Verifiable Distributed Aggregation Functions
draftirtfcfrgvdaf05
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draftirtfcfrgvdaf05
CFRG R. L. Barnes InternetDraft Cisco Intended status: Informational D. Cook Expires: 14 September 2023 ISRG C. Patton Cloudflare P. Schoppmann Google 13 March 2023 Verifiable Distributed Aggregation Functions draftirtfcfrgvdaf05 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 input that would result in an incorrect aggregate result. Discussion Venues This note is to be removed before publishing as an RFC. Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg. Source for this draft and an issue tracker can be found at https://github.com/cjpatton/vdaf. Status of This Memo This 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 Internet Drafts is at https://datatracker.ietf.org/drafts/current/. Barnes, et al. Expires 14 September 2023 [Page 1] InternetDraft VDAF March 2023 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 14 September 2023. Copyright Notice Copyright (c) 2023 IETF Trust and the persons identified as the document authors. All rights reserved. This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/ 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. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License. Table of Contents 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1. Change Log . . . . . . . . . . . . . . . . . . . . . . . 8 2. Conventions and Definitions . . . . . . . . . . . . . . . . . 11 3. Overview . . . . . . . . . . . . . . . . . . . . . . . . . . 12 4. Definition of DAFs . . . . . . . . . . . . . . . . . . . . . 14 4.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 16 4.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 17 4.3. Validity of Aggregation Parameters . . . . . . . . . . . 17 4.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 17 4.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 18 4.6. Execution of a DAF . . . . . . . . . . . . . . . . . . . 19 5. Definition of VDAFs . . . . . . . . . . . . . . . . . . . . . 21 5.1. Sharding . . . . . . . . . . . . . . . . . . . . . . . . 23 5.2. Preparation . . . . . . . . . . . . . . . . . . . . . . . 23 5.3. Validity of Aggregation Parameters . . . . . . . . . . . 26 5.4. Aggregation . . . . . . . . . . . . . . . . . . . . . . . 26 5.5. Unsharding . . . . . . . . . . . . . . . . . . . . . . . 27 5.6. Execution of a VDAF . . . . . . . . . . . . . . . . . . . 27 6. Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . 28 6.1. Finite Fields . . . . . . . . . . . . . . . . . . . . . . 29 6.1.1. Auxiliary Functions . . . . . . . . . . . . . . . . . 30 6.1.2. FFTFriendly Fields . . . . . . . . . . . . . . . . . 31 6.1.3. Parameters . . . . . . . . . . . . . . . . . . . . . 31 6.2. Pseudorandom Generators . . . . . . . . . . . . . . . . . 32 6.2.1. PrgSha3 . . . . . . . . . . . . . . . . . . . . . . . 33 Barnes, et al. Expires 14 September 2023 [Page 2] InternetDraft VDAF March 2023 6.2.2. PrgFixedKeyAes128 . . . . . . . . . . . . . . . . . . 34 6.2.3. The Customization and Binder Strings . . . . . . . . 35 7. Prio3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 7.1. Fully Linear Proof (FLP) Systems . . . . . . . . . . . . 37 7.1.1. Encoding the Input . . . . . . . . . . . . . . . . . 40 7.2. Construction . . . . . . . . . . . . . . . . . . . . . . 41 7.2.1. Sharding . . . . . . . . . . . . . . . . . . . . . . 43 7.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 46 7.2.3. Validity of Aggregation Parameters . . . . . . . . . 48 7.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 48 7.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 48 7.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 49 7.3. A GeneralPurpose FLP . . . . . . . . . . . . . . . . . . 52 7.3.1. Overview . . . . . . . . . . . . . . . . . . . . . . 52 7.3.2. Validity Circuits . . . . . . . . . . . . . . . . . . 55 7.3.3. Construction . . . . . . . . . . . . . . . . . . . . 57 7.4. Instantiations . . . . . . . . . . . . . . . . . . . . . 60 7.4.1. Prio3Count . . . . . . . . . . . . . . . . . . . . . 60 7.4.2. Prio3Sum . . . . . . . . . . . . . . . . . . . . . . 61 7.4.3. Prio3Histogram . . . . . . . . . . . . . . . . . . . 63 8. Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 8.1. Incremental Distributed Point Functions (IDPFs) . . . . . 66 8.2. Construction . . . . . . . . . . . . . . . . . . . . . . 68 8.2.1. Client . . . . . . . . . . . . . . . . . . . . . . . 69 8.2.2. Preparation . . . . . . . . . . . . . . . . . . . . . 72 8.2.3. Validity of Aggregation Parameters . . . . . . . . . 75 8.2.4. Aggregation . . . . . . . . . . . . . . . . . . . . . 75 8.2.5. Unsharding . . . . . . . . . . . . . . . . . . . . . 75 8.2.6. Auxiliary Functions . . . . . . . . . . . . . . . . . 75 8.3. The IDPF scheme of BBCGGI21 . . . . . . . . . . . . . . . 77 8.3.1. Key Generation . . . . . . . . . . . . . . . . . . . 78 8.3.2. Key Evaluation . . . . . . . . . . . . . . . . . . . 79 8.3.3. Auxiliary Functions . . . . . . . . . . . . . . . . . 81 8.4. Instantiation . . . . . . . . . . . . . . . . . . . . . . 83 9. Security Considerations . . . . . . . . . . . . . . . . . . . 83 9.1. Requirements for the Verification Key . . . . . . . . . . 84 9.2. Requirements for the Nonce . . . . . . . . . . . . . . . 85 9.3. Requirements for the Aggregation Parameters . . . . . . . 85 9.4. Additional Privacy Considerations for Aggregation Parameters . . . . . . . . . . . . . . . . . . . . . . . 86 9.5. Pseudorandom Generators and random oracles . . . . . . . 86 10. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 86 11. References . . . . . . . . . . . . . . . . . . . . . . . . . 87 11.1. Normative References . . . . . . . . . . . . . . . . . . 87 11.2. Informative References . . . . . . . . . . . . . . . . . 87 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 89 Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Prio3Count . . . . . . . . . . . . . . . . . . . . . . . . . . 89 Barnes, et al. Expires 14 September 2023 [Page 3] InternetDraft VDAF March 2023 Prio3Sum . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 Prio3Histogram . . . . . . . . . . . . . . . . . . . . . . . . 91 Poplar1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Sharding . . . . . . . . . . . . . . . . . . . . . . . . . . 93 Preparation, Aggregation, and Unsharding . . . . . . . . . . 93 Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 97 1. Introduction The ubiquity of the Internet makes it an ideal platform for measurement of 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. Most prior approaches to this problem fall under the rubric of "differential privacy (DP)" [Dwo06]. Roughly speaking, a data aggregation system that is differentially private ensures that the degree to which any individual measurement influences the value of the aggregate result can be precisely controlled. For example, in systems like RAPPOR [EPK14], each user samples noise from a well known distribution and adds it to their input before submitting to the aggregation server. The aggregation server then adds up the noisy inputs, and because it knows the distribution from whence the noise was sampled, it can estimate the true sum with reasonable precision. Barnes, et al. Expires 14 September 2023 [Page 4] InternetDraft VDAF March 2023 Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically, depending on the "amount" of noise a client adds to its input, it may be possible for a curious aggregator to make a reasonable guess of the input's true value. Indeed, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility. The ideal goal for a privacypreserving measurement system is that of secure multiparty computation (MPC): No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal. VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of noncolluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of input measurements. In addition to these MPCstyle security goals, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the aggregate result itself does not leak too much information about any one measurement. TODO(issue #94) Provide guidance for local and central DP and point to it here. The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA]. Barnes, et al. Expires 14 September 2023 [Page 5] InternetDraft VDAF March 2023 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 considerations, sometimes in subtle but consequential ways. This document therefore has two important goals: 1. Providing higherlevel protocols like [DAP] with a simple, uniform interface for accessing privacypreserving measurement schemes, and documenting relevant operational and security bounds for that interface: 1. General patterns of communications among the various actors involved in the system (clients, aggregation servers, and the collector of the aggregate result); 2. Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and 3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation. 2. Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions. This document also specifies two concrete VDAF schemes, each based on a protocol from the literature. * The aforementioned Prio system [CGB17] allows for the privacy preserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple: 1. Each client shards its measurement into a sequence of additive shares and distributes the shares among the aggregation servers. 2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate. 3. Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result. The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multiparty protocol for accomplishing this task. Barnes, et al. Expires 14 September 2023 [Page 6] InternetDraft VDAF March 2023 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 theavyhitters problem in a privacypreserving manner. Here each client holds a bitstring of length n, and the goal of the aggregation servers is to compute the set of inputs that occur at least t times. The core primitive used in their protocol is a specialized Distributed Point Function (DPF) [GI14] that allows the servers to "query" their DPF shares on any bitstring of length shorter than or equal to n. As a result of this query, each of the servers has an additive share of a bit indicating whether the string is a prefix of the client's input. The protocol also specifies a multiparty computation for verifying that at most one string among a set of candidates is a prefix of the client's input. In Section 8 we describe a VDAF called Poplar1 that implements this functionality. Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the 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. Barnes, et al. Expires 14 September 2023 [Page 7] InternetDraft VDAF March 2023 1.1. Change Log (*) Indicates a change that breaks wire compatibility with the previous draft. 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() to prep_shares_to_prep(). 04: * Align security considerations with the security analysis of [DPRS23]. * Vdaf: Pass the nonce to the sharding algorithm. * Vdaf: Rather than allow the application to choose the nonce length, have each implementation of the Vdaf interface specify the expected nonce length. (*) * Prg: Split "info string" into two components: the "customization string", intended for domain separation; and the "binder string", used to bind the output to ephemeral values, like the nonce, associated with execution of a (V)DAF. Barnes, et al. Expires 14 September 2023 [Page 8] InternetDraft VDAF March 2023 * 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. (*) Barnes, et al. Expires 14 September 2023 [Page 9] InternetDraft VDAF March 2023 * 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. (*) Barnes, et al. Expires 14 September 2023 [Page 10] InternetDraft VDAF March 2023 * Define new field Field255. 01: * Require that VDAFs specify serialization of aggregate shares. * Define Distributed Aggregation Functions (DAFs). * Prio3: Move proof verifier check from prep_next() to prep_shares_to_prep(). (*) * Remove public parameter and replace verification parameter with a "verification key" and "Aggregator ID". 2. Conventions and Definitions The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here. Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception. A variable with type Bytes is a byte string. This document defines several 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 Unsigned is defined, which algorithms are free to use as desired. Its value SHALL be 5. 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: Unsigned) > Bytes returns an array of zero bytes. The length of output MUST be len. Barnes, et al. Expires 14 September 2023 [Page 11] InternetDraft VDAF March 2023 * gen_rand(len: Unsigned) > Bytes returns an array of random bytes. The length of output MUST be len. * byte(int: Unsigned) > Bytes returns the representation of int as a byte string. The value of int MUST be in [0,256). * concat(parts: Vec[Bytes]) > Bytes returns the concatenation of the input byte strings, i.e., parts[0]  ...  parts[len(parts) 1]. * xor(left: Bytes, right: Bytes) > Bytes returns the bitwise XOR of left and right. An exception is raised if the inputs are not the same length. * to_be_bytes(val: Unsigned, length: Unsigned) > Bytes converts val to bigendian bytes; its value MUST be in range [0, 2^(8*length)). Function from_be_bytes(encoded: Bytes) > Unsigned computes the inverse. * to_le_bytes(val: Unsigned, length: Unsigned) > Bytes converts val to littleendian bytes; its value MUST be in range [0, 2^(8*length)). Function from_le_bytes(encoded: Bytes) > Unsigned computes the inverse. * next_power_of_2(n: Unsigned) > Unsigned returns the smallest integer greater than or equal to n that is also a power of two. 3. Overview Barnes, et al. Expires 14 September 2023 [Page 12] InternetDraft VDAF March 2023 ++ +> Aggregator 0 +  ++   ^      V   ++   +> Aggregator 1 +    ++   +++  ^  +>++  Client +  +> Collector > Aggregate +++ +>++  ...        V   ++  +> Aggregator N1 + ++ Input shares Aggregate shares Figure 1: Overall data flow of a (V)DAF 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 convert 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 input, at the end of this process, each aggregator holds one output share.  In VDAFs, Aggregators will need to exchange information among themselves as part of the validation process. * Each Aggregator combines the output shares across inputs in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result. Barnes, et al. Expires 14 September 2023 [Page 13] InternetDraft VDAF March 2023 * The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch. Aggregators are a new class of actor relative to traditional measurement systems where clients submit measurements to a single server. They are critical for both the privacy properties of the system and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by non collusion among Aggregators, and Aggregators are the entities that perform validation of Client measurements. Thus clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to correctly run the protocol. Within the bounds of the 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 client server 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: Barnes, et al. Expires 14 September 2023 [Page 14] InternetDraft VDAF March 2023 * Sharding  Each Client generates input shares from its measurement and distributes them among the Aggregators. * Preparation  Each Aggregator converts each input share into an output share compatible with the aggregation function. This computation involves the aggregation parameter. In general, each aggregation parameter may result in a different an output share. * Aggregation  Each Aggregator combines a sequence of output shares into its aggregate share and sends the aggregate share to the Collector. * Unsharding  The Collector combines the aggregate shares into the aggregate result. Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares). A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table. +=============+====================================+  Parameter  Description  +=============+====================================+  ID  Algorithm identifier for this DAF.  +++  SHARES  Number of input shares into which    each measurement is sharded  +++  RAND_SIZE  Size of the random byte string    passed to sharding algorithm  +++  Measurement  Type of each measurement  +++  AggParam  Type of aggregation parameter  +++  OutShare  Type of each output share  +++  AggResult  Type of the aggregate result  +++ Table 1: Constants and types defined by each concrete DAF. Barnes, et al. Expires 14 September 2023 [Page 15] InternetDraft VDAF March 2023 These types define some of the inputs and outputs of DAF methods at various stages of the computation. Observe that only the measurements, output shares, the aggregate result, and the aggregation parameter have an explicit type. All other values  in particular, the input shares and the aggregate shares  have type Bytes and are treated as opaque byte strings. This is because these values must be transmitted between parties over a network. OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. This way each method parameter has a meaningful type hint. See issue#58. Each DAF is identified by a unique, 32bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 16. 4.1. Sharding In order to protect the privacy of its measurements, a DAF Client shards its measurements into a sequence of input shares. The measurement_to_input_shares method is used for this purpose. * Daf.measurement_to_input_shares(input: Measurement, rand: Bytes[Daf.RAND_SIZE]) > (Bytes, Vec[Bytes]) is the randomized sharding algorithm run by each Client. (The input rand consists of the random coins consumed by the algorithm.) It consumes the measurement and produces a "public share", distributed to each of the Aggregators, and a corresponding sequence of input shares, one for each Aggregator. The length of the output vector MUST be SHARES. Client ====== measurement  V ++  measurement_to_input_shares  ++   ...  V V V input_share_0 input_share_1 input_share_[SHARES1]   ...  V V V Aggregator 0 Aggregator 1 Aggregator SHARES1 Barnes, et al. Expires 14 September 2023 [Page 16] InternetDraft VDAF March 2023 Figure 2: The Client divides its measurement into input shares and distributes them to the Aggregators. 4.2. Preparation Once an Aggregator has received the public share and one of the input shares, the next step is to prepare the input share for aggregation. This is accomplished using the following algorithm: * Daf.prep(agg_id: Unsigned, agg_param: AggParam, public_share: Bytes, input_share: Bytes) > OutShare is the deterministic preparation algorithm. It takes as input the public share and one of the input shares generated by a Client, the Aggregator's unique identifier, and the aggregation parameter selected by the Collector and returns an output share. The protocol in which the DAF is used MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of input_share in the sequence of input shares output by the Client. 4.3. Validity of Aggregation Parameters Concrete DAFs implementations MAY impose certain restrictions for input shares and aggregation parameters. Protocols using a DAF MUST ensure that for each input share and aggregation parameter agg_param, Daf.prep is only called if Daf.is_valid(agg_param, previous_agg_params) returns True, where previous_agg_params contains all aggregation parameters that have previously been used with the same input share. DAFs MUST implement the following function: * Daf.is_valid(agg_param: AggParam, previous_agg_params: Vec[AggParam]) > Bool: Checks if the agg_param is compatible with all elements of previous_agg_params. 4.4. Aggregation Once an Aggregator holds output shares for a batch of measurements (where batches are defined by the application), it combines them into a share of the desired aggregate result: * Daf.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare]) > agg_share: Bytes is the deterministic aggregation algorithm. It is run by each Aggregator a set of recovered output shares. Barnes, et al. Expires 14 September 2023 [Page 17] InternetDraft VDAF March 2023 Aggregator 0 Aggregator 1 Aggregator SHARES1 ============ ============ =================== out_share_0_0 out_share_1_0 out_share_[SHARES1]_0 out_share_0_1 out_share_1_1 out_share_[SHARES1]_1 out_share_0_2 out_share_1_2 out_share_[SHARES1]_2 ... ... ... out_share_0_B out_share_1_B out_share_[SHARES1]_B    V V V ++ ++ ++  out2agg   out2agg  ...  out2agg  ++ ++ ++    V V V agg_share_0 agg_share_1 agg_share_[SHARES1] Figure 3: Aggregation of output shares. `B` indicates the number of measurements in the batch. 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.agg_shares_to_result(agg_param: AggParam, agg_shares: Vec[Bytes], num_measurements: Unsigned) > AggResult is run by the Collector in order to compute the aggregate result from the Aggregators' shares. The length of agg_shares MUST be SHARES. num_measurements is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic. Barnes, et al. Expires 14 September 2023 [Page 18] InternetDraft VDAF March 2023 Aggregator 0 Aggregator 1 Aggregator SHARES1 ============ ============ =================== agg_share_0 agg_share_1 agg_share_[SHARES1]    V V V ++  agg_shares_to_result  ++  V agg_result Collector ========= Figure 4: Computation of the final aggregate result from aggregate shares. QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [DAP].) Or is this outofscope of this document? See https://github.com/ ietfwgppm/ppmspecification/issues/19. 4.6. Execution of a DAF Securely executing a DAF involves emulating the following procedure. Barnes, et al. Expires 14 September 2023 [Page 19] InternetDraft VDAF March 2023 def run_daf(Daf, agg_param: Daf.AggParam, measurements: Vec[Daf.Measurement]): out_shares = [ [] for j in range(Daf.SHARES) ] for measurement in measurements: # Each Client shards its measurement into input shares and # distributes them among the Aggregators. rand = gen_rand(Daf.RAND_SIZE) (public_share, input_shares) = \ Daf.measurement_to_input_shares(measurement, rand) # Each Aggregator prepares its input share for aggregation. for j in range(Daf.SHARES): out_shares[j].append( Daf.prep(j, agg_param, public_share, input_shares[j])) # Each Aggregator aggregates its output shares into an aggregate # share and sends it to the Collector. agg_shares = [] for j in range(Daf.SHARES): agg_share_j = Daf.out_shares_to_agg_share(agg_param, out_shares[j]) agg_shares.append(agg_share_j) # Collector unshards the aggregate result. num_measurements = len(measurements) agg_result = Daf.agg_shares_to_result(agg_param, agg_shares, num_measurements) return agg_result Figure 5: Execution of a DAF. The inputs to this procedure are the same as the aggregation function computed by the DAF: An aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a "benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure 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. Barnes, et al. Expires 14 September 2023 [Page 20] InternetDraft VDAF March 2023 5. Definition of VDAFs Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Client generated measurements. Evaluation of a VDAF involves the same four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the computation's correctness. Accommodating this interaction will require syntactic changes. Overall execution of a VDAF comprises the following stages: * Sharding  Computing input shares from an individual measurement * Preparation  Conversion and verification of input shares to output shares compatible with the aggregation function being computed * Aggregation  Combining a sequence of output shares into an aggregate share * Unsharding  Combining a sequence of aggregate shares into an aggregate result In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: Verification of the validity of the recovered output shares. This process ensures that aggregating the output shares will not lead to a garbled aggregate result. The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF. Barnes, et al. Expires 14 September 2023 [Page 21] InternetDraft VDAF March 2023 +=================+==========================================+  Parameter  Description  +=================+==========================================+  ID  Algorithm identifier for this VDAF  +++  VERIFY_KEY_SIZE  Size (in bytes) of the verification key    (Section 5.2)  +++  RAND_SIZE  Size of the random byte string passed to    sharding algorithm  +++  NONCE_SIZE  Size (in bytes) of the nonce  +++  ROUNDS  Number of rounds of communication during    the Preparation stage (Section 5.2)  +++  SHARES  Number of input shares into which each    measurement is sharded (Section 5.1)  +++  Measurement  Type of each measurement  +++  AggParam  Type of aggregation parameter  +++  Prep  State of each Aggregator during    Preparation (Section 5.2)  +++  OutShare  Type of each output share  +++  AggResult  Type of the aggregate result  +++ Table 2: Constants and types defined by each concrete VDAF. Similarly to DAFs (see {[secdaf}}), any output of a VDAF method that must be transmitted from one party to another is treated as an opaque byte string. All other quantities are given a concrete type. OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. See issue#58. Each VDAF is identified by a unique, 32bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 16. The following method is defined for every VDAF: def custom(Vdaf, usage: Unsigned) > Bytes: return format_custom(0, Vdaf.ID, usage) Barnes, et al. Expires 14 September 2023 [Page 22] InternetDraft VDAF March 2023 It is used to construct a customization string for an instance of Prg used by the VDAF. (See Section 6.2.) 5.1. Sharding Sharding transforms a measurement into input shares as it does in DAFs (cf. Section 4.1); in addition, it takes a nonce as input and produces a public share: * Vdaf.measurement_to_input_shares(measurement: Measurement, nonce: Bytes[Vdaf.NONCE_SIZE], rand: Bytes[Vdaf.RAND_SIZE]) > (Bytes, Vec[Bytes]) is the randomized sharding algorithm run by each Client. (Input rand consists of the random coins consumed by the algorithm.) It consumes the measurement and the nonce and produces a public share, distributed to each of Aggregators, and the corresponding sequence of input shares, one for each Aggregator. Depending on the VDAF, the input shares may encode additional information used to verify the recovered output shares (e.g., the "proof shares" in Prio3 Section 7). The length of the output vector MUST be SHARES. In order to ensure privacy of the measurement, the Client MUST generate the nonce using a cryptographically secure pseudorandom number generator (CSPRNG). (See Section 9 for details.) 5.2. Preparation To recover and verify output shares, the Aggregators interact with one another over ROUNDS rounds. Prior to each round, each Aggregator constructs an outbound message. Next, the sequence of outbound messages is combined into a single message, called a "preparation message". (Each of the outbound messages are called "preparation message shares".) Finally, the preparation message is distributed to the Aggregators to begin the next round. An Aggregator begins the first round with its input share and it begins each subsequent round with the previous preparation message. Its output in the last round is its output share and its output in each of the preceding rounds is a preparationmessage share. This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing inputs for aggregation. Barnes, et al. Expires 14 September 2023 [Page 23] InternetDraft VDAF March 2023 Aggregator 0 Aggregator 1 Aggregator SHARES1 ============ ============ =================== input_share_0 input_share_1 input_share_[SHARES1]   ...  V V V ++ ++ ++  prep_init   prep_init   prep_init  ++ ++ ++   ...  \ V V V  ++ ++ ++   prep_next   prep_next   prep_next   ++ ++ ++    ...   V V V  x ROUNDS ++   prep_shares_to_prep   ++    +++    ...   V V V / ... ... ...    V V V ++ ++ ++  prep_next   prep_next   prep_next  ++ ++ ++   ...  V V V out_share_0 out_share_1 out_share_[SHARES1] Figure 6: VDAF preparation process on the input shares for a single measurement. At the end of the computation, each Aggregator holds an output share or an error. To facilitate the preparation process, a concrete VDAF implements the following class methods: * Vdaf.prep_init(verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE], agg_id: Unsigned, agg_param: AggParam, nonce: Bytes[Vdaf.NONCE_SIZE], public_share: Bytes, input_share: Bytes) > Prep is the deterministic 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 Barnes, et al. Expires 14 September 2023 [Page 24] InternetDraft VDAF March 2023 (nonce, see Figure 7), the public share (public_share), and one of the input shares generated by the client (input_share). Its output is the Aggregator's initial preparation state. It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key prior to generating and processing reports. (See Section 9 for details.) Protocols using the VDAF MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of input_share in the sequence of input shares output by the Client. Protocols MUST ensure that public share consumed by each of the Aggregators is identical. This is security critical for VDAFs such as Poplar1 that require an extractable distributed point function. (See Section 8 for details.) * Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) > Union[Tuple[Prep, Bytes], OutShare] is the deterministic preparationstate update algorithm run by each Aggregator. It updates the Aggregator's preparation state (prep) and returns either its next preparation state and its message share for the current round or, if this is the last round, its output share. An exception is raised if a valid output share could not be recovered. The input of this algorithm is the inbound preparation message or, if this is the first round, None. * Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) > Bytes is the deterministic preparationmessage pre processing algorithm. It combines the preparationmessage shares generated by the Aggregators in the previous round into the preparation message consumed by each in the next round. In effect, each Aggregator moves through a linear state machine with ROUNDS+1 states. The Aggregator enters the first state on using the initialization algorithm, and the update algorithm advances the Aggregator to the next state. Thus, in addition to defining the number of rounds (ROUNDS), a VDAF instance defines the state of the Aggregator after each round. TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class. The 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 Barnes, et al. Expires 14 September 2023 [Page 25] InternetDraft VDAF March 2023 an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all. Note that it is possible for a VDAF to specify ROUNDS == 0, in which case each Aggregator runs the preparationstate update algorithm once and immediately recovers its output share without interacting with the other Aggregators. However, most, if not all, constructions will require some amount of interaction in order to ensure validity of the output shares (while also maintaining privacy). OPEN ISSUE accommodating 0round VDAFs may require syntax changes if, for example, public keys are required. On the other hand, we could consider defining this class of schemes as a different primitive. See issue#77. 