:
k_helper_seeds, seeds = front((Prio3.SHARES-1) * 2, seeds)
k_helper_meas_shares = [
k_helper_seeds[i]
for i in range(0, (Prio3.SHARES-1) * 2, 2)
]
k_helper_proof_shares = [
k_helper_seeds[i]
for i in range(1, (Prio3.SHARES-1) * 2, 2)
]
k_helper_blinds = [None] * (Prio3.SHARES-1)
k_leader_blind = None
(k_prove,), seeds = front(1, seeds)
# Finish measurement shares and joint randomness parts.
inp = Prio3.Flp.encode(measurement)
leader_meas_share = inp
k_joint_rand_parts = []
for j in range(Prio3.SHARES-1):
helper_meas_share = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_helper_meas_shares[j],
Prio3.custom(DST_MEASUREMENT_SHARE),
byte(j+1),
Prio3.Flp.INPUT_LEN
)
leader_meas_share = vec_sub(leader_meas_share,
helper_meas_share)
if use_joint_rand:
encoded = Prio3.Flp.Field.encode_vec(helper_meas_share)
k_joint_rand_part = Prio3.Prg.derive_seed(
k_helper_blinds[j],
Prio3.custom(DST_JOINT_RAND_PART),
byte(j+1) + nonce + encoded,
)
k_joint_rand_parts.append(k_joint_rand_part)
# Finish joint randomness.
if use_joint_rand:
encoded = Prio3.Flp.Field.encode_vec(leader_meas_share)
k_joint_rand_part = Prio3.Prg.derive_seed(
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k_leader_blind,
Prio3.custom(DST_JOINT_RAND_PART),
byte(0) + nonce + encoded,
)
k_joint_rand_parts.insert(0, k_joint_rand_part)
joint_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
Prio3.joint_rand(k_joint_rand_parts),
Prio3.custom(DST_JOINT_RANDOMNESS),
b'',
Prio3.Flp.JOINT_RAND_LEN,
)
else:
joint_rand = []
# Finish the proof shares.
prove_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_prove,
Prio3.custom(DST_PROVE_RANDOMNESS),
b'',
Prio3.Flp.PROVE_RAND_LEN,
)
proof = Prio3.Flp.prove(inp, prove_rand, joint_rand)
leader_proof_share = proof
for j in range(Prio3.SHARES-1):
helper_proof_share = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_helper_proof_shares[j],
Prio3.custom(DST_PROOF_SHARE),
byte(j+1),
Prio3.Flp.PROOF_LEN,
)
leader_proof_share = vec_sub(leader_proof_share,
helper_proof_share)
# Each Aggregator's input share contains its measurement share,
# proof share, and blind. The public share contains the
# Aggregators' joint randomness parts.
input_shares = []
input_shares.append(Prio3.encode_leader_share(
leader_meas_share,
leader_proof_share,
k_leader_blind,
))
for j in range(Prio3.SHARES-1):
input_shares.append(Prio3.encode_helper_share(
k_helper_meas_shares[j],
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k_helper_proof_shares[j],
k_helper_blinds[j],
))
public_share = Prio3.encode_public_share(k_joint_rand_parts)
return (public_share, input_shares)
Figure 13: Input-distribution algorithm for Prio3.
7.2.2. Preparation
This section describes the process of recovering output shares from
the input shares. The high-level idea is that each Aggregator first
queries its 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 a
"part" of this seed from its input share and the "blind" generated by
the client. The seed is derived by hashing together the parts, so
before running the query-generation algorithm, it must first gather
the parts derived by the other Aggregators.
In order to avoid extra round of communication, the Client sends each
Aggregator a "hint" consisting of the other Aggregators' parts 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 parts of the joint
randomness along with their verifier shares.
The definitions of constants and a few auxiliary functions are
defined in Section 7.2.6.
def prep_init(Prio3, verify_key, agg_id, _agg_param,
nonce, public_share, input_share):
k_joint_rand_parts = Prio3.decode_public_share(public_share)
(meas_share, proof_share, k_blind) = \
Prio3.decode_leader_share(input_share) if agg_id == 0 else \
Prio3.decode_helper_share(agg_id, input_share)
out_share = Prio3.Flp.truncate(meas_share)
# Compute joint randomness.
joint_rand = []
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k_corrected_joint_rand, k_joint_rand_part = None, None
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded = Prio3.Flp.Field.encode_vec(meas_share)
k_joint_rand_part = Prio3.Prg.derive_seed(k_blind,
Prio3.custom(DST_JOINT_RAND_PART),
byte(agg_id) + nonce + encoded)
k_joint_rand_parts[agg_id] = k_joint_rand_part
k_corrected_joint_rand = Prio3.joint_rand(k_joint_rand_parts)
joint_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
k_corrected_joint_rand,
Prio3.custom(DST_JOINT_RANDOMNESS),
b'',
Prio3.Flp.JOINT_RAND_LEN,
)
# Query the measurement and proof share.
query_rand = Prio3.Prg.expand_into_vec(
Prio3.Flp.Field,
verify_key,
Prio3.custom(DST_QUERY_RANDOMNESS),
nonce,
Prio3.Flp.QUERY_RAND_LEN,
)
verifier_share = Prio3.Flp.query(meas_share,
proof_share,
query_rand,
joint_rand,
Prio3.SHARES)
prep_msg = Prio3.encode_prep_share(verifier_share,
k_joint_rand_part)
return (out_share, k_corrected_joint_rand, prep_msg)
def prep_next(Prio3, prep, inbound):
(out_share, k_corrected_joint_rand, prep_msg) = prep
if inbound is None:
return (prep, prep_msg)
k_joint_rand_check = Prio3.decode_prep_msg(inbound)
if k_joint_rand_check != k_corrected_joint_rand:
raise ERR_VERIFY # joint randomness check failed
return out_share
def prep_shares_to_prep(Prio3, _agg_param, prep_shares):
verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN)
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k_joint_rand_parts = []
for encoded in prep_shares:
(verifier_share, k_joint_rand_part) = \
Prio3.decode_prep_share(encoded)
verifier = vec_add(verifier, verifier_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_parts.append(k_joint_rand_part)
if not Prio3.Flp.decide(verifier):
raise ERR_VERIFY # proof verifier check failed
k_joint_rand_check = None
if Prio3.Flp.JOINT_RAND_LEN > 0:
k_joint_rand_check = Prio3.joint_rand(k_joint_rand_parts)
return Prio3.encode_prep_msg(k_joint_rand_check)
Figure 14: Preparation state for Prio3.
7.2.3. Validity of Aggregation Parameters
Every input share MUST only be used once, regardless of the
aggregation parameters used.
def is_valid(agg_param, previous_agg_params):
return len(previous_agg_params) == 0
Figure 15: Validity of aggregation parameters for Prio3.
7.2.4. Aggregation
Aggregating a set of output shares is simply a matter of adding up
the vectors element-wise.
def 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 Prio3.Flp.Field.encode_vec(agg_share)
Figure 16: Aggregation algorithm for Prio3.
7.2.5. Unsharding
To unshard a set of aggregate shares, the Collector first adds up the
vectors element-wise. It then converts each element of the vector
into an integer.
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def agg_shares_to_result(Prio3, _agg_param,
agg_shares, num_measurements):
agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for agg_share in agg_shares:
agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share))
return Prio3.Flp.decode(agg, num_measurements)
Figure 17: Computation of the aggregate result for Prio3.
7.2.6. Auxiliary Functions
def joint_rand(Prio3, k_joint_rand_parts):
return Prio3.Prg.derive_seed(
zeros(Prio3.Prg.SEED_SIZE),
Prio3.custom(DST_JOINT_RAND_SEED),
concat(k_joint_rand_parts),
)
7.2.6.1. Message Serialization
def encode_leader_share(Prio3,
meas_share,
proof_share,
k_blind):
encoded = Bytes()
encoded += Prio3.Flp.Field.encode_vec(meas_share)
encoded += Prio3.Flp.Field.encode_vec(proof_share)
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
return encoded
def decode_leader_share(Prio3, encoded):
l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN
encoded_meas_share, encoded = encoded[:l], encoded[l:]
meas_share = Prio3.Flp.Field.decode_vec(encoded_meas_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
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, None)
k_blind, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, k_blind)
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def encode_helper_share(Prio3,
k_meas_share,
k_proof_share,
k_blind):
encoded = Bytes()
encoded += k_meas_share
encoded += k_proof_share
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_blind
return encoded
def decode_helper_share(Prio3, agg_id, encoded):
c_meas_share = Prio3.custom(DST_MEASUREMENT_SHARE)
c_proof_share = Prio3.custom(DST_PROOF_SHARE)
l = Prio3.Prg.SEED_SIZE
k_meas_share, encoded = encoded[:l], encoded[l:]
meas_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_meas_share,
c_meas_share,
byte(agg_id),
Prio3.Flp.INPUT_LEN)
k_proof_share, encoded = encoded[:l], encoded[l:]
proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field,
k_proof_share,
c_proof_share,
byte(agg_id),
Prio3.Flp.PROOF_LEN)
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, None)
k_blind, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return (meas_share, proof_share, k_blind)
def encode_public_share(Prio3,
k_joint_rand_parts):
encoded = Bytes()
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += concat(k_joint_rand_parts)
return encoded
def decode_public_share(Prio3, encoded):
l = Prio3.Prg.SEED_SIZE
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
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return None
k_joint_rand_parts = []
for i in range(Prio3.SHARES):
k_joint_rand_part, encoded = encoded[:l], encoded[l:]
k_joint_rand_parts.append(k_joint_rand_part)
if len(encoded) != 0:
raise ERR_DECODE
return k_joint_rand_parts
def encode_prep_share(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_prep_share(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)
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return (verifier, None)
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)
def encode_prep_msg(Prio3, k_joint_rand_check):
encoded = Bytes()
if Prio3.Flp.JOINT_RAND_LEN > 0:
encoded += k_joint_rand_check
return encoded
def decode_prep_msg(Prio3, encoded):
if Prio3.Flp.JOINT_RAND_LEN == 0:
if len(encoded) != 0:
raise ERR_DECODE
return None
l = Prio3.Prg.SEED_SIZE
k_joint_rand_check, encoded = encoded[:l], encoded[l:]
if len(encoded) != 0:
raise ERR_DECODE
return k_joint_rand_check
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7.3. A General-Purpose FLP
This section describes an FLP based on the construction from in
[BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview
of their proof system and the extensions to their proof system made
here. The construction is specified in Section 7.3.3.
