InternetDraft  VDAF  July 2024 
Barnes, et al.  Expires 9 January 2025  [Page] 
 Workgroup:
 CFRG
 InternetDraft:
 draftirtfcfrgvdaf10
 Published:
 Intended Status:
 Informational
 Expires:
Verifiable Distributed Aggregation Functions
Abstract
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multiparty protocols for computing aggregate statistics over user measurements. These protocols are designed to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers to detect if a malicious or misconfigured client submitted an measurement that would result in an invalid aggregate result.¶
Discussion Venues
This note is to be removed before publishing as an RFC.¶
Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.¶
Source for this draft and an issue tracker can be found at https://github.com/cjpatton/vdaf.¶
Status of This Memo
This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current InternetDrafts is at https://datatracker.ietf.org/drafts/current/.¶
InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress."¶
This InternetDraft will expire on 9 January 2025.¶
Copyright Notice
Copyright (c) 2024 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/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document.¶
1. Introduction
(TO BE REMOVED BY RFC EDITOR: The source for this draft and and the reference implementation can be found at https://github.com/cfrg/draftirtfcfrgvdaf.)¶
The ubiquity of the Internet makes it an ideal platform for measurement of largescale 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.¶
This problem is often formulated in terms 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 wellknown distribution and adds it to their measurement before submitting to the aggregation server. The aggregation server then adds up the noisy measurements, and because it knows the distribution from which the noise was sampled, it can estimate the true sum with reasonable accuracy.¶
However, even when noise is added to the measurements, collecting them in the clear still reveals a significant amount of information to the collector. On the one hand, depending on the "amount" of noise a client adds to its measurement, it may be possible for a curious collector to make a reasonable guess of the measurement's true value. On the other hand, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems relying solely on a DP mechanism must strike a delicate balance between privacy and utility.¶
The ideal goal for a privacypreserving measurement system is that of secure multiparty computation (MPC): No participant in the protocol should learn anything about an individual measurement beyond what it can deduce from the differentially private aggregate [MPRV09]. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of delegated MPC protocols that can be used to achieve this goal.¶
VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of noncolluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no measurement is ever accessible to any party besides the client that submitted it. VDAFs can also be composed with various DP mechanisms, thereby ensuring the aggregate result does not leak too much information about any one measurmment. At the same time, VDAFs are "verifiable" in the sense that malformed measurements that would otherwise garble the result of the computation can be detected and removed from the set of measurements. We refer to this property as "robustness".¶
The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA].¶
The VDAF abstraction laid out in Section 5 represents a class of multiparty protocols for privacypreserving measurement proposed in the literature. These protocols vary in their operational and security requirements, sometimes in subtle but consequential ways. This document therefore has two important goals:¶

Providing higherlevel protocols like [DAP] with a simple, uniform interface for accessing privacypreserving measurement schemes, documenting relevant operational and security requirements, and specifying constraints for safe usage:¶

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

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

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


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 privacypreserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶

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

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

Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result.¶
The difficult part of this system is ensuring that the servers hold shares of a valid, aggregatable value, e.g., the measurement is an integer in a specific range. Thus Prio specifies a multiparty protocol for accomplishing this task.¶
In Section 7 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶


More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar for solving the
t
heavyhitters problem in a privacypreserving manner. Here each client holds a bitstring of lengthn
, and the goal of the aggregation servers is to compute the set of strings that occur at leastt
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 bitstring of length shorter than or equal ton
. As a result of this query, each of the servers has an additive share of a bit indicating whether the string is a prefix of the client's string. The protocol also specifies a multiparty computation for verifying that at most one string among a set of candidates is a prefix of the client's string.¶In Section 8 we describe a VDAF called Poplar1 that implements this functionality.¶
Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the clientgenerated 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.¶
1.1. Change Log
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
10:¶

Define Prio3MultihotCountVec, a variant of Prio3 for aggregating bit vectors with bounded weight.¶

FLP: Allow the output of the circuit to be a vector. This makes it possible to skip joint randomness derivation in more cases.¶

Poplar1: On the first round of preparation, handle
None
as an error. Previously this message was interpreted as a length3 vector of zeros.¶ 
Prio3: Move specification of the field from the FLP validity circuit to the VDAF itself.¶

Clarify the extent to which the attacker controls the network in our threat models for privacy and robustness.¶

Clean up various aspects of the code, including: follow existing objectoriented programming patterns for Python more closely; make the type hints enforceable; and avoid shadowing variables.¶

IDPF: Add guidance for encoding byte strings as indices.¶
09:¶

Poplar1: Make prefix tree traversal stricter by requiring each node to be a child of a node that was already visited. This change is intended to make it harder for a malicious Aggregator to steer traversal towards nonheavyhitting measurements.¶

Prio3: Add more explicit guidance for choosing the field size.¶

IDPF: Define extractability and clarify (un)safe usage of intermediate prefix counts. Accordingly, add text ensuring public share consistency to security considerations.¶
08:¶

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

IdpfPoplar: Modify
extend()
by stealing each control bit from its corresponding seed. This improves performance by reducing the number of AES calls per level from 3 to 2. The cost is a slight reduction in the concrete privacy bound. (*)¶ 
Prio3: Add support for generating and verifying mutliple proofs per measurement. This enables a tradeoff between communication cost and runtime: if more proofs are used, then a smaller field can be used without impacting robustness. (*)¶

Replace SHAKE128 with TurboSHAKE128. (*)¶
07:¶

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

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

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

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

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

Daf: Align with Vdaf by adding a nonce to
shard()
andprep()
.¶ 
Vdaf: Have
prep_init()
compute the first prep share. This change is intended to simplify the interface by making the input toprep_next()
not optional.¶ 
Prio3: Split sharding into two auxiliary functions, one for sharding with joint randomness and another without. This change is intended to improve readability.¶

Fix bugs in the pingpong interface discovered after implementing it.¶
06:¶

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

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

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

Daf, Vdaf, Idpf: Add parameter
RAND_SIZE
that specifies the number of random bytes consumed by the randomized algorithm (shard()
for Daf and Vdaf andgen()
for Idpf).¶
05:¶

IdpfPoplar: Replace PrgSha3 with PrgFixedKeyAes128, a fixedkey mode for AES128 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 keygeneration 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 keygeneration 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 littleendian byteorder for field element encoding. (*)¶

Poplar1: Move the last step of sketch evaluation from
prep_next()
toprep_shares_to_prep()
.¶
04:¶

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

Vdaf: Pass the nonce to the sharding algorithm.¶

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

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

Replace PrgAes128 with PrgSha3, an implementation of the Prg interface based on SHA3, and use the new scheme as the default. Accordingly, replace Prio3Aes128Count with Prio3Count, Poplar1Aes128 with Poplar1, and so on. SHA3 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 defenseindepth by ensuring the Aggregators reject the report if the nonce does not match what the Client used for sharding. (*)¶

Poplar1: Clarify that the aggregation parameter encoding is OPTIONAL. Accordingly, update implementation considerations around crossaggregation 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 FFTfriendly fields.¶
02:¶

Complete the initial specification of Poplar1.¶

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

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

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

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

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

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

Define new field Field255.¶
01:¶

Require that VDAFs specify serialization of aggregate shares.¶

Define Distributed Aggregation Functions (DAFs).¶

Prio3: Move proof verifier check from
prep_next()
toprep_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. The type variable F
is used in signatures to
signify any type that is a subclass of Field
. 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
bytestring constants. When comprised of printable ASCII characters, they are
written as Python 3 bytestring literals (e.g., b'some constant string'
).¶
A global constant VERSION
of type int
is defined, which algorithms are
free to use as desired. Its value SHALL be 8
.¶
This document describes algorithms for multiparty 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: int) > bytes
returns an array of zero bytes. The length ofoutput
MUST belen
.¶ 
gen_rand(len: int) > bytes
returns an array of random bytes generated by a cryptographically secure pseudorandom number generator (CSPRNG). The length ofoutput
MUST belen
.¶ 
byte(int: int) > bytes
returns the representation ofint
as a byte string. The value ofint
MUST be in[0,256)
.¶ 
concat(parts: list[bytes]) > bytes
returns the concatenation of the input byte strings, i.e.,parts[0]  ...  parts[len(parts)1]
.¶ 
front(length: int, vec: list[Any]) > (list[Any], list[Any])
splitsvec
into two vectors, where the first vector is made up of the firstlength
elements of the input. I.e.,(vec[:length], vec[length:])
.¶ 
xor(left: bytes, right: bytes) > bytes
returns the bitwise XOR ofleft
andright
. An exception is raised if the inputs are not the same length.¶ 
to_be_bytes(val: int, length: int) > bytes
convertsval
to bigendian bytes; its value MUST be in range[0, 2^(8*length))
. Functionfrom_be_bytes(encoded: bytes) > int
computes the inverse.¶ 
to_le_bytes(val: int, length: int) > bytes
convertsval
to littleendian bytes; its value MUST be in range[0, 2^(8*length))
. Functionfrom_le_bytes(encoded: bytes) > int
computes the inverse.¶ 
next_power_of_2(n: int) > int
returns the smallest integer greater than or equal ton
that is also a power of two.¶ 
additive_secret_share(vec: list[Field], num_shares: int, field: type) > list[list[Field]]
takes a vector of field elements and returns multiple vectors of the same length, such that they all add up to the input vector, and each proper subset of the vectors are indistinguishable from random.¶ 
cast(typ: type, val: object) > object
returns the input value unchanged. This is only present to assist with static analysis of the Python code. Type checkers will ignore the inferred type of the input value, and assume the output value has the given type.¶ 
range(stop)
orrange(start, stop[, step])
is the range function from the Python standard library. The oneargument form returns the integers from zero (inclusive) tostop
, exclusive. The two and threeargument forms allow overriding the start of the range and overriding the step between successive output values.¶
3. Overview
In a DAF or VDAFbased private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶

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

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

Output shares are in onetoone correspondence with the input shares.¶

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

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


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

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 noncollusion 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 noncollusion 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 clientserver protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to the higherlevel 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:¶

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: int

Algorithm identifier for this DAF, in range(2**32) . 
SHARES: int

Number of input shares into which each measurement is sharded. 
NONCE_SIZE: int

Size of the nonce passed by the application. 
RAND_SIZE: int

Size of the random byte string passed to sharding algorithm. 
Measurement

Type of each measurement. 
PublicShare

Type of each public share. 
InputShare

Type of each input share. 
AggParam

Type of aggregation parameter. 
OutShare

Type of each output share. 
AggShare

Type of the aggregate share. 
AggResult

Type of the aggregate result. 
These types define the inputs and outputs of DAF methods at various stages of
the computation. Some of these values need to be written to the network in
order to carry out the computation. In particular, it is RECOMMENDED that
concrete instantiations of the Daf
interface specify a method of encoding the
PublicShare
, InputShare
, and AggShare
.¶
Each DAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 23.¶
4.1. Sharding
In order to protect the privacy of its measurements, a DAF Client shards its
measurements into a sequence of input shares. The shard
method is used for
this purpose.¶

daf.shard(measurement: Measurement, nonce: bytes, rand: bytes) > tuple[PublicShare, list[InputShare]]
is the randomized sharding algorithm run by each Client that consumes a measurement and nonce and produces a "public share" distributed to each of the Aggregate and a corresponding sequence of input shares, one for each Aggregator.¶Preconditions:¶

nonce
MUST have length equal toNONCE_SIZE
and MUST be generated using a CSPRNG.¶ 
rand
consists of the random bytes consumed by the algorithm. It MUST have length equal toRAND_SIZE
and MUST be generated using a CSPRNG.¶
Postconditions:¶