5.3. Validity of Aggregation Parameters Similar to DAFs (see Section 4.3), VDAFs MAY impose restrictions for input shares and aggregation parameters. Protocols using a VDAF MUST ensure that for each input share and aggregation parameter agg_param, the preparation phase (including Vdaf.prep_init, Vdaf.prep_next, and Vdaf.prep_shares_to_prep; see Section 5.2) is only called if Vdaf.is_valid(agg_param, previous_agg_params) returns True, where previous_agg_params contains all aggregation parameters that have previously been used with the same input share. VDAFs MUST implement the following function: * Vdaf.is_valid(agg_param: AggParam, previous_agg_params: Vec[AggParam]) > Bool: Checks if the agg_param is compatible with all elements of previous_agg_params. 5.4. Aggregation VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.4): * Vdaf.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare]) > agg_share: Bytes is the deterministic aggregation algorithm. It is run by each Aggregator over the output shares it has computed over a batch of measurement inputs. Barnes, et al. Expires 14 September 2023 [Page 26] InternetDraft VDAF March 2023 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.agg_shares_to_result(agg_param: AggParam, agg_shares: Vec[Bytes], num_measurements: Unsigned) > AggResult is run by the Collector in order to compute the aggregate result from the Aggregators' shares. The length of agg_shares MUST be SHARES. num_measurements is the number of measurements that contributed to each of the aggregate shares. This algorithm is deterministic. The data flow for this stage is illustrated in Figure 4. 5.6. Execution of a VDAF Secure execution of a VDAF involves simulating the following procedure. def run_vdaf(Vdaf, verify_key: Bytes[Vdaf.VERIFY_KEY_SIZE], agg_param: Vdaf.AggParam, nonces: Vec[Bytes[Vdaf.NONCE_SIZE]], measurements: Vec[Vdaf.Measurement]): out_shares = [] for (nonce, measurement) in zip(nonces, measurements): # Each Client shards its measurement into input shares. rand = gen_rand(Vdaf.RAND_SIZE) (public_share, input_shares) = \ Vdaf.measurement_to_input_shares(measurement, nonce, rand) # Each Aggregator initializes its preparation state. prep_states = [] for j in range(Vdaf.SHARES): state = Vdaf.prep_init(verify_key, j, agg_param, nonce, public_share, input_shares[j]) prep_states.append(state) # Aggregators recover their output shares. inbound = None Barnes, et al. Expires 14 September 2023 [Page 27] InternetDraft VDAF March 2023 for i in range(Vdaf.ROUNDS+1): outbound = [] for j in range(Vdaf.SHARES): out = Vdaf.prep_next(prep_states[j], inbound) if i < Vdaf.ROUNDS: (prep_states[j], out) = out outbound.append(out) # This is where we would send messages over the # network in a distributed VDAF computation. if i < Vdaf.ROUNDS: inbound = Vdaf.prep_shares_to_prep(agg_param, outbound) # The final outputs of prepare phase are the output shares. out_shares.append(outbound) # Each Aggregator aggregates its output shares into an # aggregate share. In a distributed VDAF computation, the # aggregate shares are sent over the network. agg_shares = [] for j in range(Vdaf.SHARES): out_shares_j = [out[j] for out in out_shares] agg_share_j = Vdaf.out_shares_to_agg_share(agg_param, out_shares_j) agg_shares.append(agg_share_j) # Collector unshards the aggregate. num_measurements = len(measurements) agg_result = Vdaf.agg_shares_to_result(agg_param, agg_shares, num_measurements) return agg_result Figure 7: Execution of a VDAF. The inputs to this algorithm are the aggregation parameter, a list of measurements, and a nonce for each measurement. This document does not specify how the nonces are chosen, but security requires that the nonces be unique. See Section 9 for details. As explained in Section 4.6, the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol participants. 6. Preliminaries This section describes the primitives that are common to the VDAFs specified in this document. Barnes, et al. Expires 14 September 2023 [Page 28] InternetDraft VDAF March 2023 6.1. Finite Fields Both Prio3 and Poplar1 use finite fields of prime order. Finite field elements are represented by a class Field with the following associated parameters: * MODULUS: Unsigned is the prime modulus that defines the field. * ENCODED_SIZE: Unsigned is the number of bytes used to encode a field element as a byte string. A concrete Field also implements the following class methods: * Field.zeros(length: Unsigned) > output: Vec[Field] returns a vector of zeros. The length of output MUST be length. * Field.rand_vec(length: Unsigned) > output: Vec[Field] returns a vector of random field elements. The length of output MUST be length. A field element is an instance of a concrete Field. The concrete class defines the usual arithmetic operations on field elements. In addition, it defines the following instance method for converting a field element to an unsigned integer: * elem.as_unsigned() > Unsigned returns the integer representation of field element elem. Likewise, each concrete Field implements a constructor for converting an unsigned integer into a field element: * Field(integer: Unsigned) returns integer represented as a field element. The value of integer MUST be less than Field.MODULUS. Finally, each concrete Field has two derived class methods, one for encoding a vector of field elements as a byte string and another for decoding a vector of field elements. Barnes, et al. Expires 14 September 2023 [Page 29] InternetDraft VDAF March 2023 def encode_vec(Field, data: Vec[Field]) > Bytes: encoded = Bytes() for x in data: encoded += to_le_bytes(x.as_unsigned(), Field.ENCODED_SIZE) return encoded def decode_vec(Field, encoded: Bytes) > Vec[Field]: L = Field.ENCODED_SIZE if len(encoded) % L != 0: raise ERR_DECODE vec = [] for i in range(0, len(encoded), L): encoded_x = encoded[i:i+L] x = from_le_bytes(encoded_x) if x >= Field.MODULUS: raise ERR_DECODE # Integer is larger than modulus vec.append(Field(x)) return vec Figure 8: Derived class methods for finite fields. 6.1.1. Auxiliary Functions The following auxiliary functions on vectors of field elements are used in the remainder of this document. Note that an exception is raised by each function if the operands are not the same length. # Compute the inner product of the operands. def inner_product(left: Vec[Field], right: Vec[Field]) > Field: return sum(map(lambda x: x[0] * x[1], zip(left, right))) # Subtract the right operand from the left and return the result. def vec_sub(left: Vec[Field], right: Vec[Field]): return list(map(lambda x: x[0]  x[1], zip(left, right))) # Add the right operand to the left and return the result. def vec_add(left: Vec[Field], right: Vec[Field]): return list(map(lambda x: x[0] + x[1], zip(left, right))) Figure 9: Common functions for finite fields. Barnes, et al. Expires 14 September 2023 [Page 30] InternetDraft VDAF March 2023 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 generic 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 least 2^20. FFTfriendly fields also define the following parameter: * GEN_ORDER: Unsigned is the order of a multiplicative subgroup generated by Field.gen(). 6.1.3. Parameters The tables below define finite fields used in the remainder of this document. +==============+================+=======================+==========+  Parameter  Field64  Field128  Field255  +==============+================+=======================+==========+  MODULUS  2^32 *  2^66 *  2^255     4294967295 + 1  4611686018427387897 +  19     1   +++++  ENCODED_SIZE  8  16  32  +++++  Generator  7^4294967295  7^4611686018427387897  n/a  +++++  GEN_ORDER  2^32  2^66  n/a  +++++ Table 3: Parameters for the finite fields used in this document. OPEN ISSUE We currently use bigendian for encoding field elements. However, for implementations of GF(2^25519), little endian is more common. See issue#90. Barnes, et al. Expires 14 September 2023 [Page 31] InternetDraft VDAF March 2023 6.2. Pseudorandom Generators A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed into a long string of pseudorandom bits. A PRG suitable for this document implements the interface specified in this section. PRGs are defined by a class Prg with the following associated parameter: * SEED_SIZE: Unsigned is the size (in bytes) of a seed. A concrete Prg implements the following class method: * Prg(seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes) constructs an instance of Prg from the given seed and customization and binder strings. (See below for definitions of these.) The seed MUST be of length SEED_SIZE and MUST be generated securely (i.e., it is either the output of gen_rand or a previous invocation of the PRG). * prg.next(length: Unsigned) returns the next length bytes of output of PRG. If the seed was securely generated, the output can be treated as pseudorandom. Each Prg has two derived class methods. The first is used to derive a fresh seed from an existing one. The second is used to compute a sequence of pseudorandom field elements. For each method, the seed MUST be of length SEED_SIZE and MUST be generated securely (i.e., it is either the output of gen_rand or a previous invocation of the PRG). Barnes, et al. Expires 14 September 2023 [Page 32] InternetDraft VDAF March 2023 # Derive a new seed. def derive_seed(Prg, seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes): prg = Prg(seed, custom, binder) return prg.next(Prg.SEED_SIZE) # Output the next `length` pseudorandom elements of `Field`. def next_vec(self, Field, length: Unsigned): m = next_power_of_2(Field.MODULUS)  1 vec = [] while len(vec) < length: x = from_le_bytes(self.next(Field.ENCODED_SIZE)) x &= m if x < Field.MODULUS: vec.append(Field(x)) return vec # Expand the input `seed` into vector of `length` field elements. def expand_into_vec(Prg, Field, seed: Bytes[Prg.SEED_SIZE], custom: Bytes, binder: Bytes, length: Unsigned): prg = Prg(seed, custom, binder) return prg.next_vec(Field, length) Figure 10: Derived class methods for PRGs. 6.2.1. PrgSha3 This section describes PrgSha3, a PRG based on the Keccak permutation of SHA3 [FIPS202]. Keccak is used in the cSHAKE128 mode of operation [SP800185]. This Prg is RECOMMENDED for all use cases within VDAFs. Barnes, et al. Expires 14 September 2023 [Page 33] InternetDraft VDAF March 2023 class PrgSha3(Prg): # Associated parameters SEED_SIZE = 16 def __init__(self, seed, custom, binder): self.l = 0 self.x = seed + binder self.s = custom def next(self, length: Unsigned) > Bytes: self.l += length # Function `cSHAKE128(x, l, n, s)` is as defined in # [SP800185, Section 3.3]. # # Implementation note: Rather than regenerate the output # stream each time `next()` is invoked, most implementations # of SHA3 will expose an "absorbthensqueeze" API that # allows stateful handling of the stream. stream = cSHAKE128(self.x, self.l, b'', self.s) return stream[length:] Figure 11: Definition of PRG PrgSha3. 6.2.2. PrgFixedKeyAes128 While PrgSha3 as described above can be securely used in all cases where a Prg is needed in the VDAFs described in this document, there are some cases where a more efficient instantiation based on fixed key AES is possible. For now, this is limited to the Prg used inside the Idpf Section 8.1 implementation in Poplar1 Section 8.3. It is NOT RECOMMENDED to use this Prg anywhere else. See Security Considerations Section 9 for a more detailed discussion. Barnes, et al. Expires 14 September 2023 [Page 34] InternetDraft VDAF March 2023 class PrgFixedKeyAes128(Prg): # Associated parameters SEED_SIZE = 16 def __init__(self, seed, custom, binder): self.length_consumed = 0 # Use SHA3 to derive a key from the binder and customization # strings. Note that the AES key does not need to be kept # secret from any party. However, when used with IdpfPoplar, # we require the binder to be a random nonce. # # Implementation note: This step can be cached across PRG # evaluations with many different seeds. self.fixed_key = cSHAKE128(binder, 16, b'', custom) self.seed = seed def next(self, length: Unsigned) > Bytes: offset = self.length_consumed % 16 new_length = self.length_consumed + length block_range = range( int(self.length_consumed / 16), int(new_length / 16) + 1) self.length_consumed = new_length hashed_blocks = [ self.hash_block(xor(self.seed, to_le_bytes(i, 16))) \ for i in block_range ] return concat(hashed_blocks)[offset:offset+length] # The multiinstance tweakable circular correlationrobust hash function of # [GKWWY20] (Section 4.2). # # Function `AES128(key, block)` is the AES128 blockcipher. def hash_block(self, block): lo, hi = block[:8], block[8:] sigma = hi + xor(hi, lo) return xor(AES128(self.fixed_key, sigma), sigma) 6.2.3. The Customization and Binder Strings PRGs are used to map a seed to a finite domain, e.g., a fresh seed or a vector of field elements. To ensure domain separation, the derivation is needs to be bound to some distinguished "customization string". The customization string encodes the following values: 1. The document version (i.e.,VERSION); Barnes, et al. Expires 14 September 2023 [Page 35] InternetDraft VDAF March 2023 2. The "class" of the algorithm using the output (e.g., VDAF); 3. A unique identifier for the algorithm; and 4. Some indication of how the output is used (e.g., for deriving the measurement shares in Prio3 Section 7). The following algorithm is used in the remainder of this document in order to format the customization string: def format_custom(algo_class: Unsigned, algo: Unsigned, usage: Unsigned) > Bytes: return to_be_bytes(VERSION, 1) + \ to_be_bytes(algo_class, 1) + \ to_be_bytes(algo, 4) + \ to_be_bytes(usage, 2) It is also sometimes necessary to bind the output to some ephemeral value that multiple parties need to agree on. We call this input the "binder string". 7. Prio3 This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure: * Each measurement is encoded as a vector over some finite field. * Input validity is determined by an arithmetic circuit evaluated over the encoded input. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The circuit's output is a single field element: if zero, then the input is said to be "valid"; otherwise, if the output is nonzero, then the input is said to be "invalid". * The aggregate result is obtained by summing up the encoded input vectors and computing some function of the sum. At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multiparty computation to determine if their shares correspond Barnes, et al. Expires 14 September 2023 [Page 36] InternetDraft VDAF March 2023 to a valid input (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its input shares locally. Finally, the Collector sums up the aggregate shares and computes the aggregate result. This VDAF does not have an aggregation parameter. Instead, the output share is derived from the input share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful use of the aggregation parameter. As the name implies, Prio3 is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction. The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating inputs. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF. The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in Appendix "Test Vectors". 7.1. Fully Linear Proof (FLP) Systems Conceptually, an FLP is a 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 input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid input. (See Section 7.2 for details.) As usual, we will describe the interface implemented by a concrete FLP in terms of an abstract base class Flp that specifies the set of methods and parameters a concrete FLP must provide. Barnes, et al. Expires 14 September 2023 [Page 37] InternetDraft VDAF March 2023 The parameters provided by a concrete FLP are listed in Table 4. +================+==========================================+  Parameter  Description  +================+==========================================+  PROVE_RAND_LEN  Length of the prover randomness, the    number of random field elements consumed    by the prover when generating a proof  +++  QUERY_RAND_LEN  Length of the query randomness, the    number of random field elements consumed    by the verifier  +++  JOINT_RAND_LEN  Length of the joint randomness, the    number of random field elements consumed    by both the prover and verifier  +++  INPUT_LEN  Length of the encoded measurement    (Section 7.1.1)  +++  OUTPUT_LEN  Length of the aggregatable output    (Section 7.1.1)  +++  PROOF_LEN  Length of the proof  +++  VERIFIER_LEN  Length of the verifier message generated    by querying the input and proof  +++  Measurement  Type of the measurement  +++  AggResult  Type of the aggregate result  +++  Field  As defined in (Section 6.1)  +++ Table 4: Constants and types defined by a concrete FLP. An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1): * Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field]) > Vec[Field] is the deterministic proofgeneration algorithm run by the prover. Its inputs are the encoded input, the "prover randomness" prove_rand, and the "joint randomness" joint_rand. The prover randomness is used only by the prover, but the joint randomness is shared by both the prover and verifier. Barnes, et al. Expires 14 September 2023 [Page 38] InternetDraft VDAF March 2023 * Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field], joint_rand: Vec[Field], num_shares: Unsigned) > Vec[Field] is the querygeneration algorithm run by the verifier. This is used to "query" the input and proof. The result of the query (i.e., the output of this function) is called the "verifier message". In addition to the input and proof, this algorithm takes as input the query randomness query_rand and the joint randomness joint_rand. The former is used only by the verifier. num_shares specifies how many input and proof shares were generated. * Flp.decide(verifier: Vec[Field]) > Bool is the deterministic decision algorithm run by the verifier. It takes as input the verifier message and outputs a boolean indicating if the input from which it was generated is valid. Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the input and proof, the querygeneration algorithm outputs a share of the verifier message. Furthermore, the "strong zeroknowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the 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 of the input and proof that were generated. If these data are not secret shared, then num_shares == 1. This parameter is useful for certain FLP constructions. For example, the FLP in Section 7.3 is defined in terms of an arithmetic circuit; when the circuit contains constants, it is sometimes necessary to normalize those constants to ensure that the circuit's output, when run on a valid input, is the same regardless of the number of shares. An FLP is executed by the prover and verifier as follows: Barnes, et al. Expires 14 September 2023 [Page 39] InternetDraft VDAF March 2023 def run_flp(Flp, inp: Vec[Flp.Field], num_shares: Unsigned): joint_rand = Flp.Field.rand_vec(Flp.JOINT_RAND_LEN) prove_rand = Flp.Field.rand_vec(Flp.PROVE_RAND_LEN) query_rand = Flp.Field.rand_vec(Flp.QUERY_RAND_LEN) # Prover generates the proof. proof = Flp.prove(inp, prove_rand, joint_rand) # Verifier queries the input and proof. verifier = Flp.query(inp, proof, query_rand, joint_rand, num_shares) # Verifier decides if the input is valid. return Flp.decide(verifier) Figure 12: Execution of an FLP. The proof system is constructed so that, if input is a valid input, then run_flp(Flp, input, 1) always returns True. On the other hand, if input is invalid, then as long as joint_rand and query_rand are generated uniform randomly, the output is False with overwhelming probability. We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5round, 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) > Vec[Field] encodes a raw measurement as a vector of field elements. The return value MUST be of length INPUT_LEN. For some FLPs, the encoded input also includes redundant field elements that are useful for checking the proof, but which are not needed after the proof has been checked. An example is the "integer sum" data type from [CGB17] in which an integer in range [0, 2^k) is encoded as a vector of k field elements (this type is also defined in Section 7.4.2). After consuming this vector, all that is needed is the integer it represents. Thus the FLP defines an algorithm for truncating the input to the length of the aggregated output: Barnes, et al. Expires 14 September 2023 [Page 40] InternetDraft VDAF March 2023 * Flp.truncate(input: Vec[Field]) > Vec[Field] maps an encoded input to an aggregatable output. The length of the input MUST be INPUT_LEN and the length of the output MUST be OUTPUT_LEN. Once the aggregate shares have been computed and combined together, their sum can be converted into the aggregate result. This could be a projection from the FLP's field to the integers, or it could include additional postprocessing. * Flp.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult maps a sum of aggregate shares to an aggregate result. The length of the input MUST be OUTPUT_LEN. num_measurements is the number of measurements that contributed to the aggregated output. We remark that, taken together, these three functionalities correspond roughly to the notion of "Affineaggregatable encodings (AFEs)" from [CGB17]. 7.2. Construction This section specifies Prio3, an implementation of the Vdaf interface (Section 5). It has two generic parameters: an Flp (Section 7.1) and a Prg (Section 6.2). The associated constants and types required by the Vdaf interface are defined in Table 5. The methods required for sharding, preparation, aggregation, and unsharding are described in the remaining subsections. These methods refer to constants enumerated in Table 6. Barnes, et al. Expires 14 September 2023 [Page 41] InternetDraft VDAF March 2023 +=================+===============================================+  Parameter  Value  +=================+===============================================+  VERIFY_KEY_SIZE  Prg.SEED_SIZE  +++  NONCE_SIZE  16  +++  ROUNDS  1  +++  SHARES  in [2, 256)  +++  Measurement  Flp.Measurement  +++  AggParam  None  +++  Prep  Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]  +++  OutShare  Vec[Flp.Field]  +++  AggResult  Flp.AggResult  +++ Table 5: VDAF parameters for Prio3. +=================================+=======+  Variable  Value  +=================================+=======+  DST_MEASUREMENT_SHARE: Unsigned  1  +++  DST_PROOF_SHARE: Unsigned  2  +++  DST_JOINT_RANDOMNESS: Unsigned  3  +++  DST_PROVE_RANDOMNESS: Unsigned  4  +++  DST_QUERY_RANDOMNESS: Unsigned  5  +++  DST_JOINT_RAND_SEED: Unsigned  6  +++  DST_JOINT_RAND_PART: Unsigned  7  +++ Table 6: Constants used by Prio3. Barnes, et al. Expires 14 September 2023 [Page 42] InternetDraft VDAF March 2023 7.2.1. Sharding Recall from Section 7.1 that the FLP syntax calls for "joint randomness" shared by the prover (i.e., the Client) and the verifier (i.e., the Aggregators). VDAFs have no such notion. Instead, the Client derives the joint randomness from its input in a way that allows the Aggregators to reconstruct it from their input shares. (This idea is based on the FiatShamir heuristic and is described in Section 6.2.3 of [BBCGGI19].) The sharding algorithm involves the following steps: 1. Encode the Client's raw measurement as an input for the FLP 2. Shard the input into a sequence of input shares 3. Derive the joint randomness from the input shares and nonce 4. Run the FLP proofgeneration algorithm using the derived joint randomness 5. Shard the proof into a sequence of proof shares The algorithm is specified below. Notice that only one set of input and proof shares (called the "leader" shares below) are vectors of field elements. The other shares (called the "helper" shares) are represented instead by PRG seeds, which are expanded into vectors of field elements. The definitions of constants and a few auxiliary functions are defined in Section 7.2.6. def measurement_to_input_shares(Prio3, measurement, nonce, rand): l = Prio3.Prg.SEED_SIZE use_joint_rand = Prio3.Flp.JOINT_RAND_LEN > 0 # Split the coins into the various seeds we'll need. if len(rand) != Prio3.RAND_SIZE: raise ERR_INPUT # unexpected length for random coins seeds = [rand[i:i+l] for i in range(0,Prio3.RAND_SIZE,l)] if use_joint_rand: k_helper_seeds, seeds = front((Prio3.SHARES1) * 3, seeds) k_helper_meas_shares = [ k_helper_seeds[i] for i in range(0, (Prio3.SHARES1) * 3, 3) ] k_helper_proof_shares = [ k_helper_seeds[i] Barnes, et al. Expires 14 September 2023 [Page 43] InternetDraft VDAF March 2023 for i in range(1, (Prio3.SHARES1) * 3, 3) ] k_helper_blinds = [ k_helper_seeds[i] for i in range(2, (Prio3.SHARES1) * 3, 3) ] (k_leader_blind,), seeds = front(1, seeds) else: k_helper_seeds, seeds = front((Prio3.SHARES1) * 2, seeds) k_helper_meas_shares = [ k_helper_seeds[i] for i in range(0, (Prio3.SHARES1) * 2, 2) ] k_helper_proof_shares = [ k_helper_seeds[i] for i in range(1, (Prio3.SHARES1) * 2, 2) ] k_helper_blinds = [None] * (Prio3.SHARES1) k_leader_blind = None (k_prove,), seeds = front(1, seeds) # Finish measurement shares and joint randomness parts. inp = Prio3.Flp.encode(measurement) leader_meas_share = inp k_joint_rand_parts = [] for j in range(Prio3.SHARES1): helper_meas_share = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, k_helper_meas_shares[j], Prio3.custom(DST_MEASUREMENT_SHARE), byte(j+1), Prio3.Flp.INPUT_LEN ) leader_meas_share = vec_sub(leader_meas_share, helper_meas_share) if use_joint_rand: encoded = Prio3.Flp.Field.encode_vec(helper_meas_share) k_joint_rand_part = Prio3.Prg.derive_seed( k_helper_blinds[j], Prio3.custom(DST_JOINT_RAND_PART), byte(j+1) + nonce + encoded, ) k_joint_rand_parts.append(k_joint_rand_part) # Finish joint randomness. if use_joint_rand: encoded = Prio3.Flp.Field.encode_vec(leader_meas_share) k_joint_rand_part = Prio3.Prg.derive_seed( Barnes, et al. Expires 14 September 2023 [Page 44] InternetDraft VDAF March 2023 k_leader_blind, Prio3.custom(DST_JOINT_RAND_PART), byte(0) + nonce + encoded, ) k_joint_rand_parts.insert(0, k_joint_rand_part) joint_rand = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, Prio3.joint_rand(k_joint_rand_parts), Prio3.custom(DST_JOINT_RANDOMNESS), b'', Prio3.Flp.JOINT_RAND_LEN, ) else: joint_rand = [] # Finish the proof shares. prove_rand = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, k_prove, Prio3.custom(DST_PROVE_RANDOMNESS), b'', Prio3.Flp.PROVE_RAND_LEN, ) proof = Prio3.Flp.prove(inp, prove_rand, joint_rand) leader_proof_share = proof for j in range(Prio3.SHARES1): helper_proof_share = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, k_helper_proof_shares[j], Prio3.custom(DST_PROOF_SHARE), byte(j+1), Prio3.Flp.PROOF_LEN, ) leader_proof_share = vec_sub(leader_proof_share, helper_proof_share) # Each Aggregator's input share contains its measurement share, # proof share, and blind. The public share contains the # Aggregators' joint randomness parts. input_shares = [] input_shares.append(Prio3.encode_leader_share( leader_meas_share, leader_proof_share, k_leader_blind, )) for j in range(Prio3.SHARES1): input_shares.append(Prio3.encode_helper_share( k_helper_meas_shares[j], Barnes, et al. Expires 14 September 2023 [Page 45] InternetDraft VDAF March 2023 k_helper_proof_shares[j], k_helper_blinds[j], )) public_share = Prio3.encode_public_share(k_joint_rand_parts) return (public_share, input_shares) Figure 13: Inputdistribution algorithm for Prio3. 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 input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept. In addition, the Aggregators must ensure that they have all used the same joint randomness for the querygeneration algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives a "part" of this seed from its input share and the "blind" generated by the client. The seed is derived by hashing together the parts, so before running the querygeneration algorithm, it must first gather the parts derived by the other Aggregators. In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the other Aggregators' parts of the joint randomness seed. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their verifier shares. The definitions of constants and a few auxiliary functions are defined in Section 7.2.6. def prep_init(Prio3, verify_key, agg_id, _agg_param, nonce, public_share, input_share): k_joint_rand_parts = Prio3.decode_public_share(public_share) (meas_share, proof_share, k_blind) = \ Prio3.decode_leader_share(input_share) if agg_id == 0 else \ Prio3.decode_helper_share(agg_id, input_share) out_share = Prio3.Flp.truncate(meas_share) # Compute joint randomness. joint_rand = [] Barnes, et al. Expires 14 September 2023 [Page 46] InternetDraft VDAF March 2023 k_corrected_joint_rand, k_joint_rand_part = None, None if Prio3.Flp.JOINT_RAND_LEN > 0: encoded = Prio3.Flp.Field.encode_vec(meas_share) k_joint_rand_part = Prio3.Prg.derive_seed(k_blind, Prio3.custom(DST_JOINT_RAND_PART), byte(agg_id) + nonce + encoded) k_joint_rand_parts[agg_id] = k_joint_rand_part k_corrected_joint_rand = Prio3.joint_rand(k_joint_rand_parts) joint_rand = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, k_corrected_joint_rand, Prio3.custom(DST_JOINT_RANDOMNESS), b'', Prio3.Flp.JOINT_RAND_LEN, ) # Query the measurement and proof share. query_rand = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, verify_key, Prio3.custom(DST_QUERY_RANDOMNESS), nonce, Prio3.Flp.QUERY_RAND_LEN, ) verifier_share = Prio3.Flp.query(meas_share, proof_share, query_rand, joint_rand, Prio3.SHARES) prep_msg = Prio3.encode_prep_share(verifier_share, k_joint_rand_part) return (out_share, k_corrected_joint_rand, prep_msg) def prep_next(Prio3, prep, inbound): (out_share, k_corrected_joint_rand, prep_msg) = prep if inbound is None: return (prep, prep_msg) k_joint_rand_check = Prio3.decode_prep_msg(inbound) if k_joint_rand_check != k_corrected_joint_rand: raise ERR_VERIFY # joint randomness check failed return out_share def prep_shares_to_prep(Prio3, _agg_param, prep_shares): verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN) Barnes, et al. Expires 14 September 2023 [Page 47] InternetDraft VDAF March 2023 k_joint_rand_parts = [] for encoded in prep_shares: (verifier_share, k_joint_rand_part) = \ Prio3.decode_prep_share(encoded) verifier = vec_add(verifier, verifier_share) if Prio3.Flp.JOINT_RAND_LEN > 0: k_joint_rand_parts.append(k_joint_rand_part) if not Prio3.Flp.decide(verifier): raise ERR_VERIFY # proof verifier check failed k_joint_rand_check = None if Prio3.Flp.JOINT_RAND_LEN > 0: k_joint_rand_check = Prio3.joint_rand(k_joint_rand_parts) return Prio3.encode_prep_msg(k_joint_rand_check) Figure 14: Preparation state for Prio3. 