OPEN ISSUE We're not yet sure if specifying this general-purpose
FLP is desirable. It might be preferable to specify specialized
FLPs for each data type that we want to standardize, for two
reasons. First, clear and concise specifications are likely
easier to write for specialized FLPs rather than the general one.
Second, we may end up tailoring each FLP to the measurement type
in a way that improves performance, but breaks compatibility with
the general-purpose FLP.
In any case, we can't make this decision until we know which data
types to standardize, so for now, we'll stick with the general-
purpose construction. The reference implementation can be found
at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc.
OPEN ISSUE Chris Wood points out that the this section reads more
like a paper than a standard. Eventually we'll want to work this
into something that is readily consumable by the CFRG.
7.3.1. Overview
In the proof system of [BBCGGI19], validity is defined via an
arithmetic circuit evaluated over the 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 7),
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
C(x_shares[0] + ... + x_shares[SHARES-1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES-1])
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(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
multiplication:
Mul(left, right) = left * right
The validity circuit can then be rewritten in terms of Mul like so:
C(x[0]) = Mul(x[0], x[0]) - x[0]
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 7.3.1.1; but for the moment it is only important that they
are distinct.)
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The prover constructs from wire and alpha a polynomial that, when
evaluated at alpha[k-1], produces the output of the kth call to the
gadget. Let us call this the "gadget polynomial". Polynomial
evaluation is linear, which means that, in the distributed setting,
the Client can disseminate additive shares of the gadget polynomial
that the Aggregators then use to compute additive shares of each
gadget output, allowing each Aggregator to compute its share of C(x)
locally.
There is one more wrinkle, however: It is still possible for a
malicious prover to produce a gadget polynomial that would result in
C(x) being computed incorrectly, potentially resulting in an invalid
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 7.3.3.
7.3.1.1. 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 7.2). This allows for circuit
optimizations that trade a small soundness error for a shorter proof.
For example, consider a circuit that recognizes the set of length-N
vectors for which each element is either one or zero. A
deterministic circuit could be constructed for this language, but it
would involve a large number of multiplications that would result in
a large proof. (See the discussion in [BBCGGI19], Section 5.2 for
details). A much shorter proof can be constructed for the following
randomized circuit:
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N-1])
(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.
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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 6.1.2.
7.3.2. Validity Circuits
The FLP described in Section 7.3.3 is defined in terms of a validity
circuit Valid that implements the interface described here.
A concrete Valid defines the following parameters:
+================+=======================================+
| 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 |
+----------------+---------------------------------------+
| AggResult | Type of the aggregate result |
+----------------+---------------------------------------+
| Field | An FFT-friendly finite field as |
| | defined in Section 6.1.2 |
+----------------+---------------------------------------+
Table 7: Validity circuit parameters.
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Each gadget G in GADGETS defines a constant DEGREE that specifies the
circuit's "arithmetic degree". This is defined to be the degree of
the polynomial that computes it. For example, the Mul circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has
degree 2. Hence, the arithmetic degree of this gadget is 2.
Each gadget also defines a parameter ARITY that specifies the
circuit's arity (i.e., the number of input wires).
A concrete Valid provides the following methods for encoding a
measurement as an input vector, truncating an input vector to the
length of an aggregatable output, and converting an aggregated output
to an aggregate result:
* Valid.encode(measurement: Measurement) -> Vec[Field] returns a
vector of length INPUT_LEN representing a measurement.
* Valid.truncate(input: Vec[Field]) -> Vec[Field] returns a vector
of length OUTPUT_LEN representing an aggregatable output.
* Valid.decode(output: Vec[Field], num_measurements: Unsigned) ->
AggResult returns an aggregate result.
Finally, the following class methods are derived for each concrete
Valid:
# 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
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Figure 18: Derived methods for validity circuits.
7.3.3. Construction
This section specifies FlpGeneric, an implementation of the Flp
interface (Section 7.1). It has as a generic parameter a validity
circuit Valid implementing the interface defined in Section 7.3.2.
NOTE A reference implementation can be found in
https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/
flp_generic.sage.
The FLP parameters for FlpGeneric are defined in Table 8. The
required methods for generating the proof, generating the verifier,
and deciding validity are specified in the remaining subsections.
In the remainder, we let [n] denote the set {1, ..., n} for positive
integer n. We also define the following constants:
* Let H = len(Valid.GADGETS)
* For each i in [H]:
- Let G_i = Valid.GADGETS[i]
- Let L_i = Valid.GADGETS[i].ARITY
- Let M_i = Valid.GADGET_CALLS[i]
- Let P_i = next_power_of_2(M_i+1)
- Let alpha_i = Field.gen()^(Field.GEN_ORDER / P_i)
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+================+============================================+
| Parameter | Value |
+================+============================================+
| PROVE_RAND_LEN | Valid.prove_rand_len() (see Section 7.3.2) |
+----------------+--------------------------------------------+
| QUERY_RAND_LEN | Valid.query_rand_len() (see Section 7.3.2) |
+----------------+--------------------------------------------+
| JOINT_RAND_LEN | Valid.JOINT_RAND_LEN |
+----------------+--------------------------------------------+
| INPUT_LEN | Valid.INPUT_LEN |
+----------------+--------------------------------------------+
| OUTPUT_LEN | Valid.OUTPUT_LEN |
+----------------+--------------------------------------------+
| PROOF_LEN | Valid.proof_len() (see Section 7.3.2) |
+----------------+--------------------------------------------+
| VERIFIER_LEN | Valid.verifier_len() (see Section 7.3.2) |
+----------------+--------------------------------------------+
| Measurement | Valid.Measurement |
+----------------+--------------------------------------------+
| Field | Valid.Field |
+----------------+--------------------------------------------+
Table 8: FLP Parameters of FlpGeneric.
7.3.3.1. Proof Generation
On input inp, prove_rand, and joint_rand, the proof is computed as
follows:
1. For each i in [H] create an empty table wire_i.
2. Partition the prover randomness prove_rand into sub-vectors
seed_1, ..., seed_H where len(seed_i) == L_i for all i in [H].
Let us call these the "wire seeds" of each gadget.
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.
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)).
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NOTE We pad w to the nearest power of 2 so that we can use FFT
for interpolating the wire polynomials. Perhaps there is some
clever math for picking wire_inp in a way that avoids having
to pad.
* Let poly_wire_i[j-1] be the lowest degree polynomial for which
poly_wire_i[j-1](alpha_i^k) == padded_w[k] for all k in [P_i].
5. Compute the "gadget polynomials". That is, for every i in [H]:
* Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-
1]). That is, evaluate the circuit G_i on the wire
polynomials for the ith gadget. (Arithmetic is in the ring of
polynomials over Field.)
The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H +
coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i
for each i in [H].
7.3.3.2. Query Generation
On input of 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 sub-vectors seed_1, coeff_1, ...,
seed_H, coeff_H defined in Section 7.3.3.1.
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]:
* Let t = query_rand[i]. Check if t^(P_i) == 1: If so, then
raise ERR_ABORT and halt. (This prevents the verifier from
inadvertently leaking a gadget output in the verifier
message.)
* Let y_i = poly_gadget_i(t).
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* For each j in [0,L_i) let x_i[j-1] = poly_wire_i[j-1](t).
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ...
+ x_H + [y_H].
7.3.3.3. Decision
On input of vector verifier, the verifier decides if the input is
valid as follows:
1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in
Section 7.3.3.2.
2. Check for well-formedness of the gadget polynomials. For every i
in [H]:
* Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i
and set z to the output.
* If z != y_i, then return False and halt.
3. Return True if v == 0 and False otherwise.
7.3.3.4. Encoding
The FLP encoding and truncation methods invoke Valid.encode,
Valid.truncate, and Valid.decode in the natural way.
7.4. Instantiations
This section specifies instantiations of Prio3 for various
measurement types. Each uses FlpGeneric as the FLP (Section 7.3) and
is determined by a validity circuit (Section 7.3.2) and a PRG
(Section 6.2). Test vectors for each can be found in Appendix "Test
Vectors".
NOTE Reference implementations of each of these VDAFs can be found
in https://github.com/cfrg/draft-irtf-cfrg-vdaf/blob/main/poc/
vdaf_prio3.sage.
7.4.1. Prio3Count
Our first instance of Prio3 is for a simple counter: Each measurement
is either one or zero and the aggregate result is the sum of the
measurements.
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This instance uses PrgSha3 (Section 6.2.1) as its PRG. Its validity
circuit, denoted Count, uses Field64 (Table 3) as its finite field.
Its gadget, denoted Mul, is the degree-2, arity-2 gadget defined as
def 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 and decoded 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) |
+----------------+--------------------------+
| AggResult | Unsigned |
+----------------+--------------------------+
| Field | Field64 (Table 3) |
+----------------+--------------------------+
Table 9: Parameters of validity circuit
Count.
7.4.2. Prio3Sum
The next instance of Prio3 supports summing of integers in a pre-
determined range. Each measurement is an integer in range [0,
2^bits), where bits is an associated parameter.