The number of input shares MUST equal
SHARES
.¶

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: int, agg_param: AggParam, nonce: bytes, public_share: PublicShare, input_share: InputShare) > OutShare
is the deterministic preparation algorithm. It takes as input the public share and one of the input shares generated by a Client, the Aggregator's unique identifier, the aggregation parameter selected by the Collector, and a nonce and returns an output share.¶Preconditions:¶
4.3. Validity of Aggregation Parameters
Concrete DAFs implementations MAY impose certain restrictions for input shares
and aggregation parameters. Protocols using a DAF MUST ensure that for each
input share and aggregation parameter agg_param
, daf.prep
is only called if
daf.is_valid(agg_param, previous_agg_params)
returns True, where
previous_agg_params
contains all aggregation parameters that have previously
been used with the same input share.¶
DAFs MUST implement the following function:¶

daf.is_valid(agg_param: AggParam, previous_agg_params: set[AggParam]) > bool
: Checks if theagg_param
is compatible with all elements ofprevious_agg_params
.¶
4.4. Aggregation
Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result:¶

daf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) > AggShare
is the deterministic aggregation algorithm. It is run by each Aggregator a set of recovered output shares.¶
For simplicity, we have written this algorithm in a "oneshot" 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 elementwise. Such is the case for Prio3 Section 7 and Poplar1 Section 8.¶
4.5. Unsharding
After the Aggregators have aggregated a sufficient number of output shares, each sends its aggregate share to the Collector, who runs the following algorithm to recover the following output:¶

daf.unshard(agg_param: AggParam, agg_shares: list[AggShare], num_measurements: int) > AggResult
is run by the Collector in order to compute the aggregate result from the Aggregators' shares. The length ofagg_shares
MUST beSHARES
.num_measurements
is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic.¶
4.6. Execution of a DAF
Securely executing a DAF involves emulating the following procedure.¶
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 pointtopoint channels. In reality, these channels need to be instantiated by some "wrapper protocol", such as [DAP], that realizes these channels using suitable cryptographic mechanisms. Moreover, some fraction of the Aggregators (or Clients) may be malicious and diverge from their prescribed behaviors. Section 9 describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.¶
5. Definition of VDAFs
Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Clientgenerated 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.¶
Parameter  Description 

ID

Algorithm identifier for this VDAF. 
VERIFY_KEY_SIZE

Size (in bytes) of the verification key (Section 5.2). 
RAND_SIZE

Size of the random byte string passed to sharding algorithm. 
NONCE_SIZE

Size (in bytes) of the nonce. 
ROUNDS

Number of rounds of communication during the Preparation stage (Section 5.2). 
SHARES

Number of input shares into which each measurement is sharded (Section 5.1). 
Measurement

Type of each measurement. 
PublicShare

Type of each public share. 
InputShare

Type of each input share. 
AggParam

Type of aggregation parameter. 
OutShare

Type of each output share. 
AggShare

Type of the aggregate share. 
AggResult

Type of the aggregate result. 
PrepState

Aggregator's state during preparation. 
PrepShare

Type of each prep share. 
PrepMessage

Type of each prep message. 
Some of these values need to be written to the network in order to carry out
the computation. In particular, it is RECOMMENDED that concrete instantiations
of the Vdaf
interface specify a method of encoding the PublicShare
,
InputShare
, AggShare
, PrepShare
, and PrepMessage
.¶
Each VDAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 23. The following
method is defined for every VDAF:¶
def domain_separation_tag(self, usage: int) > bytes: """ Format domain separation tag for this VDAF with the given usage. Preconditions:  `usage` in `range(2**16)` """ return format_dst(0, self.ID, usage)¶
It is used to construct a domain separation tag for an instance of Xof
used by
the VDAF. (See Section 6.2.)¶
5.1. Sharding
Sharding transforms a measurement into input shares as it does in DAFs (cf. Section 4.1); in addition, it takes a nonce as input and produces a public share:¶

vdaf.shard(measurement: Measurement, nonce: bytes, rand: bytes) > tuple[PublicShare, list[InputShare]]
is the randomized sharding algorithm run by each Client that consumes a measurement and nonce and produces a public share distributed to each of the Aggregate and a 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)¶Preconditions:¶

nonce
MUST have length equal toNONCE_SIZE
and MUST be generated using a CSPRNG. (See Section 9 for details.)¶ 
rand
consists of the random bytes consumed by the algorithm. It MUST have length equal toRAND_SIZE
and MUST be generated using a CSPRNG.¶
Postconditions:¶

The number of input shares MUST equal
SHARES
.¶

5.2. Preparation
To recover and verify output shares, the Aggregators interact with one another
over ROUNDS
rounds. Prior to each round, each Aggregator constructs an
outbound message. Next, the sequence of outbound messages is combined into a
single message, called a "preparation message", or "prep message" for short.
(Each of the outbound messages are called "preparationmessage shares", or
"prep shares" for short.) 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 prep message. Its output in the last round is its output share and its output in each of the preceding rounds is a prep share.¶
This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing the measurement shares for aggregation.¶
To facilitate the preparation process, a concrete VDAF implements the following methods:¶

vdaf.prep_init(verify_key: bytes, agg_id: int, agg_param: AggParam, nonce: bytes, public_share: PublicShare, input_share: InputShare) > tuple[PrepState, PrepShare]
is the deterministic preparationstate 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 (nonce
, see Figure 7), the public share (public_share
), and one of the input shares generated by the Client (input_share
). Its output is the Aggregator's initial preparation state and initial prep share.¶It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key prior to generating and processing reports. (See Section 9 for details.)¶
Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1.¶
Preconditions:¶

vdaf.prep_next(prep_state: PrepState, prep_msg: PrepMessage) > Union[tuple[PrepState, PrepShare], OutShare]
is the deterministic preparationstate update algorithm run by each Aggregator. It updates the Aggregator's preparation state (prep_state
) and returns either its next preparation state and its message share for the current round or, if this is the last round, its output share. An exception is raised if a valid output share could not be recovered. The input of this algorithm is the inbound preparation message.¶ 
vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: list[PrepShare]) > PrepMessage
is the deterministic preparationmessage preprocessing algorithm. It combines the prep shares generated by the Aggregators in the previous round into the prep message consumed by each in the next round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (ROUNDS
), a VDAF instance
defines the state of the Aggregator after each round.¶
The preparationstate update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
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: list[AggParam]) > bool
: Checks if theagg_param
is compatible with all elements ofprevious_agg_params
.¶
5.4. Aggregation
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4):¶

vdaf.aggregate(agg_param: AggParam, out_shares: list[OutShare]) > AggShare
is the deterministic aggregation algorithm. It is run by each Aggregator over the output shares it has computed for a batch of measurements.¶
The data flow for this stage is illustrated in Figure 3. Here again, we have the aggregation algorithm in a "oneshot" form, where all shares for a batch are provided at the same time. VDAFs typically also support a "streaming" form, where shares are processed one at a time.¶
5.5. Unsharding
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.5):¶

vdaf.unshard(agg_param: AggParam, agg_shares: list[AggShare], num_measurements: int) > AggResult
is run by the Collector in order to compute the aggregate result from the Aggregators' shares. The length ofagg_shares
MUST beSHARES
.num_measurements
is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic.¶
5.6. Execution of a VDAF
Secure execution of a VDAF involves simulating the following procedure.¶
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.¶
5.7. Communication Patterns for Preparation
In each round of preparation, each Aggregator writes a prep share to some
broadcast channel, which is then processed into the prep message using the
public prep_shares_to_prep()
algorithm and broadcast to the Aggregators to
start the next round. In this section we describe some approaches to realizing
this broadcast channel functionality in protocols that use VDAFs.¶
The state machine of each Aggregator is shown in Figure 8.¶
State transitions are made when the state is acted upon by the host's local
inputs and/or messages sent by the peers. The initial state is Start
. The
terminal states are Rejected
, which indicates that the report cannot be
processed any further, and Finished(out_share)
, which indicates that the
Aggregator has recovered an output share out_share
.¶
class State: pass class Start(State): pass class Continued(State): def __init__(self, prep_state): self.prep_state = prep_state class Finished(State): def __init__(self, output_share): self.output_share = output_share class Rejected(State): def __init__(self): pass¶
Note that there is no representation of the Start
state as it is never
instantiated in the pingpong topology.¶
For convenience, the methods described in this section are defined in terms of
opaque byte strings. A compatible Vdaf
MUST specify methods for encoding
public shares, input shares, prep shares, prep messages, and aggregation
parameters. Minimally:¶

vdaf.decode_public_share(encoded: bytes) > PublicShare
decodes a public share.¶ 
vdaf.decode_input_share(agg_id: int, encoded: bytes) > InputShare
decodes an input share, using the aggregator ID as optional context.¶ 
vdaf.encode_prep_share(prep_share: PrepShare) > bytes
encodes a prep share.¶ 
vdaf.decode_prep_share(prep_state: PrepState, encoded: bytes) > PrepShare
decodes a prep share, using the prep state as optional context.¶ 
vdaf.encode_prep_msg(prep_msg: PrepMessage) > bytes
encodes a prep message.¶ 
vdaf.decode_prep_msg(prep_state: PrepState, encoded: bytes) > PrepMessage
decodes a prep message, using the prep state as optional decoding context.¶ 
vdaf.decode_agg_param(encoded: bytes) > AggParam
decodes an aggregation parameter.¶ 
vdaf.encode_agg_param(agg_param: AggParam) > bytes
encodes an aggregation parameter.¶
Implementations of Prio3 and Poplar1 MUST use the encoding scheme specified in Section 7.2.7 and Section 8.2.6 respectively.¶
5.8. PingPong Topology (Only Two Aggregators)
For VDAFs with precisely two Aggregators (i.e., SHARES == 2
), the following
"ping pong" communication pattern can be used. It is compatible with any
request/response transport protocol, such as HTTP.¶
Let us call the initiating party the "Leader" and the responding party the "Helper". The highlevel idea is that the Leader and Helper will take turns running the computation locally until input from their peer is required:¶

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

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

In general, each request includes the Leader's prep share for the previous round and/or the prep message for the current round; correspondingly, each response consists of the prep message for the current round and the Helper's prep share for the next round.¶
The Aggregators proceed in this pingponging fashion until a step of the
computation fails (indicating the report is invalid and should be rejected) or
preparation is completed. All told there there are ceil((ROUNDS+1)/2)
requests sent.¶
Each message in the pingpong protocol is structured as follows (expressed in TLS syntax as defined in Section 3 of [RFC8446]):¶
enum { initialize(0), continue(1), finish(2), (255) } MessageType; struct { MessageType type; select (Message.type) { case initialize: opaque prep_share<0..2^321>; case continue: opaque prep_msg<0..2^321>; opaque prep_share<0..2^321>; case finish: opaque prep_msg<0..2^321>; }; } Message;¶
These messages are used to transition between the states described in Section 5.7. They are encoded and decoded to or from byte buffers as described Section 3 of [RFC8446]) using the following routines:¶