7.2.3. Validity of Aggregation Parameters Every input share MUST only be used once, regardless of the aggregation parameters used. def is_valid(agg_param, previous_agg_params): return len(previous_agg_params) == 0 Figure 15: Validity of aggregation parameters for Prio3. 7.2.4. Aggregation Aggregating a set of output shares is simply a matter of adding up the vectors elementwise. def out_shares_to_agg_share(Prio3, _agg_param, out_shares): agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN) for out_share in out_shares: agg_share = vec_add(agg_share, out_share) return Prio3.Flp.Field.encode_vec(agg_share) Figure 16: Aggregation algorithm for Prio3. 7.2.5. Unsharding To unshard a set of aggregate shares, the Collector first adds up the vectors elementwise. It then converts each element of the vector into an integer. Barnes, et al. Expires 14 September 2023 [Page 48] InternetDraft VDAF March 2023 def agg_shares_to_result(Prio3, _agg_param, agg_shares, num_measurements): agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN) for agg_share in agg_shares: agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share)) return Prio3.Flp.decode(agg, num_measurements) Figure 17: Computation of the aggregate result for Prio3. 7.2.6. Auxiliary Functions def joint_rand(Prio3, k_joint_rand_parts): return Prio3.Prg.derive_seed( zeros(Prio3.Prg.SEED_SIZE), Prio3.custom(DST_JOINT_RAND_SEED), concat(k_joint_rand_parts), ) 7.2.6.1. Message Serialization def encode_leader_share(Prio3, meas_share, proof_share, k_blind): encoded = Bytes() encoded += Prio3.Flp.Field.encode_vec(meas_share) encoded += Prio3.Flp.Field.encode_vec(proof_share) if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_blind return encoded def decode_leader_share(Prio3, encoded): l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN encoded_meas_share, encoded = encoded[:l], encoded[l:] meas_share = Prio3.Flp.Field.decode_vec(encoded_meas_share) l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN encoded_proof_share, encoded = encoded[:l], encoded[l:] proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share) l = Prio3.Prg.SEED_SIZE if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, None) k_blind, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, k_blind) Barnes, et al. Expires 14 September 2023 [Page 49] InternetDraft VDAF March 2023 def encode_helper_share(Prio3, k_meas_share, k_proof_share, k_blind): encoded = Bytes() encoded += k_meas_share encoded += k_proof_share if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_blind return encoded def decode_helper_share(Prio3, agg_id, encoded): c_meas_share = Prio3.custom(DST_MEASUREMENT_SHARE) c_proof_share = Prio3.custom(DST_PROOF_SHARE) l = Prio3.Prg.SEED_SIZE k_meas_share, encoded = encoded[:l], encoded[l:] meas_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field, k_meas_share, c_meas_share, byte(agg_id), Prio3.Flp.INPUT_LEN) k_proof_share, encoded = encoded[:l], encoded[l:] proof_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field, k_proof_share, c_proof_share, byte(agg_id), Prio3.Flp.PROOF_LEN) if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, None) k_blind, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (meas_share, proof_share, k_blind) def encode_public_share(Prio3, k_joint_rand_parts): encoded = Bytes() if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += concat(k_joint_rand_parts) return encoded def decode_public_share(Prio3, encoded): l = Prio3.Prg.SEED_SIZE if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE Barnes, et al. Expires 14 September 2023 [Page 50] InternetDraft VDAF March 2023 return None k_joint_rand_parts = [] for i in range(Prio3.SHARES): k_joint_rand_part, encoded = encoded[:l], encoded[l:] k_joint_rand_parts.append(k_joint_rand_part) if len(encoded) != 0: raise ERR_DECODE return k_joint_rand_parts def encode_prep_share(Prio3, verifier, k_joint_rand): encoded = Bytes() encoded += Prio3.Flp.Field.encode_vec(verifier) if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_joint_rand return encoded def decode_prep_share(Prio3, encoded): l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.VERIFIER_LEN encoded_verifier, encoded = encoded[:l], encoded[l:] verifier = Prio3.Flp.Field.decode_vec(encoded_verifier) if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return (verifier, None) l = Prio3.Prg.SEED_SIZE k_joint_rand, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (verifier, k_joint_rand) def encode_prep_msg(Prio3, k_joint_rand_check): encoded = Bytes() if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_joint_rand_check return encoded def decode_prep_msg(Prio3, encoded): if Prio3.Flp.JOINT_RAND_LEN == 0: if len(encoded) != 0: raise ERR_DECODE return None l = Prio3.Prg.SEED_SIZE k_joint_rand_check, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return k_joint_rand_check Barnes, et al. Expires 14 September 2023 [Page 51] InternetDraft VDAF March 2023 7.3. A GeneralPurpose FLP This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and the extensions to their proof system made here. The construction is specified in Section 7.3.3. OPEN ISSUE We're not yet sure if specifying this generalpurpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the generalpurpose FLP. In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the general purpose construction. The reference implementation can be found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc. OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG. 7.3.1. Overview In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is nonzero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the input. Suppose for a moment that the validity circuit C is affine, meaning its only operations are addition and multiplicationbyconstant. In particular, suppose the circuit does not contain a multiplication gate whose operands are both nonconstant. Then to decide if an input x is valid, each Aggregator could evaluate C on its share of x locally, broadcast the output share to its peers, then combine the output shares locally to recover C(x). This is true because for any SHARESway secret sharing of x it holds that C(x_shares[0] + ... + x_shares[SHARES1]) = C(x_shares[0]) + ... + C(x_shares[SHARES1]) Barnes, et al. Expires 14 September 2023 [Page 52] InternetDraft VDAF March 2023 (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 input and proof. To make this work, the proof system is restricted to validity circuits that exhibit a special structure. Specifically, an arithmetic circuit with "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 non affine operation, x[0]^2. In order to apply [BBCGGI19], Theorem 4.3, the circuit needs to be rewritten in terms of a gadget that subsumes this 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 input 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 jth wire of the kth call to the gadget during the evaluation of C(x). Suppose there are M such calls and fix distinct field elements alpha[0], ..., alpha[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.) Barnes, et al. Expires 14 September 2023 [Page 53] InternetDraft VDAF March 2023 The prover constructs from wire and alpha a polynomial that, when evaluated at alpha[k1], produces the output of the kth call to the gadget. Let us call this the "gadget polynomial". Polynomial evaluation is linear, which means that, in the distributed setting, the Client can disseminate additive shares of the gadget polynomial that the Aggregators then use to compute additive shares of each gadget output, allowing each Aggregator to compute its share of C(x) locally. There is one more wrinkle, however: It is still possible for a malicious prover to produce a gadget polynomial that would result in C(x) being computed incorrectly, potentially resulting in an invalid input being accepted. To prevent this, the verifier performs a probabilistic test to check that the gadget polynomial is well formed. This test, and the procedure for constructing the gadget polynomial, are described in detail in Section 7.3.3. 7.3.1.1. Extensions The FLP described in the next section extends the proof system [BBCGGI19], Section 4.2 in three ways. First, the validity circuit in our construction includes an additional, random input (this is the "joint randomness" derived from the input shares in Prio3; see Section 7.2). This allows for circuit optimizations that trade a small soundness error for a shorter proof. For example, consider a circuit that recognizes the set of 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(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) (Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here inp 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 inp is valid (i.e., each inp[j] is in [0,2)), then the circuit will evaluate to 0 regardless of the value of r; but if inp[j] is not in [0,2) for some j, the output will be nonzero with high probability. Barnes, et al. Expires 14 September 2023 [Page 54] InternetDraft VDAF March 2023 The second extension implemented by our FLP allows the validity circuit to contain multiple gadget types. (This generalization was suggested in [BBCGGI19], Remark 4.5.) For example, the following circuit is allowed, where Mul and Range2 are the gadgets defined above (the input has length N+1): C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) + \ 2^0 * inp[0] + ... + 2^(N1) * inp[N1]  \ Mul(inp[N], inp[N]) Finally, [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. 7.3.2. Validity Circuits The FLP described in Section 7.3.3 is defined in terms of a validity circuit Valid that implements the interface described here. A concrete Valid defines the following parameters: +================+=======================================+  Parameter  Description  +================+=======================================+  GADGETS  A list of gadgets  +++  GADGET_CALLS  Number of times each gadget is called  +++  INPUT_LEN  Length of the input  +++  OUTPUT_LEN  Length of the aggregatable output  +++  JOINT_RAND_LEN  Length of the random input  +++  Measurement  The type of measurement  +++  AggResult  Type of the aggregate result  +++  Field  An FFTfriendly finite field as    defined in Section 6.1.2  +++ Table 7: Validity circuit parameters. Barnes, et al. Expires 14 September 2023 [Page 55] InternetDraft VDAF March 2023 Each gadget G in GADGETS defines a constant DEGREE that specifies the circuit's "arithmetic degree". This is defined to be the degree of the polynomial that computes it. For example, the Mul circuit in Section 7.3.1 is defined by the polynomial Mul(x) = x * x, which has degree 2. Hence, the arithmetic degree of this gadget is 2. Each gadget also defines a parameter ARITY that specifies the circuit's arity (i.e., the number of input wires). A concrete Valid provides the following methods for encoding a measurement as an input vector, truncating an input vector to the length of an aggregatable output, and converting an aggregated output to an aggregate result: * Valid.encode(measurement: Measurement) > Vec[Field] returns a vector of length INPUT_LEN representing a measurement. * Valid.truncate(input: Vec[Field]) > Vec[Field] returns a vector of length OUTPUT_LEN representing an aggregatable output. * Valid.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult returns an aggregate result. Finally, the following class methods are derived for each concrete Valid: # Length of the prover randomness. def prove_rand_len(Valid): return sum(map(lambda g: g.ARITY, Valid.GADGETS)) # Length of the query randomness. def query_rand_len(Valid): return len(Valid.GADGETS) # Length of the proof. def proof_len(Valid): length = 0 for (g, g_calls) in zip(Valid.GADGETS, Valid.GADGET_CALLS): P = next_power_of_2(1 + g_calls) length += g.ARITY + g.DEGREE * (P  1) + 1 return length # Length of the verifier message. def verifier_len(Valid): length = 1 for g in Valid.GADGETS: length += g.ARITY + 1 return length Barnes, et al. Expires 14 September 2023 [Page 56] InternetDraft VDAF March 2023 Figure 18: Derived methods for validity circuits. 7.3.3. Construction This section specifies FlpGeneric, an implementation of the Flp interface (Section 7.1). It has as a generic parameter a validity circuit Valid implementing the interface defined in Section 7.3.2. NOTE A reference implementation can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/ flp_generic.sage. The FLP parameters for FlpGeneric are defined in Table 8. The required methods for generating the proof, generating the verifier, and deciding validity are specified in the remaining subsections. In the remainder, we let [n] denote the set {1, ..., n} for positive integer n. We also define the following constants: * Let H = len(Valid.GADGETS) * For each i in [H]:  Let G_i = Valid.GADGETS[i]  Let L_i = Valid.GADGETS[i].ARITY  Let M_i = Valid.GADGET_CALLS[i]  Let P_i = next_power_of_2(M_i+1)  Let alpha_i = Field.gen()^(Field.GEN_ORDER / P_i) Barnes, et al. Expires 14 September 2023 [Page 57] InternetDraft VDAF March 2023 +================+============================================+  Parameter  Value  +================+============================================+  PROVE_RAND_LEN  Valid.prove_rand_len() (see Section 7.3.2)  +++  QUERY_RAND_LEN  Valid.query_rand_len() (see Section 7.3.2)  +++  JOINT_RAND_LEN  Valid.JOINT_RAND_LEN  +++  INPUT_LEN  Valid.INPUT_LEN  +++  OUTPUT_LEN  Valid.OUTPUT_LEN  +++  PROOF_LEN  Valid.proof_len() (see Section 7.3.2)  +++  VERIFIER_LEN  Valid.verifier_len() (see Section 7.3.2)  +++  Measurement  Valid.Measurement  +++  Field  Valid.Field  +++ Table 8: FLP Parameters of FlpGeneric. 7.3.3.1. Proof Generation On input inp, prove_rand, and joint_rand, the proof is computed as follows: 1. For each i in [H] create an empty table wire_i. 2. Partition the prover randomness prove_rand into subvectors seed_1, ..., seed_H where len(seed_i) == L_i for all i in [H]. Let us call these the "wire seeds" of each gadget. 3. Evaluate Valid on input of inp and joint_rand, recording the inputs of each gadget in the corresponding table. Specifically, for every i in [H], set wire_i[j1,k1] to the value on the jth wire into the kth call to gadget G_i. 4. Compute the "wire polynomials". That is, for every i in [H] and j in [L_i], construct poly_wire_i[j1], the jth wire polynomial for the ith 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)). Barnes, et al. Expires 14 September 2023 [Page 58] InternetDraft VDAF March 2023 NOTE We pad w to the nearest power of 2 so that we can use FFT for interpolating the wire polynomials. Perhaps there is some clever math for picking wire_inp in a way that avoids having to pad. * Let poly_wire_i[j1] be the lowest degree polynomial for which poly_wire_i[j1](alpha_i^k) == padded_w[k] for all k in [P_i]. 