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This instance of Prio3 uses PrgSha3 (Section 6.2.1) 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]
def decode(Sum, output, _num_measurements):
return output[0].as_unsigned()
The validity circuit checks that the input consists of ones and
zeros. Its gadget, denoted Range2, is the degree-2, arity-1 gadget
defined as
def 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
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+================+================================+
| Parameter | Value |
+================+================================+
| GADGETS | [Range2] |
+----------------+--------------------------------+
| GADGET_CALLS | [bits] |
+----------------+--------------------------------+
| INPUT_LEN | bits |
+----------------+--------------------------------+
| OUTPUT_LEN | 1 |
+----------------+--------------------------------+
| JOINT_RAND_LEN | 1 |
+----------------+--------------------------------+
| Measurement | Unsigned, in range [0, 2^bits) |
+----------------+--------------------------------+
| AggResult | Unsigned |
+----------------+--------------------------------+
| Field | Field128 (Table 3) |
+----------------+--------------------------------+
Table 10: Parameters of validity circuit Sum.
7.4.3. Prio3Histogram
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 PrgSha3 (Section 6.2.1) 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
def decode(Histogram, output: Vec[Field128], _num_measurements):
return [bucket_count.as_unsigned() for bucket_count in output]
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The validity circuit uses Range2 (see Section 7.4.2) as its single
gadget. It checks for one-hotness in two steps, as follows:
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 |
+----------------+--------------------+
| AggResult | Vec[Unsigned] |
+----------------+--------------------+
| Field | Field128 (Table 3) |
+----------------+--------------------+
Table 11: Parameters of validity
circuit Histogram.
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8. Poplar1
This section specifies Poplar1, a VDAF for the following task. Each
Client holds a string of length BITS and the Aggregators hold a set
of l-bit strings, where l <= BITS. We will refer to the latter as
the set of "candidate prefixes". The Aggregators' goal is to count
how many inputs are prefixed by each candidate prefix.
This functionality is the core component of the Poplar protocol
[BBCGGI21], which was designed to compute the heavy hitters over a
set of input strings. At a high level, the protocol works as
follows.
1. Each Client splits its input string into input shares and sends
one share to each Aggregator.
2. The Aggregators agree on an initial set of candidate prefixes,
say 0 and 1.
3. The Aggregators evaluate the VDAF on each set of input shares and
aggregate the recovered output shares. The aggregation parameter
is the set of candidate prefixes.
4. The Aggregators send their aggregate shares to the Collector, who
combines them to recover the counts of each candidate prefix.
5. Let H denote the set of prefixes that occurred at least t times.
If the prefixes all have length BITS, then H is the set of t-
heavy-hitters. Otherwise compute the next set of candidate
prefixes, e.g., for each p in H, add p || 0 and p || 1 to the
set. Repeat step 3 with the new set of candidate prefixes.
Poplar1 is constructed from an "Incremental Distributed Point
Function (IDPF)", a primitive described by [BBCGGI21] that
generalizes the notion of a Distributed Point Function (DPF) [GI14].
Briefly, a DPF is used to distribute the computation of a "point
function", a function that evaluates to zero on every input except at
a programmable "point". The computation is distributed in such a way
that no one party knows either the point or what it evaluates to.
An IDPF generalizes this "point" to a path on a full binary tree from
the root to one of the leaves. It is evaluated on an "index"
representing a unique node of the tree. If the node is on the
programmed path, then the function evaluates to a non-zero value;
otherwise it evaluates to zero. This structure allows an IDPF to
provide the functionality required for the above protocol: To compute
the hit count for an index, just evaluate each set of IDPF shares at
that index and add up the results.
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Consider the sub-tree constructed from a set of input strings and a
target threshold t by including all indices that prefix at least t of
the input strings. We shall refer to this structure as the "prefix
tree" for the batch of inputs and target threshold. To compute the
t-heavy hitters for a set of inputs, the Aggregators and Collector
first compute the prefix tree, then extract the heavy hitters from
the leaves of this tree. (Note that the prefix tree may leak more
information about the set than the heavy hitters themselves; see
Section 9.4 for details.)
Poplar1 composes an IDPF with the "secure sketching" protocol of
[BBCGGI21]. This protocol ensures that evaluating a set of input
shares on a unique set of candidate prefixes results in shares of a
"one-hot" vector, i.e., a vector that is zero everywhere except for
one element, which is equal to one.
The remainder of this section is structured as follows. IDPFs are
defined in Section 8.1; a concrete instantiation is given
Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of
a generic IDPF. Finally, a concrete instantiation of Poplar1 is
specified in Section 8.4; test vectors can be found in Appendix "Test
Vectors".
8.1. Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size 2^BITS, where BITS is
constant defined by the IDPF. Indexes into the IDPF tree are encoded
as integers in range [0, 2^BITS). The Client specifies an index
alpha and a vector of values beta, one for each "level" L in range
[0, BITS). The key generation algorithm generates one IDPF "key" for
each Aggregator. When evaluated at level L and index 0 <= prefix <
2^L, each IDPF key returns an additive share of beta[L] if prefix is
the L-bit prefix of alpha and shares of zero otherwise.
An index x is defined to be a prefix of another index y as follows.
Let LSB(x, N) denote the least significant N bits of positive integer
x. By definition, a positive integer 0 <= x < 2^L is said to be the
length-L prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is
equal to the most significant L bits of LSB(y, BITS), For example, 6
(110 in binary) is the length-3 prefix of 25 (11001), but 7 (111) is
not.
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Each of the programmed points beta is a vector of elements of some
finite field. We distinguish two types of fields: One for inner
nodes (denoted Idpf.FieldInner), and one for leaf nodes
(Idpf.FieldLeaf). (Our instantiation of Poplar1 (Section 8.4) will
use a much larger field for leaf nodes than for inner nodes. This is
to ensure the IDPF is "extractable" as defined in [BBCGGI21],
Definition 1.)
A concrete IDPF defines the types and constants enumerated in
Table 12. In the remainder we write Idpf.Vec as shorthand for the
type Union[Vec[Vec[Idpf.FieldInner]], Vec[Vec[Idpf.FieldLeaf]]].
(This type denotes either a vector of inner node field elements or
leaf node field elements.) The scheme is comprised of the following
algorithms:
* Idpf.gen(alpha: Unsigned, beta_inner: Vec[Vec[Idpf.FieldInner]],
beta_leaf: Vec[Idpf.FieldLeaf], binder: Bytes, rand:
Bytes[Idpf.RAND_SIZE]) -> (Bytes, Vec[Bytes]) is the randomized
IDPF-key generation algorithm. (Input rand consists of the random
coins it consumes.) Its inputs are the index alpha the values
beta, and a binder string. The value of alpha MUST be in range
[0, 2^BITS). The output is a public part that is sent to all
Aggregators and a vector of private IDPF keys, one for each
aggregator.
* Idpf.eval(agg_id: Unsigned, public_share: Bytes, key: Bytes,
level: Unsigned, prefixes: Vec[Unsigned], binder: Bytes) ->
Idpf.Vec is the deterministic, stateless IDPF-key evaluation
algorithm run by each Aggregator. Its inputs are the Aggregator's
unique identifier, the public share distributed to all of the
Aggregators, the Aggregator's IDPF key, the "level" at which to
evaluate the IDPF, the sequence of candidate prefixes, and a
binder string. It returns the share of the value corresponding to
each candidate prefix.
The output type depends on the value of level: If level <
Idpf.BITS-1, the output is the value for an inner node, which has
type Vec[Vec[Idpf.FieldInner]]; otherwise, if level == Idpf.BITS-
1, then the output is the value for a leaf node, which has type
Vec[Vec[Idpf.FieldLeaf]].
The value of level MUST be in range [0, BITS). The indexes in
prefixes MUST all be distinct and in range [0, 2^level).
Applications MUST ensure that the Aggregator's identifier is equal
to the integer in range [0, SHARES) that matches the index of key
in the sequence of IDPF keys output by the Client.
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In addition, the following method is derived for each concrete Idpf:
def current_field(Idpf, level):
return Idpf.FieldInner if level < Idpf.BITS-1 \
else Idpf.FieldLeaf
Finally, an implementation note. The interface for IDPFs specified
here is stateless, in the sense that there is no state carried
between IDPF evaluations. This is to align the IDPF syntax with the
VDAF abstraction boundary, which does not include shared state across
across VDAF evaluations. In practice, of course, it will often be
beneficial to expose a stateful API for IDPFs and carry the state
across evaluations. See Section 8.3 for details.
+============+==================================================+
| Parameter | Description |
+============+==================================================+
| SHARES | Number of IDPF keys output by IDPF-key generator |
+------------+--------------------------------------------------+
| BITS | Length in bits of each input string |
+------------+--------------------------------------------------+
| VALUE_LEN | Number of field elements of each output value |
+------------+--------------------------------------------------+
| RAND_SIZE | Size of the random string consumed by the IDPF- |
| | key generator |
+------------+--------------------------------------------------+
| KEY_SIZE | Size in bytes of each IDPF key |
+------------+--------------------------------------------------+
| FieldInner | Implementation of Field (Section 6.1) used for |
| | values of inner nodes |
+------------+--------------------------------------------------+
| FieldLeaf | Implementation of Field used for values of leaf |
| | nodes |
+------------+--------------------------------------------------+
Table 12: Constants and types defined by a concrete IDPF.
8.2. Construction
This section specifies Poplar1, an implementation of the Vdaf
interface (Section 5). It is defined in terms of any Idpf
(Section 8.1) for which Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2 and
an implementation of Prg (Section 6.2). The associated constants and
types required by the Vdaf interface are defined in Table 13. The
methods required for sharding, preparation, aggregation, and
unsharding are described in the remaining subsections. These methods
make use of constants defined in Table 14.
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+=================+==================================+
| Parameter | Value |
+=================+==================================+
| VERIFY_KEY_SIZE | Prg.SEED_SIZE |
+-----------------+----------------------------------+
| NONCE_SIZE | 16 |
+-----------------+----------------------------------+
| ROUNDS | 2 |
+-----------------+----------------------------------+
| SHARES | 2 |
+-----------------+----------------------------------+
| Measurement | Unsigned |
+-----------------+----------------------------------+
| AggParam | Tuple[Unsigned, Vec[Unsigned]] |
+-----------------+----------------------------------+
| Prep | Tuple[Bytes, Unsigned, Idpf.Vec] |
+-----------------+----------------------------------+
| OutShare | Idpf.Vec |
+-----------------+----------------------------------+
| AggResult | Vec[Unsigned] |
+-----------------+----------------------------------+
Table 13: VDAF parameters for Poplar1.