encode_ping_pong_message(message: Message) > bytes
encodes aMessage
into an opaque byte buffer.¶ 
decode_pong_pong_message(encoded: bytes) > Message
decodes an opaque byte buffer into aMessage
, raising an error if the bytes are not a valid encoding.¶
The Leader's initial transition is computed with the following procedure:¶
def ping_pong_leader_init( vdaf, vdaf_verify_key: bytes, agg_param: bytes, nonce: bytes, public_share: bytes, input_share: bytes, ) > tuple[State, bytes]: try: (prep_state, prep_share) = vdaf.prep_init( vdaf_verify_key, 0, vdaf.decode_agg_param(agg_param), nonce, vdaf.decode_public_share(public_share), vdaf.decode_input_share(0, input_share), ) outbound = Message.initialize( vdaf.encode_prep_share(prep_share)) return ( Continued(prep_state), encode_ping_pong_message(outbound), ) except: return (Rejected(), None)¶
The output is the State
to which the Leader has transitioned and an encoded
Message
. If the Leader's state is Rejected
, then processing halts.
Otherwise, if the state is Continued
, then processing continues.¶
The Leader sends the outbound message to the Helper. The Helper's initial transition is computed using the following procedure:¶
def ping_pong_helper_init( vdaf, vdaf_verify_key: bytes, agg_param: bytes, nonce: bytes, public_share: bytes, input_share: bytes, inbound_encoded: bytes, ) > tuple[State, bytes]: try: (prep_state, prep_share) = vdaf.prep_init( vdaf_verify_key, 1, vdaf.decode_agg_param(agg_param), nonce, vdaf.decode_public_share(public_share), vdaf.decode_input_share(1, input_share), ) inbound = decode_ping_pong_message(inbound_encoded) if inbound.type != 0: # initialize return (Rejected(), None) prep_shares = [ vdaf.decode_prep_share(prep_state, inbound.prep_share), prep_share, ] return vdaf.ping_pong_transition( agg_param, prep_shares, prep_state, ) except: return (Rejected(), None)¶
Procedure ping_pong_transition()
takes in the prep shares, combines them into
the prep message, and computes the next prep state of the caller:¶
def ping_pong_transition( vdaf, agg_param: AggParam, prep_shares: list[PrepShare], prep_state: PrepState, ) > (State, bytes): prep_msg = vdaf.prep_shares_to_prep(agg_param, prep_shares) out = vdaf.prep_next(prep_state, prep_msg) if type(out) == OutShare: outbound = Message.finish(vdaf.encode_prep_msg(prep_msg)) return (Finished(out), encode_ping_pong_message(outbound)) (prep_state, prep_share) = out outbound = Message.continue( vdaf.encode_prep_msg(prep_msg), vdaf.encode_prep_share(prep_share), ) return ( Continued(prep_state), encode_ping_pong_message(outbound), )¶
The output is the State
to which the Helper has transitioned and an encoded
Message
. If the Helper's state is Finished
or Rejected
, then processing
halts. Otherwise, if the state is Continued
, then processing continues.¶
Next, the Helper sends the outbound message to the Leader. The Leader computes
its next state transition using the function ping_pong_leader_continued
:¶
def ping_pong_leader_continued( vdaf, agg_param: bytes, state: State, inbound_encoded: bytes, ) > (State, Optional[bytes]): return vdaf.ping_pong_continued( True, agg_param, state, inbound_encoded, ) def ping_pong_continued( vdaf, is_leader: bool, agg_param: bytes, state: State, inbound_encoded: bytes, ) > (State, Optional[bytes]): try: inbound = decode_ping_pong_message(inbound_encoded) if inbound.type == 0: # initialize return (Rejected(), None) if !isinstance(state, Continued): return (Rejected(), None) prep_msg = vdaf.decode_prep_msg( state.prep_state, inbound.prep_msg, ) out = vdaf.prep_next(state.prep_state, prep_msg) if type(out) == tuple[PrepState, PrepShare] \ and inbound.type == 1: # continue (prep_state, prep_share) = out prep_shares = [ vdaf.decode_prep_share( prep_state, inbound.prep_share, ), prep_share, ] if is_leader: prep_shares.reverse() return vdaf.ping_pong_transition( vdaf.decode_agg_param(agg_param), prep_shares, prep_state, ) elif type(out) == OutShare and inbound.type == 2: # finish return (Finished(out), None) else: return (Rejected(), None) except: return (Rejected(), None)¶
If the Leader's state is Finished
or Rejected
, then processing halts.
Otherwise, the Leader sends the outbound message to the Helper. The Helper
computes its next state transition using the function
ping_pong_helper_continued
:¶
def ping_pong_helper_continued( vdaf, agg_param: bytes, state: State, inbound_encoded: bytes, ) > (State, Optional[bytes]): return vdaf.ping_pong_continued( False, agg_param, state, inbound_encoded, )¶
They continue in this way until processing halts. Note that, depending on the
number of rounds of preparation that are required, there may be one more
message to send before the peer can also finish processing (i.e., outbound !=
None
).¶
5.9. Star Topology (Any Number of Aggregators)
The pingpong topology of the previous section is only suitable for VDAFs involving exactly two Aggregators. If more Aggregators are required, the star topology described in this section can be used instead.¶

TODO Describe the Leaderemulated broadcast channel architecture that was originally envisioned for DAP. (As of DAP05 we are going with the ping pong architecture described in the previous section.)¶
6. Preliminaries
This section describes the primitives that are common to the VDAFs specified in this document.¶
6.1. Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class Field
with the following associated
parameters:¶

MODULUS: int
is the prime modulus that defines the field.¶ 
ENCODED_SIZE: int
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: int) > list[Self]
returns a vector of zeros.¶Preconditions:¶

length
MUST be greater than or equal0
.¶
Postconditions:¶

The length of the output MUST be
length
.¶


Field.rand_vec(length: int) > list[Self]
returns a vector of random field elements. Same pre and postconditions as forField.zeros()
.¶
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() > int
returns the integer representation of field elementelem
.¶
Likewise, each concrete Field
implements a constructor for converting an
unsigned integer into a field element:¶

Field(integer: int)
returnsinteger
represented as a field element. The value ofinteger
MUST be nonnegative and less thanField.MODULUS
.¶
Each concrete Field
has two derived class methods, one for encoding
a vector of field elements as a byte string and another for decoding a vector of
field elements.¶
Finally, Field
implements the following methods for representing a value as
a sequence of field elements, each of which represents a bit of the input.¶
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.¶
6.1.2. FFTFriendly 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 FLP of Section 7.3.3.) Specifically, a field is said to be "FFTfriendly" 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 FFTfriendly, 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 least2^20
.¶
FFTfriendly fields also define the following parameter:¶

GEN_ORDER: int
is the order of a multiplicative subgroup generated byField.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 * 4294967295 + 1  2^66 * 4611686018427387897 + 1  2^255  19 
ENCODED_SIZE  8  16  32 
Generator  7^4294967295  7^4611686018427387897  n/a 
GEN_ORDER  2^32  2^66  n/a 
6.2. Extendable Output Functions
VDAFs in this specification use extendable output functions (XOFs) to extract short, fixedlength strings we call "seeds" from long input strings and expand seeds into long output strings. We specify a single interface that is suitable for both purposes.¶
XOFs are defined by a class Xof
with the following associated parameter and
methods:¶

SEED_SIZE: int
is the size (in bytes) of a seed.¶ 
Xof(seed: bytes, dst: bytes, binder: bytes)
constructs an instance ofXof
from the given seed, domain separation tag, and binder string. (See below for definitions of these.) The seed MUST be of lengthSEED_SIZE
and MUST be generated securely (i.e., it is either the output of a CSPRNG or a previous invocation of the XOF).¶ 
xof.next(length: int)
returns the nextlength
bytes of output ofxof
.¶
Each Xof
has three derived methods. The first is used to derive a fresh seed
from an existing one. The second is used to compute a sequence of field
elements. The third is a convenience method to construct an Xof
from a seed,
domain separation tag, and binder string, and then use it to compute a sequence
of field elements.¶
6.2.1. XofTurboShake128
This section describes XofTurboShake128, an XOF based on the
TurboSHAKE128 [TurboSHAKE]. This
XOF is RECOMMENDED for all use cases within VDAFs. The length of the
domain separation string dst
passed to XofTurboShake128 MUST NOT
exceed 255 bytes.¶
6.2.2. XofFixedKeyAes128
While XofTurboShake128 as described above can be securely used in all cases
where a XOF is needed in the VDAFs described in this document, there are some
cases where a more efficient instantiation based on fixedkey AES is possible.
For now, this is limited to the XOF used inside the Idpf Section 8.1
implementation in Poplar1 Section 8.3. It is NOT RECOMMENDED to use this
XOF anywhere else. The length of the domain separation string dst
passed to
XofFixedKeyAes128 MUST NOT exceed 255 bytes. See Security Considerations
Section 9 for a more detailed discussion.¶
class XofFixedKeyAes128(Xof): """ XOF based on a circular collisionresistant hash function from fixedkey AES. """ # Associated parameters SEED_SIZE = 16 def __init__(self, seed: bytes, dst: bytes, binder: bytes): if len(seed) != self.SEED_SIZE: raise ValueError("incorrect seed size") self.length_consumed = 0 # Use TurboSHAKE128 to derive a key from the binder string # and domain separation tag. Note that the AES key does not # need to be kept secret from any party. However, when used # with an IDPF, we require the binder to be a random nonce. # # Implementation note: This step can be cached across XOF # evaluations with many different seeds. dst_length = to_le_bytes(len(dst), 1) self.fixed_key = TurboSHAKE128( dst_length + dst + binder, 2, 16, ) self.seed = seed def next(self, length: int) > bytes: offset = self.length_consumed % 16 new_length = self.length_consumed + length block_range = range( self.length_consumed // 16, new_length // 16 + 1 ) self.length_consumed = new_length hashed_blocks = [ self.hash_block(xor(self.seed, to_le_bytes(i, 16))) for i in block_range ] return concat(hashed_blocks)[offset:offset+length] def hash_block(self, block: bytes) > bytes: """ The multiinstance tweakable circular correlationrobust hash function of [GKWWY20] (Section 4.2). The tweak here is the key that stays constant for all XOF evaluations of the same Client, but differs between Clients. Function `AES128(key, block)` is the AES128 blockcipher. """ lo, hi = block[:8], block[8:] sigma_block = concat([hi, xor(hi, lo)]) return xor(AES128(self.fixed_key, sigma_block), sigma_block)¶
6.2.3. The Domain Separation Tag and Binder String
XOFs are used to map a seed to a finite domain, e.g., a fresh seed or a vector of field elements. To ensure domain separation, the derivation is needs to be bound to some distinguished domain separation tag. The domain separation tag encodes the following values:¶

The document version (i.e.,
VERSION
);¶ 
The "class" of the algorithm using the output (e.g., VDAF);¶

A unique identifier for the algorithm; and¶

Some indication of how the output is used (e.g., for deriving the measurement shares in Prio3 Section 7).¶
The following algorithm is used in the remainder of this document in order to format the domain separation tag:¶
def format_dst(algo_class: int, algo: int, usage: int) > bytes: """ Format XOF domain separation tag for use within a (V)DAF. Preconditions:  `algo_class` in `range(0, 2 ** 8)`  `algo` in `range(0, 2 ** 32)`  `usage` in `range(0, 2 ** 16)` """ return concat([ to_be_bytes(VERSION, 1), to_be_bytes(algo_class, 1), to_be_bytes(algo, 4), to_be_bytes(usage, 2), ])¶
It is also sometimes necessary to bind the output to some ephemeral value that multiple parties need to agree on. We call this input the "binder string".¶
7. Prio3
This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶

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

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

The aggregate result is obtained by summing up the encoded measurement vectors and computing some function of the sum.¶
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 multiparty computation to determine if their shares correspond to a valid measurement (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its shares locally. Finally, the Collector sums up the aggregate shares and computes the aggregate result.¶
This VDAF does not have an aggregation parameter. Instead, the output share is derived from the measurement share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶
As the name implies, Prio3 is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction.¶
The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating measurements. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF.¶
The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in Appendix "Test Vectors".¶
7.1. Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a twoparty protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its measurement's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some computation on its measurement share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid measurement. (See Section 7.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms
of an abstract base class Flp
that specifies the set of methods and parameters
a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 4.¶
Parameter  Description 

PROVE_RAND_LEN

Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof 
QUERY_RAND_LEN

Length of the query randomness, the number of random field elements consumed by the verifier 
JOINT_RAND_LEN

Length of the joint randomness, the number of random field elements consumed by both the prover and verifier 
MEAS_LEN

Length of the encoded measurement (Section 7.1.1) 
OUTPUT_LEN

Length of the aggregatable output (Section 7.1.1) 
PROOF_LEN

Length of the proof 
VERIFIER_LEN

Length of the verifier message generated by querying the measurement and proof 
Measurement