5. Compute the "gadget polynomials". That is, for every i in [H]: * Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i 1]). That is, evaluate the circuit G_i on the wire polynomials for the ith gadget. (Arithmetic is in the ring of polynomials over Field.) The proof is the vector proof = seed_1 + coeff_1 + ... + seed_H + coeff_H, where coeff_i is the vector of coefficients of poly_gadget_i for each i in [H]. 7.3.3.2. Query Generation On input of inp, proof, query_rand, and joint_rand, the verifier message is generated as follows: 1. For every i in [H] create an empty table wire_i. 2. Partition proof into the subvectors seed_1, coeff_1, ..., seed_H, coeff_H defined in Section 7.3.3.1. 3. Evaluate Valid on input of inp and joint_rand, recording the inputs of each gadget in the corresponding table. This step is similar to the prover's step (3.) except the verifier does not evaluate the gadgets. Instead, it computes the output of the kth call to G_i by evaluating poly_gadget_i(alpha_i^k). Let v denote the output of the circuit evaluation. 4. Compute the wire polynomials just as in the prover's step (4.). 5. Compute the tests for wellformedness of the gadget polynomials. That is, for every i in [H]: * Let t = query_rand[i]. Check if t^(P_i) == 1: If so, then raise ERR_ABORT and halt. (This prevents the verifier from inadvertently leaking a gadget output in the verifier message.) * Let y_i = poly_gadget_i(t). Barnes, et al. Expires 14 September 2023 [Page 59] InternetDraft VDAF March 2023 * For each j in [0,L_i) let x_i[j1] = poly_wire_i[j1](t). The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H + [y_H]. 7.3.3.3. Decision On input of vector verifier, the verifier decides if the input is valid as follows: 1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in Section 7.3.3.2. 2. Check for wellformedness of the gadget polynomials. For every i in [H]: * Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i and set z to the output. * If z != y_i, then return False and halt. 3. Return True if v == 0 and False otherwise. 7.3.3.4. Encoding The FLP encoding and truncation methods invoke Valid.encode, Valid.truncate, and Valid.decode in the natural way. 7.4. Instantiations This section specifies instantiations of Prio3 for various measurement types. Each uses FlpGeneric as the FLP (Section 7.3) and is determined by a validity circuit (Section 7.3.2) and a PRG (Section 6.2). Test vectors for each can be found in Appendix "Test Vectors". NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/ vdaf_prio3.sage. 7.4.1. Prio3Count Our first instance of Prio3 is for a simple counter: Each measurement is either one or zero and the aggregate result is the sum of the measurements. Barnes, et al. Expires 14 September 2023 [Page 60] InternetDraft VDAF March 2023 This instance uses PrgSha3 (Section 6.2.1) as its PRG. Its validity circuit, denoted Count, uses Field64 (Table 3) as its finite field. Its gadget, denoted Mul, is the degree2, arity2 gadget defined as def Mul(x, y): return x * y The validity circuit is defined as def Count(inp: Vec[Field64]): return Mul(inp[0], inp[0])  inp[0] The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below. +================+==========================+  Parameter  Value  +================+==========================+  GADGETS  [Mul]  +++  GADGET_CALLS  [1]  +++  INPUT_LEN  1  +++  OUTPUT_LEN  1  +++  JOINT_RAND_LEN  0  +++  Measurement  Unsigned, in range [0,2)  +++  AggResult  Unsigned  +++  Field  Field64 (Table 3)  +++ Table 9: Parameters of validity circuit Count. 7.4.2. Prio3Sum The next instance of Prio3 supports summing of integers in a pre determined range. Each measurement is an integer in range [0, 2^bits), where bits is an associated parameter. Barnes, et al. Expires 14 September 2023 [Page 61] InternetDraft VDAF March 2023 This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity circuit, denoted Sum, uses Field128 (Table 3) as its finite field. The measurement is encoded as a lengthbits vector of field elements, where the lth element of the vector represents the lth bit of the summand: def encode(Sum, measurement: Integer): if 0 > measurement or measurement >= 2^Sum.INPUT_LEN: raise ERR_INPUT encoded = [] for l in range(Sum.INPUT_LEN): encoded.append(Sum.Field((measurement >> l) & 1)) return encoded def truncate(Sum, inp): decoded = Sum.Field(0) for (l, b) in enumerate(inp): w = Sum.Field(1 << l) decoded += w * b return [decoded] def decode(Sum, output, _num_measurements): return output[0].as_unsigned() The validity circuit checks that the input consists of ones and zeros. Its gadget, denoted Range2, is the degree2, arity1 gadget defined as def Range2(x): return x^2  x The validity circuit is defined as def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]): out = Field128(0) r = joint_rand[0] for x in inp: out += r * Range2(x) r *= joint_rand[0] return out Barnes, et al. Expires 14 September 2023 [Page 62] InternetDraft VDAF March 2023 +================+================================+  Parameter  Value  +================+================================+  GADGETS  [Range2]  +++  GADGET_CALLS  [bits]  +++  INPUT_LEN  bits  +++  OUTPUT_LEN  1  +++  JOINT_RAND_LEN  1  +++  Measurement  Unsigned, in range [0, 2^bits)  +++  AggResult  Unsigned  +++  Field  Field128 (Table 3)  +++ Table 10: Parameters of validity circuit Sum. 7.4.3. Prio3Histogram This instance of Prio3 allows for estimating the distribution of the measurements by computing a simple histogram. Each measurement is an arbitrary integer and the aggregate result counts the number of measurements that fall in a set of fixed buckets. This instance of Prio3 uses PrgSha3 (Section 6.2.1) as its PRG. Its validity circuit, denoted Histogram, uses Field128 (Table 3) as its finite field. The measurement is encoded as a onehot vector representing the bucket into which the measurement falls (let bucket denote a sequence of monotonically increasing integers): def encode(Histogram, measurement: Integer): boundaries = buckets + [Infinity] encoded = [Field128(0) for _ in range(len(boundaries))] for i in range(len(boundaries)): if measurement <= boundaries[i]: encoded[i] = Field128(1) return encoded def truncate(Histogram, inp: Vec[Field128]): return inp def decode(Histogram, output: Vec[Field128], _num_measurements): return [bucket_count.as_unsigned() for bucket_count in output] Barnes, et al. Expires 14 September 2023 [Page 63] InternetDraft VDAF March 2023 The validity circuit uses Range2 (see Section 7.4.2) as its single gadget. It checks for onehotness in two steps, as follows: def Histogram(inp: Vec[Field128], joint_rand: Vec[Field128], num_shares: Unsigned): # Check that each bucket is one or zero. range_check = Field128(0) r = joint_rand[0] for x in inp: range_check += r * Range2(x) r *= joint_rand[0] # Check that the buckets sum to 1. sum_check = Field128(1) * Field128(num_shares).inv() for b in inp: sum_check += b out = joint_rand[1] * range_check + \ joint_rand[1]^2 * sum_check return out Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3. +================+====================+  Parameter  Value  +================+====================+  GADGETS  [Range2]  +++  GADGET_CALLS  [buckets + 1]  +++  INPUT_LEN  buckets + 1  +++  OUTPUT_LEN  buckets + 1  +++  JOINT_RAND_LEN  2  +++  Measurement  Integer  +++  AggResult  Vec[Unsigned]  +++  Field  Field128 (Table 3)  +++ Table 11: Parameters of validity circuit Histogram. Barnes, et al. Expires 14 September 2023 [Page 64] InternetDraft VDAF March 2023 8. Poplar1 This section specifies Poplar1, a VDAF for the following task. Each Client holds a string of length BITS and the Aggregators hold a set of lbit strings, where l <= BITS. We will refer to the latter as the set of "candidate prefixes". The Aggregators' goal is to count how many inputs are prefixed by each candidate prefix. This functionality is the core component of the Poplar protocol [BBCGGI21], which was designed to compute the heavy hitters over a set of input strings. At a high level, the protocol works as follows. 1. Each Client splits its input string into input shares and sends one share to each Aggregator. 2. The Aggregators agree on an initial set of candidate prefixes, say 0 and 1. 3. The Aggregators evaluate the VDAF on each set of input shares and aggregate the recovered output shares. The aggregation parameter is the set of candidate prefixes. 4. The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix. 5. Let H denote the set of prefixes that occurred at least t times. If the prefixes all have length BITS, then H is the set of t heavyhitters. Otherwise compute the next set of candidate prefixes, e.g., for each p in H, add p  0 and p  1 to the set. Repeat step 3 with the new set of candidate prefixes. Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", a primitive described by [BBCGGI21] that generalizes the notion of a Distributed Point Function (DPF) [GI14]. Briefly, a DPF is used to distribute the computation of a "point function", a function that evaluates to zero on every input except at a programmable "point". The computation is distributed in such a way that no one party knows either the point or what it evaluates to. An IDPF generalizes this "point" to a path on a full binary tree from the root to one of the leaves. It is evaluated on an "index" representing a unique node of the tree. If the node is on the programmed path, then the function evaluates to a 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. Barnes, et al. Expires 14 September 2023 [Page 65] InternetDraft VDAF March 2023 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 theavy hitters for a set of inputs, the Aggregators and Collector first compute the prefix tree, then extract the heavy hitters from the leaves of this tree. (Note that the prefix tree may leak more information about the set than the heavy hitters themselves; see Section 9.4 for details.) Poplar1 composes an IDPF with the "secure sketching" protocol of [BBCGGI21]. This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "onehot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one. The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors". 8.1. Incremental Distributed Point Functions (IDPFs) An IDPF is defined over a domain of size 2^BITS, where BITS is constant defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range [0, 2^BITS). The Client specifies an index alpha and a vector of values beta, one for each "level" L in range [0, BITS). The key generation algorithm generates one IDPF "key" for each Aggregator. When evaluated at level L and index 0 <= prefix < 2^L, each IDPF key returns an additive share of beta[L] if prefix is the Lbit prefix of alpha and shares of zero otherwise. An index x is defined to be a prefix of another index y as follows. Let LSB(x, N) denote the least significant N bits of positive integer x. By definition, a positive integer 0 <= x < 2^L is said to be the 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. Barnes, et al. Expires 14 September 2023 [Page 66] InternetDraft VDAF March 2023 Each of the programmed points beta is a vector of elements of some finite field. We distinguish two types of fields: One for inner nodes (denoted Idpf.FieldInner), and one for leaf nodes (Idpf.FieldLeaf). (Our instantiation of Poplar1 (Section 8.4) will use a much larger field for leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as defined in [BBCGGI21], Definition 1.) A concrete IDPF defines the types and constants enumerated in Table 12. In the remainder we write Idpf.Vec as shorthand for the type Union[Vec[Vec[Idpf.FieldInner]], Vec[Vec[Idpf.FieldLeaf]]]. (This type denotes either a vector of inner node field elements or leaf node field elements.) The scheme is comprised of the following algorithms: * Idpf.gen(alpha: Unsigned, beta_inner: Vec[Vec[Idpf.FieldInner]], beta_leaf: Vec[Idpf.FieldLeaf], binder: Bytes, rand: Bytes[Idpf.RAND_SIZE]) > (Bytes, Vec[Bytes]) is the randomized IDPFkey generation algorithm. (Input rand consists of the random coins it consumes.) Its inputs are the index alpha the values beta, and a binder string. The value of alpha MUST be in range [0, 2^BITS). The output is a public part that is sent to all Aggregators and a vector of private IDPF keys, one for each aggregator. * Idpf.eval(agg_id: Unsigned, public_share: Bytes, key: Bytes, level: Unsigned, prefixes: Vec[Unsigned], binder: Bytes) > Idpf.Vec is the deterministic, stateless 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 depends on the value of level: If level < Idpf.BITS1, the output is the value for an inner node, which has type Vec[Vec[Idpf.FieldInner]]; otherwise, if level == Idpf.BITS 1, then the output is the value for a leaf node, which has type Vec[Vec[Idpf.FieldLeaf]]. The value of level MUST be in range [0, BITS). The indexes in prefixes MUST all be distinct and in range [0, 2^level). Applications MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of key in the sequence of IDPF keys output by the Client. Barnes, et al. Expires 14 September 2023 [Page 67] InternetDraft VDAF March 2023 In addition, the following method is derived for each concrete Idpf: def current_field(Idpf, level): return Idpf.FieldInner if level < Idpf.BITS1 \ else Idpf.FieldLeaf Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the VDAF abstraction boundary, which does not include shared state across across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the state across evaluations. See Section 8.3 for details. +============+==================================================+  Parameter  Description  +============+==================================================+  SHARES  Number of IDPF keys output by 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 IDPF    key generator  +++  KEY_SIZE  Size in bytes of each IDPF key  +++  FieldInner  Implementation of Field (Section 6.1) used for    values of inner nodes  +++  FieldLeaf  Implementation of Field used for values of leaf    nodes  +++ Table 12: Constants and types defined by a concrete IDPF. 8.2. Construction This section specifies Poplar1, an implementation of the Vdaf interface (Section 5). It is defined in terms of any Idpf (Section 8.1) for which Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2 and an implementation of Prg (Section 6.2). The associated constants and types required by the Vdaf interface are defined in Table 13. The methods required for sharding, preparation, aggregation, and unsharding are described in the remaining subsections. These methods make use of constants defined in Table 14. Barnes, et al. Expires 14 September 2023 [Page 68] InternetDraft VDAF March 2023 +=================+==================================+  Parameter  Value  +=================+==================================+  VERIFY_KEY_SIZE  Prg.SEED_SIZE  +++  NONCE_SIZE  16  +++  ROUNDS  2  +++  SHARES  2  +++  Measurement  Unsigned  +++  AggParam  Tuple[Unsigned, Vec[Unsigned]]  +++  Prep  Tuple[Bytes, Unsigned, Idpf.Vec]  +++  OutShare  Idpf.Vec  +++  AggResult  Vec[Unsigned]  +++ Table 13: VDAF parameters for Poplar1. +===========================+=======+  Variable  Value  +===========================+=======+  DST_SHARD_RAND: Unsigned  1  +++  DST_CORR_INNER: Unsigned  2  +++  DST_CORR_LEAF: Unsigned  3  +++  DST_VERIFY_RAND: Unsigned  4  +++ Table 14: Constants used by Poplar1. 