+===========================+=======+
| Variable | Value |
+===========================+=======+
| DST_SHARD_RAND: Unsigned | 1 |
+---------------------------+-------+
| DST_CORR_INNER: Unsigned | 2 |
+---------------------------+-------+
| DST_CORR_LEAF: Unsigned | 3 |
+---------------------------+-------+
| DST_VERIFY_RAND: Unsigned | 4 |
+---------------------------+-------+
Table 14: Constants used by Poplar1.
8.2.1. Client
The client's input is an IDPF index, denoted alpha. The programmed
IDPF values are pairs of field elements (1, k) where each k is chosen
at random. This random value is used as part of the secure sketching
protocol of [BBCGGI21], Appendix C.4. After evaluating their IDPF
key shares on a given sequence of candidate prefixes, the sketching
protocol is used by the Aggregators to verify that they hold shares
of a one-hot vector. In addition, for each level of the tree, the
prover generates random elements a, b, and c and computes
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A = -2*a + k
B = a^2 + b - k*a + c
and sends additive shares of a, b, c, A and B to the Aggregators.
Putting everything together, the sharding algorithm is defined as
follows. Function encode_input_shares is defined in Section 8.2.6.
def measurement_to_input_shares(Poplar1, measurement, nonce, rand):
l = Poplar1.Prg.SEED_SIZE
# Split the coins into coins for IDPF key generation,
# correlated randomness, and sharding.
if len(rand) != Poplar1.RAND_SIZE:
raise ERR_INPUT # unexpected length for random coins
idpf_rand, rand = front(Poplar1.Idpf.RAND_SIZE, rand)
seeds = [rand[i:i+l] for i in range(0,3*l,l)]
corr_seed, seeds = front(2, seeds)
(k_shard,), seeds = front(1, seeds)
prg = Poplar1.Prg(k_shard,
Poplar1.custom(DST_SHARD_RAND), b'')
# Construct the IDPF values for each level of the IDPF tree.
# Each "data" value is 1; in addition, the Client generates
# a random "authenticator" value used by the Aggregators to
# compute the sketch during preparation. This sketch is used
# to verify the one-hotness of their output shares.
beta_inner = [
[Poplar1.Idpf.FieldInner(1), k] \
for k in prg.next_vec(Poplar1.Idpf.FieldInner,
Poplar1.Idpf.BITS - 1) ]
beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \
prg.next_vec(Poplar1.Idpf.FieldLeaf, 1)
# Generate the IDPF keys.
(public_share, keys) = Poplar1.Idpf.gen(measurement,
beta_inner,
beta_leaf,
idpf_rand)
# Generate correlated randomness used by the Aggregators to
# compute a sketch over their output shares. PRG seeds are
# used to encode shares of the `(a, b, c)` triples.
# (See [BBCGGI21, Appendix C.4].)
corr_offsets = vec_add(
Poplar1.Prg.expand_into_vec(
Poplar1.Idpf.FieldInner,
corr_seed[0],
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Poplar1.custom(DST_CORR_INNER),
byte(0) + nonce,
3 * (Poplar1.Idpf.BITS-1),
),
Poplar1.Prg.expand_into_vec(
Poplar1.Idpf.FieldInner,
corr_seed[1],
Poplar1.custom(DST_CORR_INNER),
byte(1) + nonce,
3 * (Poplar1.Idpf.BITS-1),
),
)
corr_offsets += vec_add(
Poplar1.Prg.expand_into_vec(
Poplar1.Idpf.FieldLeaf,
corr_seed[0],
Poplar1.custom(DST_CORR_LEAF),
byte(0) + nonce,
3,
),
Poplar1.Prg.expand_into_vec(
Poplar1.Idpf.FieldLeaf,
corr_seed[1],
Poplar1.custom(DST_CORR_LEAF),
byte(1) + nonce,
3,
),
)
# For each level of the IDPF tree, shares of the `(A, B)`
# pairs are computed from the corresponding `(a, b, c)`
# triple and authenticator value `k`.
corr_inner = [[], []]
for level in range(Poplar1.Idpf.BITS):
Field = Poplar1.Idpf.current_field(level)
k = beta_inner[level][1] if level < Poplar1.Idpf.BITS - 1 \
else beta_leaf[1]
(a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:]
A = -Field(2) * a + k
B = a^2 + b - a * k + c
corr1 = prg.next_vec(Field, 2)
corr0 = vec_sub([A, B], corr1)
if level < Poplar1.Idpf.BITS - 1:
corr_inner[0] += corr0
corr_inner[1] += corr1
else:
corr_leaf = [corr0, corr1]
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# Each input share consists of the Aggregator's IDPF key
# and a share of the correlated randomness.
return (public_share,
Poplar1.encode_input_shares(
keys, corr_seed, corr_inner, corr_leaf))
Figure 19: The sharding algorithm for Poplar1.
8.2.2. Preparation
The aggregation parameter encodes a sequence of candidate prefixes.
When an Aggregator receives an input share from the Client, it begins
by evaluating its IDPF share on each candidate prefix, recovering a
data_share and auth_share for each. The Aggregators use these and
the correlation shares provided by the Client to verify that the
sequence of data_share values are additive shares of a one-hot
vector.
Aggregators MUST ensure the candidate prefixes are all unique and
appear in lexicographic order. (This is enforced in the definition
of prep_init() below.) Uniqueness is necessary to ensure the refined
measurement (i.e., the sum of the output shares) is in fact a one-hot
vector. Otherwise, sketch verification might fail, causing the
Aggregators to erroneously reject a report that is actually valid.
Note that enforcing the order is not strictly necessary, but this
does allow uniqueness to be determined more efficiently.
The algorithms below make use of the auxiliary function
decode_input_share() defined in Section 8.2.6.
def prep_init(Poplar1, verify_key, agg_id, agg_param,
nonce, public_share, input_share):
(level, prefixes) = agg_param
(key, corr_seed, corr_inner, corr_leaf) = \
Poplar1.decode_input_share(input_share)
Field = Poplar1.Idpf.current_field(level)
# Ensure that candidate prefixes are all unique and appear in
# lexicographic order.
for i in range(1,len(prefixes)):
if prefixes[i-1] >= prefixes[i]:
raise ERR_INPUT # out-of-order prefix
# Evaluate the IDPF key at the given set of prefixes.
value = Poplar1.Idpf.eval(
agg_id, public_share, key, level, prefixes)
# Get shares of the correlated randomness for computing the
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# Aggregator's share of the sketch for the given level of the IDPF
# tree.
if level < Poplar1.Idpf.BITS - 1:
corr_prg = Poplar1.Prg(corr_seed,
Poplar1.custom(DST_CORR_INNER),
byte(agg_id) + nonce)
# Fast-forward the PRG state to the current level.
corr_prg.next_vec(Field, 3 * level)
else:
corr_prg = Poplar1.Prg(corr_seed,
Poplar1.custom(DST_CORR_LEAF),
byte(agg_id) + nonce)
(a_share, b_share, c_share) = corr_prg.next_vec(Field, 3)
(A_share, B_share) = corr_inner[2*level:2*(level+1)] \
if level < Poplar1.Idpf.BITS - 1 else corr_leaf
# Compute the Aggregator's first round of the sketch. These are
# called the "masked input values" [BBCGGI21, Appendix C.4].
verify_rand_prg = Poplar1.Prg(verify_key,
Poplar1.custom(DST_VERIFY_RAND),
nonce + to_be_bytes(level, 2))
verify_rand = verify_rand_prg.next_vec(Field, len(prefixes))
sketch_share = [a_share, b_share, c_share]
out_share = []
for (i, r) in enumerate(verify_rand):
(data_share, auth_share) = value[i]
sketch_share[0] += data_share * r
sketch_share[1] += data_share * r^2
sketch_share[2] += auth_share * r
out_share.append(data_share)
prep_mem = sketch_share \
+ [A_share, B_share, Field(agg_id)] \
+ out_share
return (b'ready', level, prep_mem)
def prep_next(Poplar1, prep_state, opt_sketch):
(step, level, prep_mem) = prep_state
Field = Poplar1.Idpf.current_field(level)
# Aggregators exchange masked input values (step (3.)
# of [BBCGGI21, Appendix C.4]).
if step == b'ready' and opt_sketch == None:
sketch_share, prep_mem = prep_mem[:3], prep_mem[3:]
return ((b'sketch round 1', level, prep_mem),
Field.encode_vec(sketch_share))
# Aggregators exchange evaluated shares (step (4.)).
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elif step == b'sketch round 1' and opt_sketch != None:
prev_sketch = Field.decode_vec(opt_sketch)
if len(prev_sketch) == 0:
prev_sketch = Field.zeros(3)
elif len(prev_sketch) != 3:
raise ERR_INPUT # prep message malformed
(A_share, B_share, agg_id), prep_mem = \
prep_mem[:3], prep_mem[3:]
sketch_share = [
agg_id * (prev_sketch[0]^2 \
- prev_sketch[1]
- prev_sketch[2]) \
+ A_share * prev_sketch[0] \
+ B_share
]
return ((b'sketch round 2', level, prep_mem),
Field.encode_vec(sketch_share))
elif step == b'sketch round 2' and opt_sketch != None:
if len(opt_sketch) == 0:
return prep_mem # Output shares
else:
raise ERR_INPUT # prep message malformed
raise ERR_INPUT # unexpected input
def prep_shares_to_prep(Poplar1, agg_param, prep_shares):
if len(prep_shares) != 2:
raise ERR_INPUT # unexpected number of prep shares
(level, _) = agg_param
Field = Poplar1.Idpf.current_field(level)
sketch = vec_add(Field.decode_vec(prep_shares[0]),
Field.decode_vec(prep_shares[1]))
if len(sketch) == 3:
return Field.encode_vec(sketch)
elif len(sketch) == 1:
if sketch == Field.zeros(1):
# In order to reduce communication overhead, let the
# empty string denote a successful sketch verification.
return b''
else:
raise ERR_VERIFY # sketch verification failed
else:
return ERR_INPUT # unexpected input length
Figure 20: Preparation state for Poplar1.