Type of the measurement 
AggResult

Type of the aggregate result 
field

Class object for the field (Section 6.1) 
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1):¶

flp.prove(meas: list[F], prove_rand: list[F], joint_rand: list[F]) > list[F]
is the deterministic proofgeneration algorithm run by the prover. Its inputs are the encoded measurement, the "prover randomness"prove_rand
, and the "joint randomness"joint_rand
. The prover randomness is used only by the prover, but the joint randomness is shared by both the prover and verifier.¶ 
flp.query(meas: list[F], proof: list[F], query_rand: list[F], joint_rand: list[F], num_shares: int) > list[F]
is the querygeneration algorithm run by the verifier. This is used to "query" the measurement and proof. The result of the query (i.e., the output of this function) is called the "verifier message". In addition to the measurement and proof, this algorithm takes as input the query randomnessquery_rand
and the joint randomnessjoint_rand
. The former is used only by the verifier.num_shares
specifies how many shares were generated.¶ 
flp.decide(verifier: list[F]) > bool
is the deterministic decision algorithm run by the verifier. It takes as input the verifier message and outputs a boolean indicating if the measurement from which it was generated is valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the measurement and proof, the querygeneration algorithm outputs a share of the verifier message. Furthermore, the privacy property of the FLP system ensures that the verifier message reveals nothing about the measurement other than whether it is valid. Therefore, to decide if a measurement is valid, the Aggregators will run the querygeneration algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
The querygeneration algorithm includes a parameter num_shares
that specifies
the number of shares that were generated. If these data are not secret shared,
then num_shares == 1
. This parameter is useful for certain FLP constructions.
For example, the FLP in Section 7.3 is defined in terms of an arithmetic
circuit; when the circuit contains constants, it is sometimes necessary to
normalize those constants to ensure that the circuit's output, when run on a
valid measurement, is the same regardless of the number of shares.¶
An FLP is executed by the prover and verifier as follows:¶
The proof system is constructed so that, if meas
is valid, then run_flp(flp,
meas, 1)
always returns True
. On the other hand, if meas
is invalid, then
as long as joint_rand
and query_rand
are generated uniform randomly, the
output is False
with high probability. False positives are possible: there is
a small probability that a verifier accepts an invalid input as valid. An FLP
is said to be "sound" if this probability is sufficiently small. The soundness
of the FLP depends on a variety of parameters, like the length of the
input and the size of the field. See Section 7.3 for details.¶
Note that soundness of an FLP system is not the same as robustness for a VDAF In particular, soundness of the FLP is necessary, but insufficient for robusntess of Prio3 (Section 7). See Section 9.6 for details.¶
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 1.5round, publiccoin, 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) > list[F]
encodes a raw measurement as a vector of field elements. The return value MUST be of lengthMEAS_LEN
.¶
For some FLPs, the encoded measurement also includes redundant field elements
that are useful for checking the proof, but which are not needed after the
proof has been checked. An example is the "integer sum" data type from
[CGB17] in which an integer in range [0, 2^k)
is encoded as a vector of k
field elements, each representing a bit of the integer (this type is also
defined in Section 7.4.2). After consuming this vector, all that is needed is
the integer it represents. Thus the FLP defines an algorithm for truncating the
encoded measurement to the length of the aggregated output:¶

flp.truncate(meas: list[F]) > list[F]
maps an encoded measurement (e.g., the bitencoding of the measurement) to an aggregatable output (e.g., the singleton vector containing the measurement). The length of the input MUST beMEAS_LEN
and the length of the output MUST beOUTPUT_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 postprocessing.¶

flp.decode(output: list[F], num_measurements: int) > AggResult
maps a sum of aggregate shares to an aggregate result.¶Preconditions:¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affineaggregatable encodings (AFEs)" from [CGB17].¶
7.1.2. Multiple Proofs
It is sometimes desirable to generate and verify multiple independent proofs for the same input. First, this improves the soundness of the proof system without having to change any of its parameters. Second, it allows a smaller field to be used (e.g., replace Field128 with Field64, see Section 7.3) without sacrificing soundness. Generally, choosing a smaller field can significantly reduce communication cost. (This is a tradeoff, of course, since generating and verifying more proofs requires more time.) Given these benefits, this feature is implemented by Prio3 (Section 7).¶
To generate these proofs for a specific measurement, the prover calls
flp.prove
multiple times, each time using an independently generated prover
and joint randomness string. The verifier checks each proof independently, each
time with an independently generated query randomness string. It accepts the
measurement only if all the decision algorithm accepts on each proof.¶
See Section 9.6 below for discussions on choosing the right number of proofs.¶
7.2. Construction
This section specifies Prio3
, an implementation of the Vdaf
interface
(Section 5). It has three generic parameters: an FftField ({{fftfield}}), an
Flp ({{flp}}) and a
Xof ({{xof}}). It also has an associated constant,
PROOFS, with a value within the range of
[1, 256)`, denoting the number of
FLPs generated by the Client (Section 7.1.2).¶
The 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.¶
Parameter  Value 

VERIFY_KEY_SIZE

Xof.SEED_SIZE

RAND_SIZE

Xof.SEED_SIZE * (1 + 2 * (SHARES  1)) if flp.JOINT_RAND_LEN == 0 else Xof.SEED_SIZE * (1 + 2 * (SHARES  1) + SHARES)

NONCE_SIZE

16

ROUNDS

1

SHARES

in [2, 256)

Measurement

Flp.Measurement

AggParam

None

PublicShare

Optional[list[bytes]]

InputShare

Union[tuple[list[F], list[F], Optional[bytes]], tuple[bytes, bytes, Optional[bytes]]]

OutShare

list[F]

AggShare

list[F]

AggResult

Flp.AggResult

PrepState

tuple[list[F], Optional[bytes]]

PrepShare

tuple[list[F], Optional[bytes]]

PrepMessage

Optional[bytes]

Variable  Value 

USAGE_MEAS_SHARE: int

1 
USAGE_PROOF_SHARE: int

2 
USAGE_JOINT_RANDOMNESS: int

3 
USAGE_PROVE_RANDOMNESS: int

4 
USAGE_QUERY_RANDOMNESS: int

5 
USAGE_JOINT_RAND_SEED: int

6 
USAGE_JOINT_RAND_PART: int

7 
7.2.2. Preparation
This section describes the process of recovering output shares from the input shares. The highlevel idea is that each Aggregator first queries its measurement and share of proof(s) locally, then exchanges its share of verifier(s) with the other Aggregators. The shares of verifier(s) are then combined into the verifier message(s) used to decide whether to accept.¶
In addition, for FLPs that require joint randomness, the Aggregators must ensure that they have all used the same joint randomness for the querygeneration algorithm. To do so, they collectively rederive the joint randomness from their measurement shares just as the Client did during sharding.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the joint randomness parts. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their shares of verifier(s).¶
The definitions of constants and a few auxiliary functions are defined in Section 7.2.6.¶
7.2.3. Validity of Aggregation Parameters
Every input share MUST only be used once, regardless of the aggregation parameters used.¶
7.2.4. Aggregation
Aggregating a set of output shares is simply a matter of adding up the vectors elementwise.¶
7.2.5. Unsharding
To unshard a set of aggregate shares, the Collector first adds up the vectors elementwise. It then converts each element of the vector into an integer.¶
7.2.6. Auxiliary Functions
This section defines a number of auxiliary functions referenced by the main algorithms for Prio3 in the preceding sections.¶
The following methods are called by the sharding and preparation algorithms.¶
def helper_meas_share(self, agg_id: int, k_share: bytes) > list[F]: return self.xof.expand_into_vec( self.flp.field, k_share, self.domain_separation_tag(USAGE_MEAS_SHARE), byte(agg_id), self.flp.MEAS_LEN, ) def helper_proofs_share( self, agg_id: int, k_share: bytes) > list[F]: return self.xof.expand_into_vec( self.flp.field, k_share, self.domain_separation_tag(USAGE_PROOF_SHARE), byte(self.PROOFS) + byte(agg_id), self.flp.PROOF_LEN * self.PROOFS, ) def expand_input_share( self, agg_id: int, input_share: Prio3InputShare[F]) > tuple[ list[F], list[F], Optional[bytes]]: (meas_share, proofs_share, k_blind) = input_share if agg_id > 0: assert isinstance(meas_share, bytes) assert isinstance(proofs_share, bytes) meas_share = self.helper_meas_share(agg_id, meas_share) proofs_share = self.helper_proofs_share(agg_id, proofs_share) else: assert isinstance(meas_share, list) assert isinstance(proofs_share, list) return (meas_share, proofs_share, k_blind) def prove_rands(self, k_prove: bytes) > list[F]: return self.xof.expand_into_vec( self.flp.field, k_prove, self.domain_separation_tag(USAGE_PROVE_RANDOMNESS), byte(self.PROOFS), self.flp.PROVE_RAND_LEN * self.PROOFS, ) def query_rands(self, verify_key: bytes, nonce: bytes) > list[F]: return self.xof.expand_into_vec( self.flp.field, verify_key, self.domain_separation_tag(USAGE_QUERY_RANDOMNESS), byte(self.PROOFS) + nonce, self.flp.QUERY_RAND_LEN * self.PROOFS, ) def joint_rand_part( self, agg_id: int, k_blind: bytes, meas_share: list[F], nonce: bytes) > bytes: return self.xof.derive_seed( k_blind, self.domain_separation_tag(USAGE_JOINT_RAND_PART), byte(agg_id) + nonce + self.flp.field.encode_vec(meas_share), ) def joint_rand_seed(self, k_joint_rand_parts: list[bytes]) > bytes: """Derive the joint randomness seed from its parts.""" return self.xof.derive_seed( zeros(self.xof.SEED_SIZE), self.domain_separation_tag(USAGE_JOINT_RAND_SEED), concat(k_joint_rand_parts), ) def joint_rands(self, k_joint_rand_seed: bytes) > list[F]: """Derive the joint randomness from its seed.""" return self.xof.expand_into_vec( self.flp.field, k_joint_rand_seed, self.domain_separation_tag(USAGE_JOINT_RANDOMNESS), byte(self.PROOFS), self.flp.JOINT_RAND_LEN * self.PROOFS, )¶
7.2.7. Message Serialization
This section defines serialization formats for messages exchanged over the network while executing Prio3. It is RECOMMENDED that implementations provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the
remainder we use S
as an alias for Prio3.xof.SEED_SIZE
and F
as an alias
for Prio3.field.ENCODED_SIZE
. XOF seeds are represented as follows:¶
opaque Prio3Seed[S];¶
Field elements are encoded in littleendian byte order (as defined in Section 6.1) and represented as follows:¶
opaque Prio3Field[F];¶
7.2.7.4. Prep Message
When joint randomness is not used, the prep message is the empty string. Otherwise the prep message consists of the joint randomness seed computed by the Aggregators:¶
struct { Prio3Seed k_joint_rand; } Prio3PrepMessageWithJointRand;¶
7.2.7.5. Aggregation
Aggregate shares are structured as follows:¶
struct { Prio3Field agg_share[F * Prio3.flp.OUTPUT_LEN]; } Prio3AggShare;¶
7.3. The FLP of [BBCGGI19]
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 Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
7.3.1. Overview
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the encoded measurement: If the circuit output is zero, then the measurement is deemed valid; otherwise, if the circuit output is nonzero, then the measurement is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the measurement. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the measurement.¶
Suppose for a moment that the validity circuit C
is affine, meaning its only
operations are addition and multiplicationbyconstant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
nonconstant. Then to decide if a measurement x
is valid, each Aggregator
could evaluate C
on its share of x
locally, broadcast the output share to
its peers, then combine the output shares locally to recover C(x)
. This is
true because for any SHARES
way secret sharing of x
it holds that¶
C(x_shares[0] + ... + x_shares[SHARES1]) = C(x_shares[0]) + ... + C(x_shares[SHARES1])¶
(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
notaffine (i.e., it contains at least one multiplication gate whose operands
are nonconstant). In the proof system of [BBCGGI19], the proof is designed to
allow the (distributed) verifier to compute the nonaffine operations using only
linear operations on (its share of) the measurement and proof.¶
To make this work, the proof system is restricted to validity circuits that
exhibit a special structure. Specifically, an arithmetic circuit with "Ggates"
(see [BBCGGI19], Definition 5.2) is composed of affine gates and any number of
instances of a distinguished gate G
, which may be nonaffine. 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 length1 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 degree2 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 nonaffine 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 nonaffine 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 measurement and proof. The proof is constructed roughly as follows. Let
C
be the validity circuit and suppose the gadget is arityL
(i.e., it has
L
input wires.). Let wire[j1,k1]
denote the value of the j
th wire of
the k
th call to the gadget during the evaluation of C(x)
. Suppose there are
M
such calls and fix distinct field elements alpha[0], ..., alpha[M1]
. (We
will require these points to have a special property, as we'll discuss in
Section 7.3.1.1; but for the moment it is only important
that they are distinct.)¶
The prover constructs from wire
and alpha
a polynomial that, when evaluated
at alpha[k1]
, produces the output of the k
th call to the gadget. Let us
call this the "gadget polynomial". Polynomial evaluation is linear, which means
that, in the distributed setting, the Client can disseminate additive shares of
the gadget polynomial that the Aggregators then use to compute additive shares
of each gadget output, allowing each Aggregator to compute its share of C(x)
locally.¶
There is one more wrinkle, however: It is still possible for a malicious prover
to produce a gadget polynomial that would result in C(x)
being computed
incorrectly, potentially resulting in an invalid measurement being accepted. To
prevent this, the verifier performs a probabilistic test to check that the
gadget polynomial is wellformed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in Section 7.3.3.¶
7.3.1.1. Extensions
The FLP described in the next section extends the proof system of [BBCGGI19], Section 4.2 in a few ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the measurement shares in
Prio3; see Section 7.2). This allows for circuit optimizations that
trade a small soundness error for a shorter proof. For example, consider a
circuit that recognizes the set of lengthN
vectors for which each element is
either one or zero. A deterministic circuit could be constructed for this
language, but it would involve a large number of multiplications that would
result in a large proof. (See the discussion in [BBCGGI19], Section 5.2 for
details). A much shorter proof can be constructed for the following randomized
circuit:¶
C(meas, r) = r * Range2(meas[0]) + ... + r^N * Range2(meas[N1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here meas
is
the lengthN
input and r
is a random field element. The gadget circuit
Range2
is the "rangecheck" polynomial described above, i.e., Range2(x) =
x^2  x
. The idea is that, if meas
is valid (i.e., each meas[j]
is in
[0,2)
), then the circuit will evaluate to 0 regardless of the value of r
;
but if meas[j]
is not in [0,2)
for some j
, the output will be nonzero
with high probability.¶
The second extension implemented by our FLP allows the validity circuit to contain multiple gadget types. (This generalization was suggested in [BBCGGI19], Remark 4.5.) This provides additional flexibility for designing circuits by allowing multiple, nonaffine subcomponents. For example, the following circuit is allowed:¶
C(meas, r) = r * Range2(meas[0]) + ... + r^L * Range2(meas[L1]) + \ r^(L+1) * Range3(meas[L]) + ... + r^N * Range3(meas[N1])¶
where Range3(x) = x^3  3x^2 + 2x
. This circuit checks that the first L
inputs are in range [0,2)
and the last NL
inputs are in range [0,3)
. Of
course, the same circuit can be expressed using a subcomponent that the
gadgets have in common, namely Mul
, but the resulting proof would be longer.¶
Third, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the
fixed points alpha[0], ..., alpha[M1]
, 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 CooleyTukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFTfriendly" as defined in Section 6.1.2.¶
Finally, the validity circuit in our FLP may have any number of outputs (at least one). The input is said to be valid if each of the outputs is zero. To save bandwidth, we take a random linear combination of the outputs. If each of the outputs is zero, then the reduced output will be zero; but if one of the outputs is nonzero, then the reduced output will be nonzero with high probability.¶
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 
MEAS_LEN