8.2.1. Client The client's input is an IDPF index, denoted alpha. The programmed IDPF values are pairs of field elements (1, k) where each k is chosen at random. This random value is used as part of the secure sketching protocol of [BBCGGI21], Appendix C.4. After evaluating their IDPF key shares on a given sequence of candidate prefixes, the sketching protocol is used by the Aggregators to verify that they hold shares of a onehot vector. In addition, for each level of the tree, the prover generates random elements a, b, and c and computes Barnes, et al. Expires 14 September 2023 [Page 69] InternetDraft VDAF March 2023 A = 2*a + k B = a^2 + b  k*a + c and sends additive shares of a, b, c, A and B to the Aggregators. Putting everything together, the sharding algorithm is defined as follows. Function encode_input_shares is defined in Section 8.2.6. def measurement_to_input_shares(Poplar1, measurement, nonce, rand): l = Poplar1.Prg.SEED_SIZE # Split the coins into coins for IDPF key generation, # correlated randomness, and sharding. if len(rand) != Poplar1.RAND_SIZE: raise ERR_INPUT # unexpected length for random coins idpf_rand, rand = front(Poplar1.Idpf.RAND_SIZE, rand) seeds = [rand[i:i+l] for i in range(0,3*l,l)] corr_seed, seeds = front(2, seeds) (k_shard,), seeds = front(1, seeds) prg = Poplar1.Prg(k_shard, Poplar1.custom(DST_SHARD_RAND), b'') # Construct the IDPF values for each level of the IDPF tree. # Each "data" value is 1; in addition, the Client generates # a random "authenticator" value used by the Aggregators to # compute the sketch during preparation. This sketch is used # to verify the onehotness of their output shares. beta_inner = [ [Poplar1.Idpf.FieldInner(1), k] \ for k in prg.next_vec(Poplar1.Idpf.FieldInner, Poplar1.Idpf.BITS  1) ] beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \ prg.next_vec(Poplar1.Idpf.FieldLeaf, 1) # Generate the IDPF keys. (public_share, keys) = Poplar1.Idpf.gen(measurement, beta_inner, beta_leaf, idpf_rand) # Generate correlated randomness used by the Aggregators to # compute a sketch over their output shares. PRG seeds are # used to encode shares of the `(a, b, c)` triples. # (See [BBCGGI21, Appendix C.4].) corr_offsets = vec_add( Poplar1.Prg.expand_into_vec( Poplar1.Idpf.FieldInner, corr_seed[0], Barnes, et al. Expires 14 September 2023 [Page 70] InternetDraft VDAF March 2023 Poplar1.custom(DST_CORR_INNER), byte(0) + nonce, 3 * (Poplar1.Idpf.BITS1), ), Poplar1.Prg.expand_into_vec( Poplar1.Idpf.FieldInner, corr_seed[1], Poplar1.custom(DST_CORR_INNER), byte(1) + nonce, 3 * (Poplar1.Idpf.BITS1), ), ) corr_offsets += vec_add( Poplar1.Prg.expand_into_vec( Poplar1.Idpf.FieldLeaf, corr_seed[0], Poplar1.custom(DST_CORR_LEAF), byte(0) + nonce, 3, ), Poplar1.Prg.expand_into_vec( Poplar1.Idpf.FieldLeaf, corr_seed[1], Poplar1.custom(DST_CORR_LEAF), byte(1) + nonce, 3, ), ) # For each level of the IDPF tree, shares of the `(A, B)` # pairs are computed from the corresponding `(a, b, c)` # triple and authenticator value `k`. corr_inner = [[], []] for level in range(Poplar1.Idpf.BITS): Field = Poplar1.Idpf.current_field(level) k = beta_inner[level][1] if level < Poplar1.Idpf.BITS  1 \ else beta_leaf[1] (a, b, c), corr_offsets = corr_offsets[:3], corr_offsets[3:] A = Field(2) * a + k B = a^2 + b  a * k + c corr1 = prg.next_vec(Field, 2) corr0 = vec_sub([A, B], corr1) if level < Poplar1.Idpf.BITS  1: corr_inner[0] += corr0 corr_inner[1] += corr1 else: corr_leaf = [corr0, corr1] Barnes, et al. Expires 14 September 2023 [Page 71] InternetDraft VDAF March 2023 # Each input share consists of the Aggregator's IDPF key # and a share of the correlated randomness. return (public_share, Poplar1.encode_input_shares( keys, corr_seed, corr_inner, corr_leaf)) Figure 19: The sharding algorithm for Poplar1. 8.2.2. Preparation The aggregation parameter encodes a sequence of candidate prefixes. When an Aggregator receives an input share from the Client, it begins by evaluating its IDPF share on each candidate prefix, recovering a data_share and auth_share for each. The Aggregators use these and the correlation shares provided by the Client to verify that the sequence of data_share values are additive shares of a 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. The algorithms below make use of the auxiliary function decode_input_share() defined in Section 8.2.6. def prep_init(Poplar1, verify_key, agg_id, agg_param, nonce, public_share, input_share): (level, prefixes) = agg_param (key, corr_seed, corr_inner, corr_leaf) = \ Poplar1.decode_input_share(input_share) Field = Poplar1.Idpf.current_field(level) # Ensure that candidate prefixes are all unique and appear in # lexicographic order. for i in range(1,len(prefixes)): if prefixes[i1] >= prefixes[i]: raise ERR_INPUT # outoforder prefix # Evaluate the IDPF key at the given set of prefixes. value = Poplar1.Idpf.eval( agg_id, public_share, key, level, prefixes) # Get shares of the correlated randomness for computing the Barnes, et al. Expires 14 September 2023 [Page 72] InternetDraft VDAF March 2023 # Aggregator's share of the sketch for the given level of the IDPF # tree. if level < Poplar1.Idpf.BITS  1: corr_prg = Poplar1.Prg(corr_seed, Poplar1.custom(DST_CORR_INNER), byte(agg_id) + nonce) # Fastforward the PRG state to the current level. corr_prg.next_vec(Field, 3 * level) else: corr_prg = Poplar1.Prg(corr_seed, Poplar1.custom(DST_CORR_LEAF), byte(agg_id) + nonce) (a_share, b_share, c_share) = corr_prg.next_vec(Field, 3) (A_share, B_share) = corr_inner[2*level:2*(level+1)] \ if level < Poplar1.Idpf.BITS  1 else corr_leaf # Compute the Aggregator's first round of the sketch. These are # called the "masked input values" [BBCGGI21, Appendix C.4]. verify_rand_prg = Poplar1.Prg(verify_key, Poplar1.custom(DST_VERIFY_RAND), nonce + to_be_bytes(level, 2)) verify_rand = verify_rand_prg.next_vec(Field, len(prefixes)) sketch_share = [a_share, b_share, c_share] out_share = [] for (i, r) in enumerate(verify_rand): (data_share, auth_share) = value[i] sketch_share[0] += data_share * r sketch_share[1] += data_share * r^2 sketch_share[2] += auth_share * r out_share.append(data_share) prep_mem = sketch_share \ + [A_share, B_share, Field(agg_id)] \ + out_share return (b'ready', level, prep_mem) def prep_next(Poplar1, prep_state, opt_sketch): (step, level, prep_mem) = prep_state Field = Poplar1.Idpf.current_field(level) # Aggregators exchange masked input values (step (3.) # of [BBCGGI21, Appendix C.4]). if step == b'ready' and opt_sketch == None: sketch_share, prep_mem = prep_mem[:3], prep_mem[3:] return ((b'sketch round 1', level, prep_mem), Field.encode_vec(sketch_share)) # Aggregators exchange evaluated shares (step (4.)). Barnes, et al. Expires 14 September 2023 [Page 73] InternetDraft VDAF March 2023 elif step == b'sketch round 1' and opt_sketch != None: prev_sketch = Field.decode_vec(opt_sketch) if len(prev_sketch) == 0: prev_sketch = Field.zeros(3) elif len(prev_sketch) != 3: raise ERR_INPUT # prep message malformed (A_share, B_share, agg_id), prep_mem = \ prep_mem[:3], prep_mem[3:] sketch_share = [ agg_id * (prev_sketch[0]^2 \  prev_sketch[1]  prev_sketch[2]) \ + A_share * prev_sketch[0] \ + B_share ] return ((b'sketch round 2', level, prep_mem), Field.encode_vec(sketch_share)) elif step == b'sketch round 2' and opt_sketch != None: if len(opt_sketch) == 0: return prep_mem # Output shares else: raise ERR_INPUT # prep message malformed raise ERR_INPUT # unexpected input def prep_shares_to_prep(Poplar1, agg_param, prep_shares): if len(prep_shares) != 2: raise ERR_INPUT # unexpected number of prep shares (level, _) = agg_param Field = Poplar1.Idpf.current_field(level) sketch = vec_add(Field.decode_vec(prep_shares[0]), Field.decode_vec(prep_shares[1])) if len(sketch) == 3: return Field.encode_vec(sketch) elif len(sketch) == 1: if sketch == Field.zeros(1): # In order to reduce communication overhead, let the # empty string denote a successful sketch verification. return b'' else: raise ERR_VERIFY # sketch verification failed else: return ERR_INPUT # unexpected input length Figure 20: Preparation state for Poplar1. Barnes, et al. Expires 14 September 2023 [Page 74] InternetDraft VDAF March 2023 8.2.3. Validity of Aggregation Parameters Aggregation parameters are valid for a given input share if no aggregation parameter with the same level has been used with the same input share before. The whole preparation phase MUST NOT be run more than once for a given combination of input share and level. def is_valid(agg_param, previous_agg_params): (level, _) = agg_param return all( level != other_level for (other_level, _) in previous_agg_params ) Figure 21: Validity of aggregation parameters for Poplar1. 8.2.4. Aggregation Aggregation involves simply adding up the output shares. def out_shares_to_agg_share(Poplar1, agg_param, out_shares): (level, prefixes) = agg_param Field = Poplar1.Idpf.current_field(level) agg_share = Field.zeros(len(prefixes)) for out_share in out_shares: agg_share = vec_add(agg_share, out_share) return Field.encode_vec(agg_share) Figure 22: Aggregation algorithm for Poplar1. 8.2.5. Unsharding Finally, the Collector unshards the aggregate result by adding up the aggregate shares. def agg_shares_to_result(Poplar1, agg_param, agg_shares, _num_measurements): (level, prefixes) = agg_param Field = Poplar1.Idpf.current_field(level) agg = Field.zeros(len(prefixes)) for agg_share in agg_shares: agg = vec_add(agg, Field.decode_vec(agg_share)) return list(map(lambda x: x.as_unsigned(), agg)) Figure 23: Computation of the aggregate result for Poplar1. 8.2.6. Auxiliary Functions Barnes, et al. Expires 14 September 2023 [Page 75] InternetDraft VDAF March 2023 8.2.6.1. Message Serialization This section defines methods for serializing input shares, as required by the Vdaf interface. Optional serialization of the aggregation parameter is also specified below. Implementation note: The aggregation parameter includes the level of the IDPF tree and the sequence of indices to evaluate. For implementations that perform 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. def encode_input_shares(Poplar1, keys, corr_seed, corr_inner, corr_leaf): input_shares = [] for (key, seed, inner, leaf) in zip(keys, corr_seed, corr_inner, corr_leaf): encoded = Bytes() encoded += key encoded += seed encoded += Poplar1.Idpf.FieldInner.encode_vec(inner) encoded += Poplar1.Idpf.FieldLeaf.encode_vec(leaf) input_shares.append(encoded) return input_shares def decode_input_share(Poplar1, encoded): l = Poplar1.Idpf.KEY_SIZE key, encoded = encoded[:l], encoded[l:] l = Poplar1.Prg.SEED_SIZE corr_seed, encoded = encoded[:l], encoded[l:] l = Poplar1.Idpf.FieldInner.ENCODED_SIZE \ * 2 * (Poplar1.Idpf.BITS  1) encoded_corr_inner, encoded = encoded[:l], encoded[l:] corr_inner = Poplar1.Idpf.FieldInner.decode_vec( encoded_corr_inner) l = Poplar1.Idpf.FieldLeaf.ENCODED_SIZE * 2 encoded_corr_leaf, encoded = encoded[:l], encoded[l:] corr_leaf = Poplar1.Idpf.FieldLeaf.decode_vec( encoded_corr_leaf) if len(encoded) != 0: raise ERR_INPUT return (key, corr_seed, corr_inner, corr_leaf) def encode_agg_param(Poplar1, level, prefixes): Barnes, et al. Expires 14 September 2023 [Page 76] InternetDraft VDAF March 2023 if level > 2^16  1: raise ERR_INPUT # level too deep if len(prefixes) > 2^32  1: raise ERR_INPUT # too many prefixes encoded = Bytes() encoded += to_be_bytes(level, 2) encoded += to_be_bytes(len(prefixes), 4) packed = 0 for (i, prefix) in enumerate(prefixes): packed = prefix << ((level+1) * i) l = floor(((level+1) * len(prefixes) + 7) / 8) encoded += to_be_bytes(packed, l) return encoded def decode_agg_param(Poplar1, encoded): encoded_level, encoded = encoded[:2], encoded[2:] level = from_be_bytes(encoded_level) encoded_prefix_count, encoded = encoded[:4], encoded[4:] prefix_count = from_be_bytes(encoded_prefix_count) l = floor(((level+1) * prefix_count + 7) / 8) encoded_packed, encoded = encoded[:l], encoded[l:] packed = from_be_bytes(encoded_packed) prefixes = [] m = 2^(level+1)  1 for i in range(prefix_count): prefixes.append(packed >> ((level+1) * i) & m) if len(encoded) != 0: raise ERR_INPUT return (level, prefixes) 8.3. The IDPF scheme of [BBCGGI21] TODO(issue#32) Consider replacing the generic Prg object here with some fixedkey mode for AES (something along the lines of ia.cr/2019/074). This would allow us to take advantage of hardware acceleration, which would significantly improve performance. We use SHA3 primarily to instantiate random oracles, but the random oracle model may not be required for IDPF. More investigation is needed. In this section we specify a concrete IDPF, called IdpfPoplar, suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21]. TODO We should consider giving IdpfPoplar a more distinctive name. The constant and type definitions required by the Idpf interface are given in Table 15. Barnes, et al. Expires 14 September 2023 [Page 77] InternetDraft VDAF March 2023 IdpfPoplar requires a PRG for deriving the output shares, as well as a variety of other artifacts used internally. For performance reasons, we instantiate this object using PrgFixedKeyAes128 (Section 6.2.2). See Section 9.5 for justification of this choice. +============+======================+  Parameter  Value  +============+======================+  SHARES  2  +++  BITS  any positive integer  +++  VALUE_LEN  any positive integer  +++  KEY_SIZE  Prg.SEED_SIZE  +++  FieldInner  Field64 (Table 3)  +++  FieldLeaf  Field255 (Table 3)  +++ Table 15: Constants and type definitions for IdpfPoplar. 8.3.1. Key Generation TODO Describe the construction in prose, beginning with a gentle introduction to the high level idea. The description of the 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). def gen(IdpfPoplar, alpha, beta_inner, beta_leaf, binder, rand): if alpha >= 2^IdpfPoplar.BITS: raise ERR_INPUT # alpha too long if len(beta_inner) != IdpfPoplar.BITS  1: raise ERR_INPUT # beta_inner vector is the wrong size if len(rand) != IdpfPoplar.RAND_SIZE: raise ERR_INPUT # unexpected length for random coins init_seed = [ rand[:PrgFixedKeyAes128.SEED_SIZE], rand[PrgFixedKeyAes128.SEED_SIZE:], ] seed = init_seed.copy() Barnes, et al. Expires 14 September 2023 [Page 78] InternetDraft VDAF March 2023 ctrl = [Field2(0), Field2(1)] correction_words = [] for level in range(IdpfPoplar.BITS): Field = IdpfPoplar.current_field(level) keep = (alpha >> (IdpfPoplar.BITS  level  1)) & 1 lose = 1  keep bit = Field2(keep) (s0, t0) = IdpfPoplar.extend(seed[0], binder) (s1, t1) = IdpfPoplar.extend(seed[1], binder) seed_cw = xor(s0[lose], s1[lose]) ctrl_cw = ( t0[0] + t1[0] + bit + Field2(1), t0[1] + t1[1] + bit, ) x0 = xor(s0[keep], ctrl[0].conditional_select(seed_cw)) x1 = xor(s1[keep], ctrl[1].conditional_select(seed_cw)) (seed[0], w0) = IdpfPoplar.convert(level, x0, binder) (seed[1], w1) = IdpfPoplar.convert(level, x1, binder) ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep] ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep] b = beta_inner[level] if level < IdpfPoplar.BITS1 \ else beta_leaf if len(b) != IdpfPoplar.VALUE_LEN: raise ERR_INPUT # beta too long or too short w_cw = vec_add(vec_sub(b, w0), w1) # Implementation note: Here we negate the correction word if # the control bit `ctrl[1]` is set. We avoid branching on the # value in order to reduce leakage via timing side channels. mask = Field(1)  Field(2) * Field(ctrl[1].as_unsigned()) for i in range(len(w_cw)): w_cw[i] *= mask correction_words.append((seed_cw, ctrl_cw, w_cw)) public_share = IdpfPoplar.encode_public_share(correction_words) return (public_share, init_seed) Figure 24: IDPFkey generation algorithm of IdpfPoplar. 