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8.2.3. Validity of Aggregation Parameters
Aggregation parameters are valid for a given input share if no
aggregation parameter with the same level has been used with the same
input share before. The whole preparation phase MUST NOT be run more
than once for a given combination of input share and level.
def is_valid(agg_param, previous_agg_params):
(level, _) = agg_param
return all(
level != other_level
for (other_level, _) in previous_agg_params
)
Figure 21: Validity of aggregation parameters for Poplar1.
8.2.4. Aggregation
Aggregation involves simply adding up the output shares.
def out_shares_to_agg_share(Poplar1, agg_param, out_shares):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg_share = Field.zeros(len(prefixes))
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return Field.encode_vec(agg_share)
Figure 22: Aggregation algorithm for Poplar1.
8.2.5. Unsharding
Finally, the Collector unshards the aggregate result by adding up the
aggregate shares.
def agg_shares_to_result(Poplar1, agg_param,
agg_shares, _num_measurements):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg = Field.zeros(len(prefixes))
for agg_share in agg_shares:
agg = vec_add(agg, Field.decode_vec(agg_share))
return list(map(lambda x: x.as_unsigned(), agg))
Figure 23: Computation of the aggregate result for Poplar1.
8.2.6. Auxiliary Functions
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8.2.6.1. Message Serialization
This section defines methods for serializing input shares, as
required by the Vdaf interface. Optional serialization of the
aggregation parameter is also specified below.
Implementation note: The aggregation parameter includes the level of
the IDPF tree and the sequence of indices to evaluate. For
implementations that perform per-report caching across executions of
the VDAF, this may be more information than is strictly needed. In
particular, it may be sufficient to convey which indices from the
previous execution will have their children included in the next.
This would help reduce communication overhead.
def encode_input_shares(Poplar1, keys,
corr_seed, corr_inner, corr_leaf):
input_shares = []
for (key, seed, inner, leaf) in zip(keys,
corr_seed,
corr_inner,
corr_leaf):
encoded = Bytes()
encoded += key
encoded += seed
encoded += Poplar1.Idpf.FieldInner.encode_vec(inner)
encoded += Poplar1.Idpf.FieldLeaf.encode_vec(leaf)
input_shares.append(encoded)
return input_shares
def decode_input_share(Poplar1, encoded):
l = Poplar1.Idpf.KEY_SIZE
key, encoded = encoded[:l], encoded[l:]
l = Poplar1.Prg.SEED_SIZE
corr_seed, encoded = encoded[:l], encoded[l:]
l = Poplar1.Idpf.FieldInner.ENCODED_SIZE \
* 2 * (Poplar1.Idpf.BITS - 1)
encoded_corr_inner, encoded = encoded[:l], encoded[l:]
corr_inner = Poplar1.Idpf.FieldInner.decode_vec(
encoded_corr_inner)
l = Poplar1.Idpf.FieldLeaf.ENCODED_SIZE * 2
encoded_corr_leaf, encoded = encoded[:l], encoded[l:]
corr_leaf = Poplar1.Idpf.FieldLeaf.decode_vec(
encoded_corr_leaf)
if len(encoded) != 0:
raise ERR_INPUT
return (key, corr_seed, corr_inner, corr_leaf)
def encode_agg_param(Poplar1, level, prefixes):
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if level > 2^16 - 1:
raise ERR_INPUT # level too deep
if len(prefixes) > 2^32 - 1:
raise ERR_INPUT # too many prefixes
encoded = Bytes()
encoded += to_be_bytes(level, 2)
encoded += to_be_bytes(len(prefixes), 4)
packed = 0
for (i, prefix) in enumerate(prefixes):
packed |= prefix << ((level+1) * i)
l = floor(((level+1) * len(prefixes) + 7) / 8)
encoded += to_be_bytes(packed, l)
return encoded
def decode_agg_param(Poplar1, encoded):
encoded_level, encoded = encoded[:2], encoded[2:]
level = from_be_bytes(encoded_level)
encoded_prefix_count, encoded = encoded[:4], encoded[4:]
prefix_count = from_be_bytes(encoded_prefix_count)
l = floor(((level+1) * prefix_count + 7) / 8)
encoded_packed, encoded = encoded[:l], encoded[l:]
packed = from_be_bytes(encoded_packed)
prefixes = []
m = 2^(level+1) - 1
for i in range(prefix_count):
prefixes.append(packed >> ((level+1) * i) & m)
if len(encoded) != 0:
raise ERR_INPUT
return (level, prefixes)
8.3. The IDPF scheme of [BBCGGI21]
TODO(issue#32) Consider replacing the generic Prg object here with
some fixed-key mode for AES (something along the lines of
ia.cr/2019/074). This would allow us to take advantage of
hardware acceleration, which would significantly improve
performance. We use SHA-3 primarily to instantiate random
oracles, but the random oracle model may not be required for IDPF.
More investigation is needed.
In this section we specify a concrete IDPF, called IdpfPoplar,
suitable for instantiating Poplar1. The scheme gets its name from
the name of the protocol of [BBCGGI21].
TODO We should consider giving IdpfPoplar a more distinctive name.
The constant and type definitions required by the Idpf interface are
given in Table 15.
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IdpfPoplar requires a PRG for deriving the output shares, as well as
a variety of other artifacts used internally. For performance
reasons, we instantiate this object using PrgFixedKeyAes128
(Section 6.2.2). See Section 9.5 for justification of this choice.
+============+======================+
| Parameter | Value |
+============+======================+
| SHARES | 2 |
+------------+----------------------+
| BITS | any positive integer |
+------------+----------------------+
| VALUE_LEN | any positive integer |
+------------+----------------------+
| KEY_SIZE | Prg.SEED_SIZE |
+------------+----------------------+
| FieldInner | Field64 (Table 3) |
+------------+----------------------+
| FieldLeaf | Field255 (Table 3) |
+------------+----------------------+
Table 15: Constants and type
definitions for IdpfPoplar.
8.3.1. Key Generation
TODO Describe the construction in prose, beginning with a gentle
introduction to the high level idea.
The description of the IDPF-key generation algorithm makes use of
auxiliary functions extend(), convert(), and encode_public_share()
defined in Section 8.3.3. In the following, we let Field2 denote the
field GF(2).
def gen(IdpfPoplar, alpha, beta_inner, beta_leaf, binder, rand):
if alpha >= 2^IdpfPoplar.BITS:
raise ERR_INPUT # alpha too long
if len(beta_inner) != IdpfPoplar.BITS - 1:
raise ERR_INPUT # beta_inner vector is the wrong size
if len(rand) != IdpfPoplar.RAND_SIZE:
raise ERR_INPUT # unexpected length for random coins
init_seed = [
rand[:PrgFixedKeyAes128.SEED_SIZE],
rand[PrgFixedKeyAes128.SEED_SIZE:],
]
seed = init_seed.copy()
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ctrl = [Field2(0), Field2(1)]
correction_words = []
for level in range(IdpfPoplar.BITS):
Field = IdpfPoplar.current_field(level)
keep = (alpha >> (IdpfPoplar.BITS - level - 1)) & 1
lose = 1 - keep
bit = Field2(keep)
(s0, t0) = IdpfPoplar.extend(seed[0], binder)
(s1, t1) = IdpfPoplar.extend(seed[1], binder)
seed_cw = xor(s0[lose], s1[lose])
ctrl_cw = (
t0[0] + t1[0] + bit + Field2(1),
t0[1] + t1[1] + bit,
)
x0 = xor(s0[keep], ctrl[0].conditional_select(seed_cw))
x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw))
(seed[0], w0) = IdpfPoplar.convert(level, x0, binder)
(seed[1], w1) = IdpfPoplar.convert(level, x1, binder)
ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep]
ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep]
b = beta_inner[level] if level < IdpfPoplar.BITS-1 \
else beta_leaf
if len(b) != IdpfPoplar.VALUE_LEN:
raise ERR_INPUT # beta too long or too short
w_cw = vec_add(vec_sub(b, w0), w1)
# Implementation note: Here we negate the correction word if
# the control bit `ctrl[1]` is set. We avoid branching on the
# value in order to reduce leakage via timing side channels.
mask = Field(1) - Field(2) * Field(ctrl[1].as_unsigned())
for i in range(len(w_cw)):
w_cw[i] *= mask
correction_words.append((seed_cw, ctrl_cw, w_cw))
public_share = IdpfPoplar.encode_public_share(correction_words)
return (public_share, init_seed)
Figure 24: IDPF-key generation algorithm of IdpfPoplar.
8.3.2. Key Evaluation
TODO Describe in prose how IDPF-key evaluation algorithm works.