Length of the measurement 
OUTPUT_LEN

Length of the aggregatable output 
JOINT_RAND_LEN

Length of the random input 
EVAL_OUTPUT_LEN

Length of the circuit output 
Measurement

The type of measurement 
AggResult

Type of the aggregate result 
field

Class object for the field 
Each gadget G
in GADGETS
defines a constant DEGREE
that specifies the
circuit's "arithmetic degree". This is defined to be the degree of the
polynomial that computes it. For example, the Mul
circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x
, which
has degree 2
. Hence, the arithmetic degree of this gadget is 2
.¶
Each gadget also defines a parameter ARITY
that specifies the circuit's arity
(i.e., the number of input wires).¶
Gadgets provide a method to evaluate their circuit on a list of inputs,
eval()
. The inputs can either belong to the validity circuit's field, or the
polynomial ring over that field.¶
A concrete Valid
provides the following methods for encoding a measurement as
an input vector, truncating an input vector to the length of an aggregatable
output, and converting an aggregated output to an aggregate result:¶

valid.encode(measurement: Measurement) > list[F]
returns a vector of lengthMEAS_LEN
representing a measurement.¶ 
valid.truncate(meas: list[F]) > list[F]
returns a vector of lengthOUTPUT_LEN
representing an aggregatable output.¶ 
valid.decode(output: list[F], num_measurements: int) > AggResult
returns an aggregate result.¶
Finally, the following methods are derived for each concrete Valid
:¶
7.3.3. Construction
This section specifies 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.¶
The parameters 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:¶
Parameter  Value 

PROVE_RAND_LEN

valid.prove_rand_len() (see Section 7.3.2) 
QUERY_RAND_LEN

valid.query_rand_len() (see Section 7.3.2) 
JOINT_RAND_LEN

valid.JOINT_RAND_LEN

MEAS_LEN

valid.MEAS_LEN

OUTPUT_LEN

valid.OUTPUT_LEN

PROOF_LEN

valid.proof_len() (see Section 7.3.2) 
VERIFIER_LEN

valid.verifier_len() (see Section 7.3.2) 
Measurement

valid.Measurement

7.3.3.1. Proof Generation
On input of meas
, prove_rand
, and joint_rand
, the proof is computed as
follows:¶

For each
i
in[H]
create an empty tablewire_i
.¶ 
Partition the prover randomness
prove_rand
into subvectorsseed_1, ..., seed_H
wherelen(seed_i) == L_i
for alli
in[H]
. Let us call these the "wire seeds" of each gadget.¶ 
Evaluate
Valid
on input ofmeas
andjoint_rand
, recording the inputs of each gadget in the corresponding table. Specifically, for everyi
in[H]
, setwire_i[j1,k1]
to the value on thej
th wire into thek
th call to gadgetG_i
.¶ 
Compute the "wire polynomials". That is, for every
i
in[H]
andj
in[L_i]
, constructpoly_wire_i[j1]
, thej
th wire polynomial for thei
th gadget, as follows:¶
Let
w = [seed_i[j1], wire_i[j1,0], ..., wire_i[j1,M_i1]]
.¶ 
Let
padded_w = w + field.zeros(P_i  len(w))
.¶

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

Let
poly_wire_i[j1]
be the lowest degree polynomial for whichpoly_wire_i[j1](alpha_i^k) == padded_w[k]
for allk
in[P_i]
.¶


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_i1])
. That is, evaluate the circuitG_i
on the wire polynomials for thei
th gadget. (Arithmetic is in the ring of polynomials overfield
.)¶

The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H
,
where coeff_i
is the vector of coefficients of poly_gadget_i
for each i
in
[H]
.¶
7.3.3.2. Query Generation
On input of meas
, proof
, query_rand
, and joint_rand
, the verifier message
is generated as follows:¶

For every
i
in[H]
create an empty tablewire_i
.¶ 
Partition
proof
into the subvectorsseed_1
,coeff_1
, ...,seed_H
,coeff_H
defined in Section 7.3.3.1.¶ 
Evaluate
Valid
on input ofmeas
andjoint_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 thek
th call toG_i
by evaluatingpoly_gadget_i(alpha_i^k)
. Letout
denote the output of the circuit evaluation.¶ 
Next, reduce
out
as follows. IfEVAL_OUTPUT_LEN > 1
, then consume the first element ofquery_rand
by letting[r], query_rand = front(1, query_rand)
. Then letv = r*out[0] + r**2*out[1] + r**3*out[2] + ...
. That is, interpret the outputs as coefficients of a polynomialf(x)
and evaluate polynomialf(x)*x
at a random pointr
.¶ 
Compute the wire polynomials just as in the prover's step (4.).¶

Compute the tests for wellformedness of the gadget polynomials. That is, for every
i
in[H]
:¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H]
.¶
7.3.3.3. Decision
On input of vector verifier
, the verifier decides if the measurement is valid
as follows:¶

Parse
verifier
intov
,x_1
,y_1
, ...,x_H
,y_H
as defined in Section 7.3.3.2.¶ 
Check for wellformedness of the gadget polynomials. For every
i
in[H]
:¶ 
Return
True
ifv == 0
andFalse
otherwise.¶
7.4. Instantiations
This section specifies instantiations of Prio3 for various measurement types. Each is determined by a field (Section 6.1), a validity circuit (Section 7.3.2), an XOF (Section 6.2). and the number of proofs to generate and verify. Test vectors for each can be found in Appendix "Test Vectors".¶
7.4.1. Prio3Count
Parameter  Value 

Valid

Count(Field64) (this section) 
Field

Field64 (Table 3) 
PROOFS

1

Xof

XofTurboShake128 (Section 6.2.1) 
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.¶
Its validity circuit, denoted Count
, uses the following degree2, arity2
gadget, denoted Mul
:¶
def eval(self, _field: type[F], inp: list[F]) > F: self.check_gadget_eval(inp) return inp[0] * inp[1]¶
The call to check_gadget_eval()
raises an error if the length of the input is
not equal to the gadget's ARITY
parameter.¶
The Count
validity circuit is defined as¶
def eval( self, meas: list[F], joint_rand: list[F], _num_shares: int) > list[F]: self.check_valid_eval(meas, joint_rand) squared = self.GADGETS[0].eval(self.field, [meas[0], meas[0]]) return [squared  meas[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]

MEAS_LEN

1

OUTPUT_LEN

1

JOINT_RAND_LEN

0

EVAL_OUTPUT_LEN

1

Measurement

int in range(2)

AggResult

int

7.4.2. Prio3Sum
Parameter  Value 

Valid

Sum(Field128, bits) (this section) 
Field

Field128 (Table 3) 
PROOFS

1

Xof

XofTurboShake128 (Section 6.2.1) 
The next instance of Prio3 supports summing of integers in a predetermined
range. Each measurement is an integer in range [0, 2^bits)
, where bits
is an
associated parameter.¶
The validity circuit is denoted Sum
. The measurement is encoded as a
lengthbits
vector of field elements, where the l
th element of the vector
represents the l
th bit of the summand:¶
def encode(self, measurement: int) > list[F]: if 0 > measurement or measurement >= 2 ** self.MEAS_LEN: raise ValueError('measurement out of range') return self.field.encode_into_bit_vector(measurement, self.MEAS_LEN) def truncate(self, meas: list[F]) > list[F]: return [self.field.decode_from_bit_vector(meas)] def decode( self, output: list[F], _num_measurements: int) > int: return output[0].as_unsigned()¶
The validity circuit checks that the input consists of ones and zeros. Its
gadget, denoted Range2
, is the degree2, arity1 gadget defined as¶
def eval(self, _field: type[F], inp: list[F]) > F: self.check_gadget_eval(inp) return inp[0] * inp[0]  inp[0]¶
The Sum
validity circuit is defined as¶
def eval( self, meas: list[F], joint_rand: list[F], _num_shares: int) > list[F]: self.check_valid_eval(meas, joint_rand) out = self.field(0) r = joint_rand[0] for b in meas: out += r * self.GADGETS[0].eval(self.field, [b]) r *= joint_rand[0] return [out]¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[bits]

MEAS_LEN

bits

OUTPUT_LEN

1

JOINT_RAND_LEN

1

EVAL_OUTPUT_LEN

1

Measurement

int in range(2**bits)

AggResult

int

7.4.3. Prio3SumVec
Parameter  Value 

Valid

Sum(Field128, length, bits, chunk_lengh) (this section) 
Field

Field128 (Table 3) 
PROOFS

1

Xof

XofTurboShake128 (Section 6.2.1) 
This instance of Prio3 supports summing a vector of integers. It has three
parameters, length
, bits
, and chunk_length
. Each measurement is a vector
of positive integers with length equal to the length
parameter. Each element
of the measurement is an integer in the range [0, 2^bits)
. It is RECOMMENDED
to set chunk_length
to an integer near the square root of length * bits
(see Section 7.4.3.1).¶
The validity circuit is denoted SumVec
. Measurements are encoded as a vector
of field elements with length length * bits
. The field elements in the
encoded vector represent all the bits of the measurement vector's elements,
consecutively, in LSB to MSB order:¶
def encode(self, measurement: list[int]) > list[F]: if len(measurement) != self.length: raise ValueError('incorrect measurement length') encoded = [] for val in measurement: if val not in range(2**self.bits): raise ValueError( 'entry of measurement vector is out of range' ) encoded += self.field.encode_into_bit_vector(val, self.bits) return encoded def truncate(self, meas: list[F]) > list[F]: truncated = [] for i in range(self.length): truncated.append(self.field.decode_from_bit_vector( meas[i * self.bits: (i + 1) * self.bits] )) return truncated def decode( self, output: list[F], _num_measurements: int) > list[int]: return [x.as_unsigned() for x in output]¶
This validity circuit uses a ParallelSum
gadget to achieve a smaller proof
size. This optimization for "parallelsum circuits" is described in
[BBCGGI19], section 4.4. Briefly, for circuits that add up the output of
multiple identical subcircuits, it is possible to achieve smaller proof sizes
(on the order of O(sqrt(MEAS_LEN)) instead of O(MEAS_LEN)) by packaging more
than one such subcircuit into a gadget.¶
The ParallelSum
gadget is parameterized with an arithmetic subcircuit, and a
count
of how many times it evaluates that subcircuit. It takes in a list of
inputs and passes them through to instances of the subcircuit in the same order.
It returns the sum of the subcircuit outputs. Note that only the ParallelSum
gadget itself, and not its subcircuit, participates in the FLP's wire recording
during evaluation, gadget consistency proofs, and proof validation, even though
the subcircuit is provided to ParallelSum
as an implementation of the
Gadget
interface.¶
def eval(self, field: type[F], inp: list[F]) > F: self.check_gadget_eval(inp) out = field(0) for i in range(self.count): start_index = i * self.subcircuit.ARITY end_index = (i + 1) * self.subcircuit.ARITY out += self.subcircuit.eval( field, inp[start_index:end_index], ) return out¶
The SumVec
validity circuit checks that the encoded measurement consists of
ones and zeros. Rather than use the Range2
gadget on each element, as in the
Sum
validity circuit, it instead uses Mul
subcircuits and "free" constant
multiplication and addition gates to simultaneously evaluate the same range
check polynomial on each element, and multiply by a constant. One of the two
Mul
subcircuit inputs is equal to a measurement element multiplied by a power
of the joint randomness value, and the other is equal to the same measurement
element minus one. These Mul
subcircuits are evaluated by a ParallelSum
gadget, and the results are added up both within the ParallelSum
gadget and
after it.¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) > list[F]: self.check_valid_eval(meas, joint_rand) out = self.field(0) r = joint_rand[0] r_power = r shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem  shares_inv r_power *= r out += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) return [out]¶
Parameter  Value 

GADGETS

[ParallelSum(Mul(), chunk_length)]

GADGET_CALLS

[(length * bits + chunk_length  1) // chunk_length]

MEAS_LEN

length * bits

OUTPUT_LEN

length

JOINT_RAND_LEN

1

EVAL_OUTPUT_LEN

1

Measurement

list[int] , each element in range(2**bits)

AggResult

list[int]

7.4.3.1. Selection of ParallelSum
chunk length
The chunk_length
parameter provides a tradeoff between the arity of the
ParallelSum
gadget and the number of times the gadget is called. The proof
length is asymptotically minimized when the chunk length is near the square root
of the length of the measurement. However, the relationship between VDAF
parameters and proof length is complicated, involving two forms of rounding (the
circuit pads the inputs to its last ParallelSum
gadget call, up to the chunk
length, and proof system rounds the degree of wire polynomials  determined by
the number of times a gadget is called  up to the next power of two).
Therefore, the optimal choice of chunk_length
for a concrete measurement size
will vary, and must be found through trial and error. Setting chunk_length
equal to the square root of the appropriate measurement length will result in
proofs up to 50% larger than the optimal proof size.¶
7.4.4. Prio3Histogram
Parameter  Value 

Valid

Sum(Field128, length, chunk_lengh) (this section) 
Field

Field128 (Table 3) 
PROOFS

1

Xof

XofTurboShake128 (Section 6.2.1) 
This instance of Prio3 allows for estimating the distribution of some quantity
by computing a simple histogram. Each measurement increments one histogram
bucket, out of a set of fixed buckets. (Bucket indexing begins at 0
.) For
example, the buckets might quantize the real numbers, and each measurement would
report the bucket that the corresponding client's realnumbered value falls
into. The aggregate result counts the number of measurements in each bucket.¶
The validity circuit is denoted Histogram
. It has two parameters, length
,
the number of histogram buckets, and chunk_length
, which is used by by a
circuit optimization described below. It is RECOMMENDED to set chunk_length
to an integer near the square root of length
(see
Section 7.4.3.1).¶
The measurement is encoded as a onehot vector representing the bucket into which the measurement falls:¶
def encode(self, measurement: int) > list[F]: encoded = [self.field(0)] * self.length encoded[measurement] = self.field(1) return encoded def truncate(self, meas: list[F]) > list[F]: return meas def decode( self, output: list[F], _num_measurements: int) > list[int]: return [bucket_count.as_unsigned() for bucket_count in output]¶
The Histogram
validity circuit checks for onehotness in two steps, by
checking that the encoded measurement consists of ones and zeros, and by
checking that the sum of all elements in the encoded measurement is equal to
one. All the individual checks are combined together in a random linear
combination.¶
As in the SumVec
validity circuit (Section 7.4.3), the first part of the
validity circuit uses the ParallelSum
gadget to perform range checks while
achieving a smaller proof size. The ParallelSum
gadget uses Mul
subcircuits
to evaluate a range check polynomial on each element, and includes an additional
constant multiplication. One of the two Mul
subcircuit inputs is equal to a
measurement element multiplied by a power of the first joint randomness value,
and the other is equal to the same measurement element minus one. The results
are added up both within the ParallelSum
gadget and after it.¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) > list[F]: self.check_valid_eval(meas, joint_rand) # Check that each bucket is one or zero. range_check = self.field(0) r = joint_rand[0] r_power = r shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem  shares_inv r_power *= r range_check += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) # Check that the buckets sum to 1. sum_check = shares_inv for b in meas: sum_check += b out = joint_rand[1] * range_check + \ joint_rand[1] ** 2 * sum_check return [out]¶
Note that this circuit depends on the number of shares into which the measurement is sharded. This is provided to the FLP by Prio3.¶
Parameter  Value 

GADGETS

[ParallelSum(Mul(), chunk_length)]

GADGET_CALLS

[(length + chunk_length  1) // chunk_length]

MEAS_LEN

length

OUTPUT_LEN

length

JOINT_RAND_LEN

2

EVAL_OUTPUT_LEN

1

Measurement

int

AggResult

list[int]

7.4.5. Prio3MultihotCountVec
Parameter  Value 

Valid

Sum(Field128, length, max_eight chunk_lengh) (this section) 
Field

Field128 (Table 3) 
PROOFS

1

Xof

XofTurboShake128 (Section 6.2.1) 
For this instance of Prio3, each measurement is a vector of ones and zeros, where the number of ones is bounded. This provides a functionality similar to Prio3Histogram except that more than one entry may be nonzero. This allows Prio3MultihotCountVec to be composed with a randomized response mechanism, like [EPK14], for providing differential privacy. (For example, each Client would set each entry to one with some small probability.)¶
Prio3MultihotCountVec uses XofTurboShake128 (Section 6.2.1) as its XOF.
Its validity circuit is denoted MultihotCountVec
. It has three parameters:
length
, the number of of entries in the count vector; max_weight
, the
maximum number of nonzero entries (i.e., the weight must be at most
max_weight
); and chunk_length
, used the same way as in Section 7.4.3 and
Section 7.4.4.¶
Validation works as follows. Let¶
The Client reports the weight of the count vector by adding offset
to it and
bitencoding the result. Observe that only a weight of at most max_weight
can
be encoded with bits_for_weight
bits.¶
The verifier checks that each entry of the encoded measurement is a bit (i.e.,
either one or zero). It then decodes the reported weight and subtracts it from
offset + sum(count_vec)
, where count_vec
is the count vector. The result is
zero if and only if the reported weight is equal to the true weight.¶
Encoding, truncation, and decoding are defined as follows:¶
def encode(self, measurement: list[int]) > list[F]: if len(measurement) != self.length: raise ValueError('invalid Client measurement length') # The first part is the vector of counters. count_vec = list(map(self.field, measurement)) # The second part is the reported weight. weight_reported = sum(count_vec, self.field(0)) encoded = [] encoded += count_vec encoded += self.field.encode_into_bit_vector( (self.offset + weight_reported).as_unsigned(), self.bits_for_weight) return encoded def truncate(self, meas: list[F]) > list[F]: return meas[:self.length] def decode( self, output: list[F], _num_measurements: int) > list[int]: return [bucket_count.as_unsigned() for bucket_count in output]¶
Circuit evaluation is defined as follows:¶
def eval( self, meas: list[F], joint_rand: list[F], num_shares: int) > list[F]: self.check_valid_eval(meas, joint_rand) # Check that each entry in the input vector is one or zero. range_check = self.field(0) r = joint_rand[0] r_power = r shares_inv = self.field(num_shares).inv() for i in range(self.GADGET_CALLS[0]): inputs: list[Optional[F]] inputs = [None] * (2 * self.chunk_length) for j in range(self.chunk_length): index = i * self.chunk_length + j if index < len(meas): meas_elem = meas[index] else: meas_elem = self.field(0) inputs[j * 2] = r_power * meas_elem inputs[j * 2 + 1] = meas_elem  shares_inv r_power *= r range_check += self.GADGETS[0].eval( self.field, cast(list[F], inputs), ) # Check that the weight `offset` plus the sum of the counters # is equal to the value claimed by the Client. count_vec = meas[:self.length] weight = sum(count_vec, self.field(0)) weight_reported = \ self.field.decode_from_bit_vector(meas[self.length:]) weight_check = self.offset*shares_inv + weight  \ weight_reported out = joint_rand[1] * range_check + \ joint_rand[1] ** 2 * weight_check return [out]¶
Parameter  Value 

GADGETS

[ParallelSum(Mul(), chunk_length)]

GADGET_CALLS

[(length + bits_for_weight + chunk_length  1) // chunk_length]

MEAS_LEN

length + bits_for_weight

OUTPUT_LEN

length

JOINT_RAND_LEN

2

Measurement

list[int]

AggResult

list[int]

8. Poplar1
This section specifies Poplar1, a VDAF for the following task. Each Client
holds a bitstring of length BITS
and the Aggregators hold a sequence of
L
bit strings, where L <= BITS
. We will refer to the latter as the set of
"candidate prefixes". The Aggregators' goal is to count how many measurements
are prefixed by each candidate prefix.¶
This functionality is the core component of the Poplar protocol [BBCGGI21], which was designed to compute the heavy hitters over a set of input strings. At a high level, the protocol works as follows.¶

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

The Aggregators agree on an initial set of candidate prefixes, say
0
and1
.¶ 
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.¶

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

Let
H
denote the set of prefixes that occurred at leastt
times. If the prefixes all have lengthBITS
, thenH
is the set oft
heavyhitters. Otherwise compute the next set of candidate prefixes, e.g., for eachp
inH
, addp  0
andp  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 nonzero value; otherwise it evaluates to zero. This structure allows an IDPF to provide the functionality required for the above protocol: To compute the hit count for an index, just evaluate each set of IDPF shares at that index and add up the results.¶
Consider the subtree 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 9.4.1 for details.)¶
Poplar1 composes an IDPF with the arithmetic sketch of [BBCGGI21], Section 4.2. (The paper calls this a "secure sketch", but the underlying technique was later generalized in [BBCGGI23], where it is called "arithmetic sketching".) This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "onehot" vector, i.e., a vector that is zero everywhere except for at most 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 a constant.
Indices into the IDPF tree are encoded as integers in range [0, 2^BITS)
. (In
Poplar1, each Client's bit string is encoded as an index; see
Section 8.1.1 for details.) 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, L)
denote the least significant L
bits of positive integer x
. A
positive integer 0 <= x < 2^L
is defined to be the lengthL
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
length3 prefix of 25 (11001), but 7 (111) is not.¶
Each of the programmed points beta
is a vector of elements of some finite
field. We distinguish two types of fields: One for inner nodes (denoted
FieldInner
), and one for leaf nodes (FieldLeaf
). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger field for
leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as
defined in [BBCGGI21], Definition 1.)¶
A concrete IDPF defines the types and constants enumerated in Table 19.
In the remainder we write Output
as shorthand for the type
Union[list[list[FieldInner]], list[list[FieldLeaf]]]
. (This type
denotes either a vector of inner node field elements or leaf node field
elements.) The scheme is comprised of the following algorithms:¶

idpf.gen(alpha: int, beta_inner: list[list[FieldInner]], beta_leaf: list[FieldLeaf], binder: bytes, rand: bytes) > tuple[bytes, list[bytes]]
is the randomized IDPFkey generation algorithm. Its inputs are the indexalpha
the valuesbeta
, and a binder string.¶The output is a public part that is sent to all Aggregators and a vector of private IDPF keys, one for each aggregator. The binder string is used to derive the key in the underlying XofFixedKeyAes128 XOF that is used for expanding seeds at each level.¶
Preconditions:¶

alpha
MUST be inrange(2**BITS)
.¶ 
beta_inner
MUST have lengthBITS  1
.¶ 
beta_inner[level]
MUST have lengthVALUE_LEN
for eachlevel
inrange(BITS  1)
.¶ 
beta_leaf
MUST have lengthVALUE_LEN
.¶ 
rand
MUST be generated by a CSPRNG and have lengthRAND_SIZE
.¶ 
binder
MUST be chosen uniformly at random by the Client (see Section 9.2).¶
TODO If the binder needs to be random, then we need to specify its length so that the user knows how many bytes to sample.¶