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The description of the IDPF-evaluation algorithm makes use of
auxiliary functions extend(), convert(), and decode_public_share()
defined in Section 8.3.3.
def eval(IdpfPoplar, agg_id, public_share, init_seed,
level, prefixes, binder):
if agg_id >= IdpfPoplar.SHARES:
raise ERR_INPUT # invalid aggregator ID
if level >= IdpfPoplar.BITS:
raise ERR_INPUT # level too deep
if len(set(prefixes)) != len(prefixes):
raise ERR_INPUT # candidate prefixes are non-unique
correction_words = IdpfPoplar.decode_public_share(public_share)
out_share = []
for prefix in prefixes:
if prefix >= 2^(level+1):
raise ERR_INPUT # prefix too long
# The Aggregator's output share is the value of a node of
# the IDPF tree at the given `level`. The node's value is
# computed by traversing the path defined by the candidate
# `prefix`. Each node in the tree is represented by a seed
# (`seed`) and a set of control bits (`ctrl`).
seed = init_seed
ctrl = Field2(agg_id)
for current_level in range(level+1):
bit = (prefix >> (level - current_level)) & 1
# Implementation note: Typically the current round of
# candidate prefixes would have been derived from
# aggregate results computed during previous rounds. For
# example, when using `IdpfPoplar` to compute heavy
# hitters, a string whose hit count exceeded the given
# threshold in the last round would be the prefix of each
# `prefix` in the current round. (See [BBCGGI21,
# Section 5.1].) In this case, part of the path would
# have already been traversed.
#
# Re-computing nodes along previously traversed paths is
# wasteful. Implementations can eliminate this added
# complexity by caching nodes (i.e., `(seed, ctrl)`
# pairs) output by previous calls to `eval_next()`.
(seed, ctrl, y) = IdpfPoplar.eval_next(seed, ctrl,
correction_words[current_level], current_level, bit, binder)
out_share.append(y if agg_id == 0 else vec_neg(y))
return out_share
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# Compute the next node in the IDPF tree along the path determined by
# a candidate prefix. The next node is determined by `bit`, the bit
# of the prefix corresponding to the next level of the tree.
#
# TODO Consider implementing some version of the optimization
# discussed at the end of [BBCGGI21, Appendix C.2]. This could on
# average reduce the number of AES calls by a constant factor.
def eval_next(IdpfPoplar, prev_seed, prev_ctrl,
correction_word, level, bit, binder):
Field = IdpfPoplar.current_field(level)
(seed_cw, ctrl_cw, w_cw) = correction_word
(s, t) = IdpfPoplar.extend(prev_seed, binder)
s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw))
s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw))
t[0] += ctrl_cw[0] * prev_ctrl
t[1] += ctrl_cw[1] * prev_ctrl
next_ctrl = t[bit]
(next_seed, y) = IdpfPoplar.convert(level, s[bit], binder)
# Implementation note: Here we add the correction word to the
# output if `next_ctrl` is set. We avoid branching on the value of
# the control bit in order to reduce side channel leakage.
mask = Field(next_ctrl.as_unsigned())
for i in range(len(y)):
y[i] += w_cw[i] * mask
return (next_seed, next_ctrl, y)
Figure 25: IDPF-evaluation generation algorithm of IdpfPoplar.
8.3.3. Auxiliary Functions
def extend(IdpfPoplar, seed, binder):
prg = PrgFixedKeyAes128(seed, format_custom(1, 0, 0), binder)
s = [
prg.next(PrgFixedKeyAes128.SEED_SIZE),
prg.next(PrgFixedKeyAes128.SEED_SIZE),
]
b = prg.next(1)[0]
t = [Field2(b & 1), Field2((b >> 1) & 1)]
return (s, t)
def convert(IdpfPoplar, level, seed, binder):
prg = PrgFixedKeyAes128(seed, format_custom(1, 0, 1), binder)
next_seed = prg.next(PrgFixedKeyAes128.SEED_SIZE)
Field = IdpfPoplar.current_field(level)
w = prg.next_vec(Field, IdpfPoplar.VALUE_LEN)
return (next_seed, w)
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def encode_public_share(IdpfPoplar, correction_words):
encoded = Bytes()
control_bits = list(itertools.chain.from_iterable(
cw[1] for cw in correction_words
))
encoded += pack_bits(control_bits)
for (level, (seed_cw, _, w_cw)) \
in enumerate(correction_words):
Field = IdpfPoplar.current_field(level)
encoded += seed_cw
encoded += Field.encode_vec(w_cw)
return encoded
def decode_public_share(IdpfPoplar, encoded):
l = floor((2*IdpfPoplar.BITS + 7) / 8)
encoded_ctrl, encoded = encoded[:l], encoded[l:]
control_bits = unpack_bits(encoded_ctrl, 2 * IdpfPoplar.BITS)
correction_words = []
for level in range(IdpfPoplar.BITS):
Field = IdpfPoplar.current_field(level)
ctrl_cw = (
control_bits[level * 2],
control_bits[level * 2 + 1],
)
l = PrgFixedKeyAes128.SEED_SIZE
seed_cw, encoded = encoded[:l], encoded[l:]
l = Field.ENCODED_SIZE * IdpfPoplar.VALUE_LEN
encoded_w_cw, encoded = encoded[:l], encoded[l:]
w_cw = Field.decode_vec(encoded_w_cw)
correction_words.append((seed_cw, ctrl_cw, w_cw))
if len(encoded) != 0:
raise ERR_DECODE
return correction_words
Figure 26: Helper functions for IdpfPoplar.
Here, pack_bits() takes a list of bits, packs each group of eight
bits into a byte, in LSB to MSB order, padding the most significant
bits of the last byte with zeros as necessary, and returns the byte
array. unpack_bits() performs the reverse operation: it takes in a
byte array and a number of bits, and returns a list of bits,
extracting eight bits from each byte in turn, in LSB to MSB order,
and stopping after the requested number of bits. If the byte array
has an incorrect length, or if unused bits in the last bytes are not
zero, it throws an error.
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8.4. Instantiation
By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2)
and PrgSha3 (Section 6.2.1). This VDAF is suitable for any positive
value of BITS. Test vectors can be found in Appendix "Test Vectors".
9. Security Considerations
VDAFs have two essential security goals:
1. Privacy: An attacker that controls the network, the Collector,
and a subset of Clients and Aggregators learns nothing about the
measurements of honest Clients beyond what it can deduce from the
aggregate result.
2. Robustness: An attacker that controls the network and a subset of
Clients cannot cause the Collector to compute anything other than
the aggregate of the measurements of honest Clients.
Formal definitions of privacy and robustness can be found in
[DPRS23]. A VDAF is the core cryptographic primitive of a protocol
that achieves the above privacy and robustness goals. It is not
sufficient on its own, however. The application will need to assure
a few security properties, for example:
* Securely distributing the long-lived parameters, in particular the
verification key.
* 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.
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On their own, VDAFs do not mitigate Sybil attacks [Dou02]. In this
attack, the adversary observes a subset of input shares transmitted
by a Client it is interested in. It allows the input shares to be
processed, but corrupts and picks bogus measurements for the
remaining Clients. Applications can guard against these risks by
adding additional controls on report submission, such as client
authentication and rate limits.
VDAFs do not inherently provide differential privacy [Dwo06]. The
VDAF approach to private measurement can be viewed as complementary
to differential privacy, relying on non-collusion instead of
statistical noise to protect the privacy of the inputs. It is
possible that a future VDAF could incorporate differential privacy
features, e.g., by injecting noise before the sharding stage and
removing it after unsharding.
9.1. Requirements for the Verification Key
The Aggregators are responsible for exchanging the verification key
in advance of executing the VDAF. Any procedure is acceptable as
long as the following conditions are met:
1. To ensure robustness of the computation, the Aggregators MUST NOT
reveal the verification key to the Clients. Otherwise, a
malicious Client might be able to exploit knowledge of this key
to craft an invalid report that would be accepted by the
Aggregators.
2. To ensure privacy of the measurements, the Aggregators MUST
commit to the verification key prior to processing reports
generated by Clients. Otherwise, a malicious Aggregator may be
able to craft a verification key that, for a given report, causes
an honest Aggregator to leak information about the measurement
during preparation.
Meeting these conditions is required in order to leverage security
analysis in the framework of [DPRS23]. Their definition of
robustness allows the attacker, playing the role of a cohort of
malicious Clients, to submit arbitrary reports to the Aggregators and
eavesdrop on their communications as they process them. Security in
this model is achievable as long as the verification key is kept
secret from the attacker.
The privacy definition of [DPRS23] considers an active attacker that
controls the network and a subset of Aggregators; in addition, the
attacker is allowed to choose the verification key used by each
honest Aggregator over the course of the experiment. Security is
achievable in this model as long as the key is picked at the start of
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the experiment, prior to any reports being generated. (The model
also requires nonces to be generated at random; see Section 9.2
below.)
Meeting these requirements is relatively straightforward. For
example, the Aggregators may designate one of their peers to generate
the verification key and distribute it to the others. To assure
Clients of key commitment, the Clients and (honest) Aggregators could
bind reports to a shared context string derived from the key. For
instance, the "task ID" of DAP [DAP] could be set to the hash of the
verification key; then as long as honest Aggregators only consume
reports for the task indicated by the Client, forging a new key after
the fact would reduce to finding collisions in the underlying hash
function. (Keeping the key secret from the Clients would require the
hash function to be one-way.) However, since rotating the key
implies rotating the task ID, this scheme would not allow key
rotation over the lifetime of a task.
9.2. Requirements for the Nonce
The sharding and preparation steps of VDAF execution depend on a
nonce associated with the Client's report. To ensure privacy of the
underlying measurement, the Client MUST generate this nonce using a
CSPRNG. This is required in order to leverage security analysis for
the privacy definition of [DPRS23], which assumes the nonce is chosen
at random prior to generating the report.
Other security considerations may require the nonce to be non-
repeating. For example, to achieve differential privacy it is
necessary to avoid "over exposing" a measurement by including it too
many times in a single batch or across multiple batches. It is
RECOMMENDED that the nonce generated by the Client be used by the
Aggregators for replay protection.
9.3. Requirements for the Aggregation Parameters
As described in Section 4.3 and Section 5.3 respectively, DAFs and
VDAFs may impose restrictions on the re-use of input shares. This is
to ensure that correlated randomness provided by the Client through
the input share is not used more than once, which might compromise
confidentiality of the Client's measurements.
Protocols that make use of VDAFs therefore MUST call Vdaf.is_valid on
the set of all aggregation parameters used for a Client's input
share, and only proceed with the preparation and aggregation phases
if that function call returns True.
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9.4. Additional Privacy Considerations for Aggregation Parameters
Aggregating a batch of reports multiple times, each time with a
different aggregation parameter, could result in information leakage
beyond what is used by the application.
For example, when Poplar1 is used for heavy hitters, the Aggregators
learn not only the heavy hitters themselves, but also the prefix tree
(as defined in Section 8) computed along the way. Indeed, this
leakage is inherent to any construction that uses an IDPF
(Section 8.1) in the same way. Depending on the distribution of the
measurements, the prefix tree can leak a significant amount of
information about unpopular inputs. For instance, it is possible
(though perhaps unlikely) for a large set of non-heavy-hitter values
to share a common prefix, which would be leaked by a prefix tree with
a sufficiently small threshold.
The only known, general-purpose approach to mitigating this leakage
is via differential privacy.
TODO(issue #94) Describe (or point to some description of) the
central DP mechanism for Poplar described in [BBCGGI21].
9.5. Pseudorandom Generators and random oracles
The objects we describe in Section 6.2 share a common interface,
which we have called Prg. However, these are not necessarily all
modeled as cryptographic Pseudorandom Generators in the security
analyses of our protocols. Instead, most of them are modeled as
random oracles. For these use cases, we want to be conservative in
our assumptions, and hence prescribe PrgSha3 as the only RECOMMENDED
Prg instantiation.
The one exception is the PRG used in the Idpf implementation
IdpfPoplar Section 8.3. Here, a random oracle is not needed to prove
security, and hence a construction based on fixed-key AES
Section 6.2.2 can be used. However, as PrgFixedKeyAes128 has been
shown to be differentiable from a random oracle [GKWWY20], it is NOT
RECOMMENDED to use it anywhere else.
OPEN ISSUE: We may want to drop the common interface for PRGs and
random oracles. See issue #159.
10. IANA Considerations
A codepoint for each (V)DAF in this document is defined in the table
below. Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for
private use.
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+==========================+================+======+===============+
| Value | Scheme | Type | Reference |
+==========================+================+======+===============+
| 0x00000000 | Prio3Count | VDAF | Section 7.4.1 |
+--------------------------+----------------+------+---------------+
| 0x00000001 | Prio3Sum | VDAF | Section 7.4.2 |
+--------------------------+----------------+------+---------------+
| 0x00000002 | Prio3Histogram | VDAF | Section 7.4.3 |
+--------------------------+----------------+------+---------------+
| 0x00000003 to 0x00000FFF | reserved for | VDAF | n/a |
| | Prio3 | | |
+--------------------------+----------------+------+---------------+
| 0x00001000 | Poplar1 | VDAF | Section 8.4 |
+--------------------------+----------------+------+---------------+
| 0xFFFF0000 to 0xFFFFFFFF | reserved | n/a | n/a |
+--------------------------+----------------+------+---------------+
Table 16: Unique identifiers for (V)DAFs.
TODO Add IANA considerations for the codepoints summarized in
Table 16.
11. References
11.1. Normative References
[FIPS202] "SHA-3 Standard: Permutation-Based Hash and Extendable-
Output Functions", NIST FIPS PUB 202 , August 2015.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119,
DOI 10.17487/RFC2119, March 1997,
<https://www.rfc-editor.org/rfc/rfc2119>.
[RFC8174] Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC
2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174,
May 2017, <https://www.rfc-editor.org/rfc/rfc8174>.
[SP800-185]
"SHA-3 Derived Functions: cSHAKE, KMAC, TupleHash and
ParallelHash", NIST Special Publication 800-185 , December
2016.
11.2. Informative References
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[AGJOP21] Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A.
Polychroniadou, "Prio+: Privacy Preserving Aggregate
Statistics via Boolean Shares", 2021,
<https://ia.cr/2021/576>.
[BBCGGI19] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and
Y. Ishai, "Zero-Knowledge Proofs on Secret-Shared Data via
Fully Linear PCPs", CRYPTO 2019 , 2019,
<https://ia.cr/2019/188>.
[BBCGGI21] Boneh, D., Boyle, E., Corrigan-Gibbs, H., Gilboa, N., and
Y. Ishai, "Lightweight Techniques for Private Heavy
Hitters", IEEE S&P 2021 , 2021, <https://ia.cr/2021/017>.
[CGB17] Corrigan-Gibbs, H. and D. Boneh, "Prio: Private, Robust,
and Scalable Computation of Aggregate Statistics", NSDI
2017 , 2017,
<https://dl.acm.org/doi/10.5555/3154630.3154652>.
[DAP] Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood,
"Distributed Aggregation Protocol for Privacy Preserving
Measurement", Work in Progress, Internet-Draft, draft-
ietf-ppm-dap-03, 9 December 2022,
<https://datatracker.ietf.org/doc/html/draft-ietf-ppm-dap-
03>.
[Dou02] Douceur, J., "The Sybil Attack", IPTPS 2002 , 2002,
<https://doi.org/10.1007/3-540-45748-8_24>.
[DPRS23] Davis, H., Patton, C., Rosulek, M., and P. Schoppmann,
"Verifiable Distributed Aggregation Functions", n.d.,
<https://ia.cr/2023/130>.
[Dwo06] Dwork, C., "Differential Privacy", ICALP 2006 , 2006,
<https://link.springer.com/chapter/10.1007/11787006_1>.
[ENPA] "Exposure Notification Privacy-preserving Analytics (ENPA)
White Paper", 2021, <https://covid19-static.cdn-
apple.com/applications/covid19/current/static/contact-
tracing/pdf/ENPA_White_Paper.pdf>.
[EPK14] Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR:
Randomized Aggregatable Privacy-Preserving Ordinal
Response", CCS 2014 , 2014,
<https://dl.acm.org/doi/10.1145/2660267.2660348>.
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[GI14] Gilboa, N. and Y. Ishai, "Distributed Point Functions and
Their Applications", EUROCRYPT 2014 , 2014,
<https://link.springer.com/
chapter/10.1007/978-3-642-55220-5_35>.
[GKWWY20] "Better concrete security for half-gates garbling (in the
multi-instance setting)", CRYPTO 2020 , 2020,
<https://link.springer.com/
chapter/10.1007/978-3-030-56880-1_28>.
[OriginTelemetry]
"Origin Telemetry", 2020, <https://firefox-source-
docs.mozilla.org/toolkit/components/telemetry/collection/
origin.html>.
Acknowledgments
The security considerations in Section 9 are based largely on the
security analysis of [DPRS23]. Thanks to Hannah Davis and Mike
Rosulek, who lent their time to developing definitions and security
proofs.
Thanks to Henry Corrigan-Gibbs, Armando Faz-Hernández, Simon
Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, Xiao
Wang, and Christopher Wood for useful feedback on and contributions
to the spec.
Test Vectors
NOTE Machine-readable test vectors can be found at
https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/
test_vec.
Test vectors cover the generation of input shares and the conversion
of input shares into output shares. Vectors specify the verification
key, measurements, aggregation parameter, and any parameters needed
to construct the VDAF. (For example, for Prio3Sum, the user
specifies the number of bits for representing each summand.)
Byte strings are encoded in hexadecimal. To make the tests
deterministic, the random inputs of randomized algorithms were fixed
to the byte sequence starting with 0, incrementing by 1, and wrapping
at 256:
0, 1, 2, ..., 255, 0, 1, 2, ...
Prio3Count
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verify_key: "000102030405060708090a0b0c0d0e0f"
upload_0:
measurement: 1
nonce: "000102030405060708090a0b0c0d0e0f"
public_share: >-
input_share_0: >-
e7a225b76420dd6dd0682380363bd782c8ca9ace6e7abc559dd873bafb503e3cc8b7
9f3b2b8b0a14676172e46be2ce2f
input_share_1: >-
000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f
round_0:
prep_share_0: >-
56df20acd68725eab85e44c706629df555d635a5b899df767fe28effd0c45f60
prep_share_1: >-
ab20df532878da1518b04178e08a39d0d40fdbfe1876e479dc5e94b43afc68d5
prep_message: >-
out_share_0:
- e7a225b76420dd6d
out_share_1:
- 1b5dda489adf2292
agg_share_0: >-
e7a225b76420dd6d
agg_share_1: >-
1b5dda489adf2292
agg_result: 1
Prio3Sum
bits: 8
verify_key: "000102030405060708090a0b0c0d0e0f"
upload_0:
measurement: 100
nonce: "000102030405060708090a0b0c0d0e0f"
public_share: >-
da6f5f000d95651e628464f5d086fde6a544464ac776c3812e2299cce1d29095
input_share_0: >-
2d97ed6ebc5f6748db3017211c7a31ceeba9500b8317a7f61578b85c08a7f7d95c63
7bb14b95c062c74bd5b08625bdafbfd91fd01d22291b22b76e54a9be1f7e7015a396