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


idpf.eval(agg_id: int, public_share: bytes, key: bytes, level: int, prefixes: tuple[int, ...], binder: bytes) > Output
is the deterministic, stateless IDPFkey evaluation algorithm run by each Aggregator. Its inputs are the Aggregator's unique identifier, the public share distributed to all of the Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the IDPF, the sequence of candidate prefixes, and a binder string. It returns the share of the value corresponding to each candidate prefix.¶The output type (i.e.,
Output
) depends on the value oflevel
: Iflevel < BITS1
, the output is the value for an inner node, which has typelist[list[FieldInner]]
; otherwise, iflevel == BITS1
, then the output is the value for a leaf node, which has typelist[list[FieldLeaf]]
.¶Preconditions:¶
In addition, the following method is derived for each concrete Idpf
:¶
def current_field( self, level: int) > Union[type[FieldInner], type[FieldLeaf]]: if level < self.BITS  1: return self.field_inner return self.field_leaf¶
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 IDPFkey 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 IDPFkey generator. Equal to twice the XOF's seed size. 
KEY_SIZE  Size in bytes of each IDPF key 
FieldInner  Implementation of Field (Section 6.1) used for values of inner nodes 
FieldLeaf  Implementation of Field used for values of leaf nodes 
Output  Alias of Union[list[list[FieldInner]], list[list[FieldLeaf]]]

FieldVec  Alias of Union[list[FieldInner], list[FieldLeaf]]

8.1.1. Encoding inputs as indices
How data are represented as IDPF indices is up to the application. When the
inputs are fixedlength byte strings, the most natural choice of encoder is
from_be_bytes()
. This ensures that, when a string is a prefix of another, so
too is its index. (Index prefixes are defined in Section 8.1). For example,¶
from_be_bytes(b"\x01\x02") == 0x0102¶
is a prefix of¶
from_be_bytes(b"\x01\x02\x03") == 0x010203¶
When the inputs are variable length, it is necessary to pad each input to some
fixed length. Further, the padding scheme must be nonambiguous. For example,
each input could be padded with b"\x01"
followed by as many b"\x00"
bytes
as needed.¶
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
SHARES == 2
and VALUE_LEN == 2
and an implementation of Xof
(Section 6.2). The associated constants and types required by the Vdaf
interface
are defined in Table 20. The methods required for sharding,
preparation, aggregation, and unsharding are described in the remaining
subsections. These methods make use of constants defined in Table 21.¶
Parameter  Value 

VERIFY_KEY_SIZE

Xof.SEED_SIZE

RAND_SIZE

Xof.SEED_SIZE * 3 + Idpf.RAND_SIZE

NONCE_SIZE

16

ROUNDS

2

SHARES

2

Measurement

int

AggParam

tuple[int, Sequence[int]]

PublicShare

bytes (IDPF public share) 
InputShare

tuple[bytes, bytes, list[FieldInner], list[FieldLeaf]]

OutShare

FieldVec

AggShare

FieldVec

AggResult

list[int]

PrepState

tuple[bytes, int, FieldVec]

PrepShare

FieldVec

PrepMessage

Optional[FieldVec]

Variable  Value 

USAGE_SHARD_RAND: int  1 
USAGE_CORR_INNER: int  2 
USAGE_CORR_LEAF: int  3 
USAGE_VERIFY_RAND: int  4 
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 onehot 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 onehot 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.¶
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. This function checks that levels are increasing between calls, and also enforces that the prefixes at each level are suffixes of the previous level's prefixes.¶
8.2.4. Aggregation
Aggregation involves simply adding up the output shares.¶
8.2.5. Unsharding
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
8.2.6. Message Serialization
This section defines serialization formats for messages exchanged over the
network while executing Poplar1
. It is RECOMMENDED that implementations
provide serialization methods for them.¶
Message structures are defined following Section 3 of [RFC8446]). In the
remainder we use S
as an alias for Poplar1.xof.SEED_SIZE
, Fi
as an alias
for Poplar1.idpf.field_inner
and Fl
as an alias for
Poplar1.idpf.field_leaf
. XOF seeds are represented as follows:¶
opaque Poplar1Seed[S];¶
Elements of the inner field are encoded in littleendian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldInner[Fi];¶
Likewise, elements of the leaf field are encoded in littleendian byte order (as defined in Section 6.1) and are represented as follows:¶
opaque Poplar1FieldLeaf[Fl];¶
8.2.6.4. Prep Message
Likewise, the structure of the prep message for Poplar1 depends on the sketching round and field. For the first round and inner field:¶
struct { Poplar1FieldInner[Fi * 3]; } Poplar1PrepMessageRoundOneInner;¶
For the first round and leaf field:¶
struct { Poplar1FieldLeaf sketch[Fl * 3]; } Poplar1PrepMessageRoundOneLeaf;¶
Note that these messages have the same structures as the prep shares for the first round.¶
The secondround prep message is the empty string. This is because the sketch shares are expected to sum to a particular value if the output shares are valid; we represent a successful preparation with the empty string and otherwise return an error.¶
8.2.6.6. Aggregation Parameter
The aggregation parameter is encoded as follows:¶

TODO(issue #255) Express the aggregation parameter encoding in TLS syntax. Decide whether to RECOMMEND this encoding, and if so, add it to test vectors.¶
def encode_agg_param(self, agg_param: Poplar1AggParam) > bytes: level, prefixes = agg_param if level not in range(2 ** 16): raise ValueError('level out of range') if len(prefixes) not in range(2 ** 32): raise ValueError('number of prefixes out of range') encoded = bytes() encoded += to_be_bytes(level, 2) encoded += to_be_bytes(len(prefixes), 4) packed = 0 for (i, prefix) in enumerate(prefixes): packed = prefix << ((level + 1) * i) l = ((level + 1) * len(prefixes) + 7) // 8 encoded += to_be_bytes(packed, l) return encoded def decode_agg_param(self, encoded: bytes) > Poplar1AggParam: encoded_level, encoded = encoded[:2], encoded[2:] level = from_be_bytes(encoded_level) encoded_prefix_count, encoded = encoded[:4], encoded[4:] prefix_count = from_be_bytes(encoded_prefix_count) l = ((level + 1) * prefix_count + 7) // 8 encoded_packed, encoded = encoded[:l], encoded[l:] packed = from_be_bytes(encoded_packed) prefixes = [] m = 2 ** (level + 1)  1 for i in range(prefix_count): prefixes.append(packed >> ((level + 1) * i) & m) if len(encoded) != 0: raise ValueError('trailing bytes') return (level, tuple(prefixes))¶
Implementation note: The aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform perreport caching across executions of the VDAF, this may be more information than is strictly needed. In particular, it may be sufficient to convey which indices from the previous execution will have their children included in the next. This would help reduce communication overhead.¶
8.3. The IDPF scheme of [BBCGGI21]
In this section we specify a concrete IDPF suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
The constant and type definitions required by the Idpf
interface are given in
Table 22.¶
Our IDPF requires an XOF for deriving the output shares, as well as a variety of other artifacts used internally. For performance reasons, we instantiate this object using XofFixedKeyAes128 (Section 6.2.2). See Section 9.5 for justification of this choice.¶
Parameter  Value 

SHARES 
2

BITS  any positive integer 
VALUE_LEN  any positive integer 
KEY_SIZE 
Xof.SEED_SIZE

FieldInner 
Field64 (Table 3) 
FieldLeaf 
Field255 (Table 3) 
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 IDPFkey 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)
.¶
8.3.2. Key Evaluation

TODO Describe in prose how IDPFkey evaluation algorithm works.¶
The description of the IDPFevaluation algorithm makes use of auxiliary
functions extend()
, convert()
, and decode_public_share()
defined in
Section 8.3.3.¶
8.3.3. Auxiliary Functions
Here, pack_bits()
takes a list of bits, packs each group of eight bits into a
byte, in LSB to MSB order, padding the most significant bits of the last byte
with zeros as necessary, and returns the byte array. unpack_bits()
performs
the reverse operation: it takes in a byte array and a number of bits, and
returns a list of bits, extracting eight bits from each byte in turn, in LSB to
MSB order, and stopping after the requested number of bits. If the byte array
has an incorrect length, or if unused bits in the last bytes are not zero, it
throws an error.¶
8.4. Instantiation
By default, Poplar1 is instantiated with the IDPF in Section 8.3 (VALUE_LEN
== 2
) and XofTurboShake128 (Section 6.2.1). This VDAF is suitable for
any positive value of BITS
. Test vectors can be found in Appendix "Test Vectors".¶
9. Security Considerations
VDAFs have two essential security goals:¶

Privacy: An attacker that controls 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. We assume the attacker controls the entire network except for channels between honest Clients and honest Aggregators. In particular, it cannot forge or prevent transmission of messages on these channels.¶

Robustness: An attacker that controls a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients. We assume the attacker eavesdrops on the network but does not control transmission of messages between honest parties.¶
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 longlived parameters, in particular the verification key.¶

Establishing secure channels:¶

Enforcing the noncollusion properties required of the specific VDAF in use.¶
In such an environment, a VDAF provides the highlevel privacy property described above: The Collector learns only the aggregate measurement, and nothing about individual measurements aside from what can be inferred from the aggregate result. The Aggregators learn neither individual measurements nor the aggregate result. The Collector is assured that the aggregate statistic accurately reflects the inputs as long as the Aggregators correctly executed their role in the VDAF.¶
On their own, VDAFs do not provide:¶

Mitigation of 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.¶

Differential privacy [Dwo06]. Depending on the distribution of the measurements, the aggregate result itself can still leak a significant amount of information about an individual measurement or the person that generated it.¶

Robustness in the presence of a malicious Aggregator. An Aggregator can, without detection, manipulate the aggregate result by modifying its own aggregate share.¶

Guaranteed output delivery [GSZ20]. An attacker that controls transmission of messages between honest parties can prevent computation of the aggregate result by dropping messages.¶
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:¶