ae0d6d034b3d22795b3d98db2a9aa51b4ea56c247496d380367deb786a9b34d5cb88
f0f43e5cb4ab71910c8fa20b0b518f7dd9ce11ec9bf4ce460c81d3e9db05c34e0f05
14d53f5eadea231dbc1fc575ad52e01622671eaf49a66533a6511710eeea024df0da
7e7d21451ff5fd9d38e36eee8b78bf7c0767b91e747c6b80d6ca6b726286edebeae4
ff377040d87f84288f4d0e5960e27b79387caef816c27481be166b71d38d1695e5e5
061c26f0f667f8fff3133c2ae3ac1853ac542a9af0b5e422764609abd882dedf6aa8
c835228bbc0cb99a695e81c45a2876a961af0c680662de4400ec96004c45bdc32726
68b01d7f861f2e03fc18adcc352bdeb7b06b39c2de749da45e9b48f46c45a390cb59
455dbf3e8be5938203def1c45af5eb5bf2037be005768d74faf11efd255b699f5ae9
bce8b3bf4688907d11ca89e7609b77c04abde845f63091b934017cf9a83f80eb86fd
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aa063630f86ee982d812062db0d1d760d84cfebbf99927cad516dd028bf2f9e9b7d2
4aa0bd9c0cb16a46c469eacfe1c48b8d7bf626e5d3eedb3f3cc4972f14a4ba605565
ec795dd9e68af4ac6185f370f1f77610c64b04d295007b34ed59ed7bceace7b9dea8
8346a2ee47dfec257ede5cd6985c34abaf5983d067786a695ae1039b631555bbe876
7baeedfe4f30df6b47479543c58a87e0aaf371234fc9094c4b2b79d025229634a3fd
f68a36be0e8661a4f043504b7025f03f02e57704b4e410c60e69e88050d0c33ea71b
c37f306c225f457fbab4929c65ba9401dc57a2dc31bd6d3f53099a00303132333435
363738393a3b3c3d3e3f
input_share_1: >-
000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021
22232425262728292a2b2c2d2e2f
round_0:
prep_share_0: >-
7ae72bff1c063181bfea222dd77bcd40c5d15f62174d4af984d728effb82edd302
74c927a9ffa51b90c167c60b40c3cdda6f5f000d95651e628464f5d086fde6
prep_share_1: >-
8718d400e3f9ce7e2415ddd2288432bf85623812cb8a7cdfed57f5a417c422e25e
84a67ab22c0b8ef7b0500c366feb95a544464ac776c3812e2299cce1d29095
prep_message: >-
4f351791f4d706a31c54ce699ef1af97
out_share_0:
- a18d0c7e2a74470727d4f628ff934e52
out_share_1:
- c472f381d58bb8f8bc2b09d7006cb1ad
agg_share_0: >-
a18d0c7e2a74470727d4f628ff934e52
agg_share_1: >-
c472f381d58bb8f8bc2b09d7006cb1ad
agg_result: 100
Prio3Histogram
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buckets: [1, 10, 100]
verify_key: "000102030405060708090a0b0c0d0e0f"
upload_0:
measurement: 50
nonce: "000102030405060708090a0b0c0d0e0f"
public_share: >-
5e015517900cfc204138c24f808ddf4ee85eca87ba246cd715d116195172e500
input_share_0: >-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_share_1: >-
000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021
22232425262728292a2b2c2d2e2f
round_0:
prep_share_0: >-
ac872edaaaa532de61bcd63b168e522230b35064b09b6b35c1352bc354560fdeac
684da645d3e509c6db8ba31f58dabb5e015517900cfc204138c24f808ddf4e
prep_share_1: >-
5578d125555acd21824329c4e971adddc07187f0f25a30ceba8407513f60f8e3d1
5c17270c5631a718b3b4f5702aff54e85eca87ba246cd715d116195172e500
prep_message: >-
890bfbdf0c619d5c853a92e28bce41b3
out_share_0:
- c269f2189db431ca2ebe7230bacd692a
- c02a7790818681b6ee5ecbd509587b29
- 76c7897326840bb70083ee4df8ffe3dc
- 65dbd71f56735a7dcc3a150da8e77171
out_share_1:
- 3f960de7624bce35b5418dcf453296d5
- 41d5886f7e797e49f5a0342af6a784d6
- 8c38768cd97bf448e37c11b207001c23
- 9c2428e0a98ca58217c5eaf257188e8e
agg_share_0: >-
c269f2189db431ca2ebe7230bacd692ac02a7790818681b6ee5ecbd509587b2976c789
7326840bb70083ee4df8ffe3dc65dbd71f56735a7dcc3a150da8e77171
agg_share_1: >-
3f960de7624bce35b5418dcf453296d541d5886f7e797e49f5a0342af6a784d68c3876
8cd97bf448e37c11b207001c239c2428e0a98ca58217c5eaf257188e8e
agg_result: [0, 0, 1, 0]
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Poplar1
Sharding
bits: 4
upload_0:
measurement: 13
nonce: "000102030405060708090a0b0c0d0e0f"
public_share: >-
8b332c8b09b84f5801ad1cfd30b0df3fba69d11ddf6065a9cdb87cb31579ba0443a4
82f4ed21ac079b96f5944d1669a1159406148c00878ac9953bcfbadc0273042653fd
0fae8c43ce746fd31f248d1cbf5d1bc01fe02a883571cf38a1a4c10f54d41979b6fc
35ef1fb051f968ba4397b8e61a81d835fa7ee54b949a35abf76c13e0a9daf90f1528
c40db5904b6e24d70116538b28c16d9dbfee45000fe14db45cafbb76768b28250e74
6af691f4bb1619
input_share_0: >-
000102030405060708090a0b0c0d0e0f202122232425262728292a2b2c2d2e2f0734
d55dcb0dec9599b96f38ca694d46056dbe5bb24ab41ca10faaa8249839632c6503c9
795300a160e8c378762f3b329e9ebe6a081eaed2d72aaff0aeb52415211c314f609a
0377e66b7be10a3d8d4e75890da4c93f62172bc07e54be31c060c5970b52af8e8633
d63560f59fa0bb36
input_share_1: >-
101112131415161718191a1b1c1d1e1f303132333435363738393a3b3c3d3e3f9558
af2229b50110d88e17b5b3a81902194a5eab5f8ad69f5c061f16ac1e92deadc3c6ed
32714a3756fa68e580d523d6c0e4b92ba2277033a7adb060e7a2ee3b2e44591d7c99
b1cbcbfc58e4ae9ef7613a1ff24e53201e7264940bed2cd318e8e80ab7aae630f342
911d75c85980691d
Preparation, Aggregation, and Unsharding
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verify_key: "000102030405060708090a0b0c0d0e0f"
agg_param: (0, [0, 1])
upload_0:
round_0:
prep_share_0: >-
a2d369b7ee44e5f8b903e889b37ea7ef4d29ee36d1abdb13
prep_share_1: >-
31182ff6e075b925a06c802eac3b14c1ad66ce991ca2ffc2
prep_message: >-
d2eb98add0ba9e1e587068b860babbb0fa8fbcd0ed4ddbd6
round_1:
prep_share_0: >-
f4445265dc801ab5
prep_share_1: >-
0dbbad9a227fe54a
prep_message: >-
out_share_0:
- d47170d4f804449e
- e16add75c1d3f0c4
out_share_1:
- 2d8e8f2b06fbbb61
- 2195228a3d2c0f3b
agg_share_0: >-
d47170d4f804449ee16add75c1d3f0c4
agg_share_1: >-
2d8e8f2b06fbbb612195228a3d2c0f3b
agg_result: [0, 1]
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verify_key: "000102030405060708090a0b0c0d0e0f"
agg_param: (1, [0, 1, 2, 3])
upload_0:
round_0:
prep_share_0: >-
94352b588fbd978be7ef2cacb83b1b64c1d11af699253242
prep_share_1: >-
4a3529a2b34159be1666064ba0a3a9cc4e72223ef6d0ac79
prep_message: >-
dd6a54fa43fff049fc5533f759dfc4300f443d3490f6debb
round_1:
prep_share_0: >-
3dc2833b1f9010ee
prep_share_1: >-
c43d7cc4df6fef11
prep_message: >-
out_share_0:
- c03ede9261e7f441
- 8ef9fde3f50eb967
- de3bb4acf0b9dff8
- 24687334644cc809
out_share_1:
- 41c1216d9d180bbe
- 7306021c09f14698
- 23c44b530e462007
- de978ccb9ab337f6
agg_share_0: >-
c03ede9261e7f4418ef9fde3f50eb967de3bb4acf0b9dff824687334644cc809
agg_share_1: >-
41c1216d9d180bbe7306021c09f1469823c44b530e462007de978ccb9ab337f6
agg_result: [0, 0, 0, 1]
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verify_key: "000102030405060708090a0b0c0d0e0f"
agg_param: (2, [0, 2, 4, 6])
upload_0:
round_0:
prep_share_0: >-
b640798e11bc37ac1f4981c99f0de15569b2bcc187d2f0f2
prep_share_1: >-
37a720e1739caca1b39b8fe97ce96e09277eee9cc694c2cf
prep_message: >-
ece7996f8658e44dd2e410b31cf74f5f8f30ab5e4f67b3c2
round_1:
prep_share_0: >-
52a1848f964385f8
prep_share_1: >-
af5e7b7068bc7a07
prep_message: >-
out_share_0:
- 352395a6c5e5804e
- f9b61274892e76dd
- b27e3eb4accbaa0a
- 3b1c4a7f78d59935
out_share_1:
- ccdc6a59391a7fb1
- 0849ed8b75d18922
- 4f81c14b523455f5
- c7e3b580862a66ca
agg_share_0: >-
352395a6c5e5804ef9b61274892e76ddb27e3eb4accbaa0a3b1c4a7f78d59935
agg_share_1: >-
ccdc6a59391a7fb10849ed8b75d189224f81c14b523455f5c7e3b580862a66ca
agg_result: [0, 0, 0, 1]
verify_key: "000102030405060708090a0b0c0d0e0f"
agg_param: (3, [1, 3, 5, 7, 9, 13, 15])
upload_0:
round_0:
prep_share_0: >-
08efd12285f6e61599b2f9d941577fd843597d3b15f747d15b0b02daf6147752c6
a7a72723fa0cad98c7c38a0716ff8833dd7f97d40ed374f963dd2bced921614ccd
4dca993e6c4f5d1ab0ce9bf37df76b41e2c111868322dfc39b3145dfc832
prep_share_1: >-
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prep_message: >-
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Barnes, et al. Expires 14 September 2023 [Page 96]
Internet-Draft VDAF March 2023
round_1:
prep_share_0: >-
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prep_share_1: >-
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prep_message: >-
out_share_0:
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out_share_1:
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- 63d69b576328f28459a44ad0f0ea04d24823e3acb28bc1c80ddc08b47b763722
agg_share_0: >-
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agg_share_1: >-
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agg_result: [0, 0, 0, 0, 0, 1, 0]
Authors' Addresses
Richard L. Barnes
Cisco
Email: rlb@ipv.sx
Barnes, et al. Expires 14 September 2023 [Page 97]
Internet-Draft VDAF March 2023
David Cook
ISRG
Email: divergentdave@gmail.com
Christopher Patton
Cloudflare
Email: chrispatton+ietf@gmail.com
Phillipp Schoppmann
Google
Email: schoppmann@google.com
Barnes, et al. Expires 14 September 2023 [Page 98]