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.¶

To ensure privacy of the measurements, the Aggregators MUST commit to the verification key prior to processing reports generated by Clients. Otherwise, a malicious Aggregator may be able to craft a verification key that, for a given report, causes an honest Aggregator to leak information about the measurement during preparation.¶
Meeting these conditions is required in order to leverage security analysis in the framework of [DPRS23]. Their definition of robustness allows the attacker, playing the role of a cohort of malicious Clients, to submit arbitrary reports to the Aggregators and eavesdrop on their communications as they process them. Security in this model is achievable as long as the verification key is kept secret from the attacker.¶
The privacy definition of [DPRS23] considers an active attacker that controls the network and a subset of Aggregators; in addition, the attacker is allowed to choose the verification key used by each honest Aggregator over the course of the experiment. Security is achievable in this model as long as the key is picked at the start of the experiment, prior to any reports being generated. (The model also requires nonces to be generated at random; see Section 9.2 below.)¶
Meeting these requirements is relatively straightforward. For example, the Aggregators may designate one of their peers to generate the verification key and distribute it to the others. To assure Clients of key commitment, the Clients and (honest) Aggregators could bind reports to a shared context string derived from the key. For instance, the "task ID" of DAP [DAP] could be set to the hash of the verification key; then as long as honest Aggregators only consume reports for the task indicated by the Client, forging a new key after the fact would reduce to finding collisions in the underlying hash function. (Keeping the key secret from the Clients would require the hash function to be oneway.) 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 nonrepeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a report 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.¶
9.4. Requirements for Aggregation Parameters
As described in Section 4.3 and Section 5.3 respectively, DAFs and VDAFs may impose restrictions on the reuse of input shares. This is to ensure that correlated randomness provided by the Client through the input share is not used more than once, which might compromise confidentiality of the Client's measurements.¶
Protocols that make use of VDAFs therefore MUST call vdaf.is_valid
on the set of all aggregation parameters used for a Client's input share, and
only proceed with the preparation and aggregation phases if that function call
returns True
.¶
9.4.1. Additional Privacy Considerations
Aggregating a batch of reports multiple times, each time with a different aggregation parameter, could result in information leakage beyond what is used by the application.¶
For example, when Poplar1 is used for heavy hitters, the Aggregators learn not only the heavy hitters themselves, but also the prefix tree (as defined in Section 8) computed along the way. Indeed, this leakage is inherent to any construction that uses an IDPF (Section 8.1) in the same way. Depending on the distribution of the measurements, the prefix tree can leak a significant amount of information about unpopular inputs. For instance, it is possible (though perhaps unlikely) for a large set of nonheavyhitter values to share a common prefix, which would be leaked by a prefix tree with a sufficiently small threshold.¶
A malicious adversary controlling the Collector and one of the Aggregators can further turn arbitrary nonheavy prefixes into heavy ones by tampering with the IDPF output at any position. While our construction ensures that the nodes evaluated at one level are children of the nodes evaluated at the previous level, this still may allow an adversary to discover individual nonheavy strings.¶
The only practical, generalpurpose approach to mitigating these leakages is via differential privacy, which is RECOMMENDED for all protocols using Poplar1 for heavyhitter type applications.¶
9.4.2. Safe Usage of IDPF Outputs
The arithmetic sketch described in Section 8 is used by the Aggregators to check
that the shares of the vector obtained by evaluating a Client's IDPF at a
sequence of candidate prefixes has at most one nonzero value, and that the
nonzero value is 1
. Depending on how the values are used, the arithmetic sketch
on its own may not be sufficient for robustness of the application. In
particular, a malicious Client may attempt to influence the computation by
choosing an IDPF that evaluates to 1
at more than one node at a given
level of the tree.¶
This issue can be mitigated by using an IDPF that is extractable as defined in in Appendix D of [BBCGGI21]. Extractability ensures that, for a particular level of the tree, it is infeasible for an attacker to control values of the IDPF such that it takes on chosen nonzero values at more than one node. (It can practically only achieve the zero function, a point function, or a pseudorandom function.)¶
The IDPF specified in Section 8.1 only guarantees extractability at the last level of the tree. (This is by virtue of using a larger field for the leaves than for the inner nodes.) For intermediate levels, it is feasible for a client to produce IDPF shares with two controlled nonzero nodes.¶
This is not an issue for running heavy hitters, since (1) each node in the prefix tree is a child of a previously traversed node, (2) the arithmetic sketch would detect double voting at every level of the prefix tree, and (3) the IDPF is extractable at the last level of the tree. However, the lack of extractability at intermediate levels may result in attacks on the robustness of certain applications.¶
Thus applications SHOULD NOT use prefix counts for intermediate levels for any purpose beyond the heavyhitters tree traversal.¶
9.5. Requirements for XOFs
As described in Section 6.2, our constructions rely on eXtendable Output Functions (XOFs). In the security analyses of our protocols, these are usually modeled as random oracles. XofTurboShake128 is designed to be indifferentiable from a random oracle [MRH04], making it a suitable choice for most situations.¶
The one exception is the IDPF of Section 8.3. Here, a random oracle is not
needed to prove privacy, since the analysis of [BBCGGI21], Proposition 1, only
requires a Pseudorandom Generator (PRG). As observed in [GKWY20], a PRG can be
instantiated from a correlationrobust hash function H
. Informally,
correlation robustness requires that for a random r
, H(xor(r, x))
is
computationally indistinguishable from a random function of x
. A PRG can
therefore be constructed as¶
PRG(r) = H(xor(r, 1))  H(xor(r, 2))  ...¶
since each individual hash function evaluation is indistinguishable from a random function.¶
Our construction at Section 6.2.2 implements a correlationrobust hash function using fixedkey AES. For security, it assumes that AES with a fixed key can be modeled as a random permutation [GKWY20]. Additionally, we use a different AES key for every client, which in the ideal cipher model leads to better concrete security [GKWWY20].¶
We note that for robustness, the analysis of [BBCGGI21] still assumes a random oracle to make the Idpf extractable. While XofFixedKeyAes128 has been shown to be differentiable from a random oracle [GKWWY20], there are no known attacks exploiting this difference. We also stress that even if the Idpf is not extractable, Poplar1 guarantees that every client can contribute to at most one prefix among the ones being evaluated by the helpers.¶
9.6. Choosing the Field Size
Prio3 and other systems built from FLPs (Section 7.3 in particular) may benefit from choosing a field size that is as small as possible. Generally speaking, a smaller field results in lower communication and storage costs. Care must be taken, however, since a smaller field also results in degraded (or even vacuous) robustness.¶
Different variants of Prio3 (Section 7) use different field sizes: Prio3Count uses Field64; but Prio3Sum, Prio3SumVec, and Prio3Histogram use Field128, a field that is twice as large as Field64. This is due to the use of joint randomness (Section 7.1) in the latter variants. Joint randomness allows for more flexible circuit design (see Section 7.3.1.1), but opens up Prio3 to precomputation attacks, which the larger field mitigates. (See [DPRS23], Theorem 1.) Note that privacy is not susceptible to such attacks.¶
Another way to mitigate this issue (or improve robustness in general) is to
generate and verify multiple, independent proofs. (See Section 7.1.2.) For
Prio3, the PROOFS
parameter controls the number of proofs (at least one) that
are generated and verified.¶
In general, Field128 is RECOMMENDED for use in Prio3 when the circuit uses
joint randomness (JOINT_RAND_LEN > 0
) and PROOFS == 1
. Field64 MAY be used
instead, but PROOFS
MUST be set to at least 3
. Breaking robustness for
PROOFS == 2
is feasible, if impractical; but PROOFS == 1
is completely
broken for such a small field.¶
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.¶
Value  Scheme  Type  Reference 

0x00000000

Prio3Count  VDAF  Section 7.4.1 
0x00000001

Prio3Sum  VDAF  Section 7.4.2 
0x00000002

Prio3SumVec  VDAF  Section 7.4.3 
0x00000003

Prio3Histogram  VDAF  Section 7.4.4 
0x00000004

Prio3MultihotCountVec  VDAF  Section 7.4.5 
0x00000005 to 0x00000FFF

reserved for Prio3  VDAF  n/a 
0x00001000

Poplar1  VDAF  Section 8.4 
0xFFFF0000 to 0xFFFFFFFF

reserved  n/a  n/a 
11. References
11.1. Normative References
 [RFC2119]
 Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, , <https://www.rfceditor.org/rfc/rfc2119>.
 [RFC8174]
 Leiba, B., "Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words", BCP 14, RFC 8174, DOI 10.17487/RFC8174, , <https://www.rfceditor.org/rfc/rfc8174>.
 [RFC8446]
 Rescorla, E., "The Transport Layer Security (TLS) Protocol Version 1.3", RFC 8446, DOI 10.17487/RFC8446, , <https://www.rfceditor.org/rfc/rfc8446>.
 [TurboSHAKE]
 Viguier, B., Wong, D., Van Assche, G., Dang, Q., and J. Daemen, "KangarooTwelve and TurboSHAKE", Work in Progress, InternetDraft, draftirtfcfrgkangarootwelve14, , <https://datatracker.ietf.org/doc/html/draftirtfcfrgkangarootwelve14>.
11.2. Informative References
 [AGJOP21]
 Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A. Polychroniadou, "Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares", , <https://ia.cr/2021/576>.
 [BBCGGI19]
 Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via Fully Linear PCPs", CRYPTO 2019 , , <https://ia.cr/2019/188>.
 [BBCGGI21]
 Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "Lightweight Techniques for Private Heavy Hitters", IEEE S&P 2021 , , <https://ia.cr/2021/017>.
 [BBCGGI23]
 Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "Arithmetic Sketching", CRYPTO 2023 , , <https://ia.cr/2023/1012>.
 [CGB17]
 CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust, and Scalable Computation of Aggregate Statistics", NSDI 2017 , , <https://dl.acm.org/doi/10.5555/3154630.3154652>.
 [DAP]
 Geoghegan, T., Patton, C., Pitman, B., Rescorla, E., and C. A. Wood, "Distributed Aggregation Protocol for Privacy Preserving Measurement", Work in Progress, InternetDraft, draftietfppmdap11, , <https://datatracker.ietf.org/doc/html/draftietfppmdap11>.
 [Dou02]
 Douceur, J., "The Sybil Attack", IPTPS 2002 , , <https://doi.org/10.1007/3540457488_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 , , <https://link.springer.com/chapter/10.1007/11787006_1>.
 [ENPA]
 "Exposure Notification Privacypreserving Analytics (ENPA) White Paper", , <https://covid19static.cdnapple.com/applications/covid19/current/static/contacttracing/pdf/ENPA_White_Paper.pdf>.
 [EPK14]
 Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR: Randomized Aggregatable PrivacyPreserving Ordinal Response", CCS 2014 , , <https://dl.acm.org/doi/10.1145/2660267.2660348>.
 [GI14]
 Gilboa, N. and Y. Ishai, "Distributed Point Functions and Their Applications", EUROCRYPT 2014 , , <https://link.springer.com/chapter/10.1007/9783642552205_35>.
 [GKWWY20]
 Guo, C., Katz, J., Wang, X., Weng, C., and Y. Yu, "Better concrete security for halfgates garbling (in the multiinstance setting)", CRYPTO 2020 , , <https://link.springer.com/chapter/10.1007/9783030568801_28>.
 [GKWY20]
 Guo, C., Katz, J., Wang, X., and Y. Yu, "Efficient and Secure Multiparty Computation from FixedKey Block Ciphers", S&P 2020 , , <https://eprint.iacr.org/2019/074>.
 [GSZ20]
 Goyal, V., Song, Y., and C. Zhu, "Guaranteed Output Delivery Comes Free in Honest Majority MPC", CRYPTO 2020 , , <https://link.springer.com/chapter/10.1007/9783030568801_22>.
 [MPRV09]
 Mironov, I., Pandey, O., Reingold, O., and S. Vadhan, "Computational Differential Privacy", CRYPTO 2009 , n.d., <https://link.springer.com/chapter/10.1007/9783642033568_8>.
 [MRH04]
 Maurer, U., Renner, R., and C. Holenstein, "Indifferentiability, impossibility results on reductions, and applications to the random oracle methodology", In TCC 2004: Theory of Cryptography, pages 2139, DOI 10.1007/9783540246381_2, , <https://doi.org/10.1007/9783540246381_2>.
 [OriginTelemetry]
 "Origin Telemetry", , <https://firefoxsourcedocs.mozilla.org/toolkit/components/telemetry/collection/origin.html>.
Acknowledgments
The security considerations in Section 9 are based largely on the security analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their time to developing definitions and security proofs.¶
Thanks to Junye Chen, Henry CorriganGibbs, Armando FazHernández, Simon Friedberger, Tim Geoghegan, Albert Liu, Brandon Pitman, Mariana Raykova, Jacob Rothstein, Shan Wang, Xiao Wang, Bas Westerbaan, and Christopher Wood for useful feedback on and contributions to the spec.¶
Test Vectors
(TO BE REMOVED BY RFC EDITOR: Machinereadable test vectors can be found at https://github.com/cfrg/draftirtfcfrgvdaf/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

TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/test_vec) and format them for humans.¶
Prio3Sum

TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/test_vec) and format them for humans.¶
Prio3SumVec

TODO Copy the machine readable vectors from the source repository (https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/test_vec) and format them for humans.¶
Prio3Histogram

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