8.3.2. Key Evaluation TODO Describe in prose how IDPFkey evaluation algorithm works. Barnes, et al. Expires 14 September 2023 [Page 79] InternetDraft VDAF March 2023 The description of the IDPFevaluation algorithm makes use of auxiliary functions extend(), convert(), and decode_public_share() defined in Section 8.3.3. def eval(IdpfPoplar, agg_id, public_share, init_seed, level, prefixes, binder): if agg_id >= IdpfPoplar.SHARES: raise ERR_INPUT # invalid aggregator ID if level >= IdpfPoplar.BITS: raise ERR_INPUT # level too deep if len(set(prefixes)) != len(prefixes): raise ERR_INPUT # candidate prefixes are nonunique correction_words = IdpfPoplar.decode_public_share(public_share) out_share = [] for prefix in prefixes: if prefix >= 2^(level+1): raise ERR_INPUT # prefix too long # The Aggregator's output share is the value of a node of # the IDPF tree at the given `level`. The node's value is # computed by traversing the path defined by the candidate # `prefix`. Each node in the tree is represented by a seed # (`seed`) and a set of control bits (`ctrl`). seed = init_seed ctrl = Field2(agg_id) for current_level in range(level+1): bit = (prefix >> (level  current_level)) & 1 # Implementation note: Typically the current round of # candidate prefixes would have been derived from # aggregate results computed during previous rounds. For # example, when using `IdpfPoplar` to compute heavy # hitters, a string whose hit count exceeded the given # threshold in the last round would be the prefix of each # `prefix` in the current round. (See [BBCGGI21, # Section 5.1].) In this case, part of the path would # have already been traversed. # # Recomputing nodes along previously traversed paths is # wasteful. Implementations can eliminate this added # complexity by caching nodes (i.e., `(seed, ctrl)` # pairs) output by previous calls to `eval_next()`. (seed, ctrl, y) = IdpfPoplar.eval_next(seed, ctrl, correction_words[current_level], current_level, bit, binder) out_share.append(y if agg_id == 0 else vec_neg(y)) return out_share Barnes, et al. Expires 14 September 2023 [Page 80] InternetDraft VDAF March 2023 # Compute the next node in the IDPF tree along the path determined by # a candidate prefix. The next node is determined by `bit`, the bit # of the prefix corresponding to the next level of the tree. # # TODO Consider implementing some version of the optimization # discussed at the end of [BBCGGI21, Appendix C.2]. This could on # average reduce the number of AES calls by a constant factor. def eval_next(IdpfPoplar, prev_seed, prev_ctrl, correction_word, level, bit, binder): Field = IdpfPoplar.current_field(level) (seed_cw, ctrl_cw, w_cw) = correction_word (s, t) = IdpfPoplar.extend(prev_seed, binder) s[0] = xor(s[0], prev_ctrl.conditional_select(seed_cw)) s[1] = xor(s[1], prev_ctrl.conditional_select(seed_cw)) t[0] += ctrl_cw[0] * prev_ctrl t[1] += ctrl_cw[1] * prev_ctrl next_ctrl = t[bit] (next_seed, y) = IdpfPoplar.convert(level, s[bit], binder) # Implementation note: Here we add the correction word to the # output if `next_ctrl` is set. We avoid branching on the value of # the control bit in order to reduce side channel leakage. mask = Field(next_ctrl.as_unsigned()) for i in range(len(y)): y[i] += w_cw[i] * mask return (next_seed, next_ctrl, y) Figure 25: IDPFevaluation generation algorithm of IdpfPoplar. 8.3.3. Auxiliary Functions def extend(IdpfPoplar, seed, binder): prg = PrgFixedKeyAes128(seed, format_custom(1, 0, 0), binder) s = [ prg.next(PrgFixedKeyAes128.SEED_SIZE), prg.next(PrgFixedKeyAes128.SEED_SIZE), ] b = prg.next(1)[0] t = [Field2(b & 1), Field2((b >> 1) & 1)] return (s, t) def convert(IdpfPoplar, level, seed, binder): prg = PrgFixedKeyAes128(seed, format_custom(1, 0, 1), binder) next_seed = prg.next(PrgFixedKeyAes128.SEED_SIZE) Field = IdpfPoplar.current_field(level) w = prg.next_vec(Field, IdpfPoplar.VALUE_LEN) return (next_seed, w) Barnes, et al. Expires 14 September 2023 [Page 81] InternetDraft VDAF March 2023 def encode_public_share(IdpfPoplar, correction_words): encoded = Bytes() control_bits = list(itertools.chain.from_iterable( cw[1] for cw in correction_words )) encoded += pack_bits(control_bits) for (level, (seed_cw, _, w_cw)) \ in enumerate(correction_words): Field = IdpfPoplar.current_field(level) encoded += seed_cw encoded += Field.encode_vec(w_cw) return encoded def decode_public_share(IdpfPoplar, encoded): l = floor((2*IdpfPoplar.BITS + 7) / 8) encoded_ctrl, encoded = encoded[:l], encoded[l:] control_bits = unpack_bits(encoded_ctrl, 2 * IdpfPoplar.BITS) correction_words = [] for level in range(IdpfPoplar.BITS): Field = IdpfPoplar.current_field(level) ctrl_cw = ( control_bits[level * 2], control_bits[level * 2 + 1], ) l = PrgFixedKeyAes128.SEED_SIZE seed_cw, encoded = encoded[:l], encoded[l:] l = Field.ENCODED_SIZE * IdpfPoplar.VALUE_LEN encoded_w_cw, encoded = encoded[:l], encoded[l:] w_cw = Field.decode_vec(encoded_w_cw) correction_words.append((seed_cw, ctrl_cw, w_cw)) if len(encoded) != 0: raise ERR_DECODE return correction_words Figure 26: Helper functions for IdpfPoplar. Here, pack_bits() takes a list of bits, packs each group of eight bits into a byte, in LSB to MSB order, padding the most significant bits of the last byte with zeros as necessary, and returns the byte array. unpack_bits() performs the reverse operation: it takes in a byte array and a number of bits, and returns a list of bits, extracting eight bits from each byte in turn, in LSB to MSB order, and stopping after the requested number of bits. If the byte array has an incorrect length, or if unused bits in the last bytes are not zero, it throws an error. Barnes, et al. Expires 14 September 2023 [Page 82] InternetDraft VDAF March 2023 8.4. Instantiation By default, Poplar1 is instantiated with IdpfPoplar (VALUE_LEN == 2) and PrgSha3 (Section 6.2.1). This VDAF is suitable for any positive value of BITS. Test vectors can be found in Appendix "Test Vectors". 9. Security Considerations VDAFs have two essential security goals: 1. Privacy: An attacker that controls the network, the Collector, and a subset of Clients and Aggregators learns nothing about the measurements of honest Clients beyond what it can deduce from the aggregate result. 2. Robustness: An attacker that controls the network and a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients. Formal definitions of privacy and robustness can be found in [DPRS23]. A VDAF is the core cryptographic primitive of a protocol that achieves the above privacy and robustness goals. It is not sufficient on its own, however. The application will need to assure a few security properties, for example: * Securely distributing the longlived parameters, in particular the verification key. * Establishing secure channels:  Confidential and authentic channels among Aggregators, and between the Aggregators and the Collector; and  Confidential and Aggregatorauthenticated channels between Clients and Aggregators. * 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. Barnes, et al. Expires 14 September 2023 [Page 83] InternetDraft VDAF March 2023 On their own, VDAFs do not mitigate Sybil attacks [Dou02]. In this attack, the adversary observes a subset of input shares transmitted by a Client it is interested in. It allows the input shares to be processed, but corrupts and picks bogus measurements for the remaining Clients. Applications can guard against these risks by adding additional controls on report submission, such as client authentication and rate limits. VDAFs do not inherently provide differential privacy [Dwo06]. The VDAF approach to private measurement can be viewed as complementary to differential privacy, relying on noncollusion instead of statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and removing it after unsharding. 9.1. Requirements for the Verification Key The Aggregators are responsible for exchanging the verification key in advance of executing the VDAF. Any procedure is acceptable as long as the following conditions are met: 1. To ensure robustness of the computation, the Aggregators MUST NOT reveal the verification key to the Clients. Otherwise, a malicious Client might be able to exploit knowledge of this key to craft an invalid report that would be accepted by the Aggregators. 2. To ensure privacy of the measurements, the Aggregators MUST commit to the verification key prior to processing reports generated by Clients. Otherwise, a malicious Aggregator may be able to craft a verification key that, for a given report, causes an honest Aggregator to leak information about the measurement during preparation. Meeting these conditions is required in order to leverage security analysis in the framework of [DPRS23]. Their definition of robustness allows the attacker, playing the role of a cohort of malicious Clients, to submit arbitrary reports to the Aggregators and eavesdrop on their communications as they process them. Security in this model is achievable as long as the verification key is kept secret from the attacker. The privacy definition of [DPRS23] considers an active attacker that controls the network and a subset of Aggregators; in addition, the attacker is allowed to choose the verification key used by each honest Aggregator over the course of the experiment. Security is achievable in this model as long as the key is picked at the start of Barnes, et al. Expires 14 September 2023 [Page 84] InternetDraft VDAF March 2023 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 non repeating. For example, to achieve differential privacy it is necessary to avoid "over exposing" a measurement by including it too many times in a single batch or across multiple batches. It is RECOMMENDED that the nonce generated by the Client be used by the Aggregators for replay protection. 9.3. Requirements for the Aggregation Parameters As described in Section 4.3 and Section 5.3 respectively, DAFs and VDAFs may impose restrictions on the 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. Barnes, et al. Expires 14 September 2023 [Page 85] InternetDraft VDAF March 2023 9.4. Additional Privacy Considerations for Aggregation Parameters Aggregating a batch of reports multiple times, each time with a different aggregation parameter, could result in information leakage beyond what is used by the application. For example, when Poplar1 is used for heavy hitters, the Aggregators learn not only the heavy hitters themselves, but also the prefix tree (as defined in Section 8) computed along the way. Indeed, this leakage is inherent to any construction that uses an IDPF (Section 8.1) in the same way. Depending on the distribution of the measurements, the prefix tree can leak a significant amount of information about unpopular inputs. For instance, it is possible (though perhaps unlikely) for a large set of nonheavyhitter values to share a common prefix, which would be leaked by a prefix tree with a sufficiently small threshold. The only known, generalpurpose approach to mitigating this leakage is via differential privacy. TODO(issue #94) Describe (or point to some description of) the central DP mechanism for Poplar described in [BBCGGI21]. 9.5. Pseudorandom Generators and random oracles The objects we describe in Section 6.2 share a common interface, which we have called Prg. However, these are not necessarily all modeled as cryptographic Pseudorandom Generators in the security analyses of our protocols. Instead, most of them are modeled as random oracles. For these use cases, we want to be conservative in our assumptions, and hence prescribe PrgSha3 as the only RECOMMENDED Prg instantiation. The one exception is the PRG used in the Idpf implementation IdpfPoplar Section 8.3. Here, a random oracle is not needed to prove security, and hence a construction based on fixedkey AES Section 6.2.2 can be used. However, as PrgFixedKeyAes128 has been shown to be differentiable from a random oracle [GKWWY20], it is NOT RECOMMENDED to use it anywhere else. OPEN ISSUE: We may want to drop the common interface for PRGs and random oracles. See issue #159. 10. IANA Considerations A codepoint for each (V)DAF in this document is defined in the table below. Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for private use. Barnes, et al. Expires 14 September 2023 [Page 86] InternetDraft VDAF March 2023 +==========================+================+======+===============+  Value  Scheme  Type  Reference  +==========================+================+======+===============+  0x00000000  Prio3Count  VDAF  Section 7.4.1  +++++  0x00000001  Prio3Sum  VDAF  Section 7.4.2  +++++  0x00000002  Prio3Histogram  VDAF  Section 7.4.3  +++++  0x00000003 to 0x00000FFF  reserved for  VDAF  n/a    Prio3    +++++  0x00001000  Poplar1  VDAF  Section 8.4  +++++  0xFFFF0000 to 0xFFFFFFFF  reserved  n/a  n/a  +++++ Table 16: Unique identifiers for (V)DAFs. TODO Add IANA considerations for the codepoints summarized in Table 16. 11. References 11.1. Normative References [FIPS202] "SHA3 Standard: PermutationBased Hash and Extendable Output Functions", NIST FIPS PUB 202 , August 2015. [RFC2119] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", BCP 14, RFC 2119, DOI 10.17487/RFC2119, March 1997, <https://www.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, May 2017, <https://www.rfceditor.org/rfc/rfc8174>. [SP800185] "SHA3 Derived Functions: cSHAKE, KMAC, TupleHash and ParallelHash", NIST Special Publication 800185 , December 2016. 11.2. Informative References Barnes, et al. Expires 14 September 2023 [Page 87] InternetDraft VDAF March 2023 [AGJOP21] Addanki, S., Garbe, K., Jaffe, E., Ostrovsky, R., and A. Polychroniadou, "Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares", 2021, <https://ia.cr/2021/576>. [BBCGGI19] Boneh, D., Boyle, E., CorriganGibbs, H., Gilboa, N., and Y. Ishai, "ZeroKnowledge Proofs on SecretShared Data via Fully Linear PCPs", CRYPTO 2019 , 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 , 2021, <https://ia.cr/2021/017>. [CGB17] CorriganGibbs, H. and D. Boneh, "Prio: Private, Robust, and Scalable Computation of Aggregate Statistics", NSDI 2017 , 2017, <https://dl.acm.org/doi/10.5555/3154630.3154652>. [DAP] Geoghegan, T., Patton, C., Rescorla, E., and C. A. Wood, "Distributed Aggregation Protocol for Privacy Preserving Measurement", Work in Progress, InternetDraft, draft ietfppmdap03, 9 December 2022, <https://datatracker.ietf.org/doc/html/draftietfppmdap 03>. [Dou02] Douceur, J., "The Sybil Attack", IPTPS 2002 , 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 , 2006, <https://link.springer.com/chapter/10.1007/11787006_1>. [ENPA] "Exposure Notification Privacypreserving Analytics (ENPA) White Paper", 2021, <https://covid19static.cdn apple.com/applications/covid19/current/static/contact tracing/pdf/ENPA_White_Paper.pdf>. [EPK14] Erlingsson, Ú., Pihur, V., and A. Korolova, "RAPPOR: Randomized Aggregatable PrivacyPreserving Ordinal Response", CCS 2014 , 2014, <https://dl.acm.org/doi/10.1145/2660267.2660348>. Barnes, et al. Expires 14 September 2023 [Page 88] InternetDraft VDAF March 2023 [GI14] Gilboa, N. and Y. Ishai, "Distributed Point Functions and Their Applications", EUROCRYPT 2014 , 2014, <https://link.springer.com/ chapter/10.1007/9783642552205_35>. [GKWWY20] "Better concrete security for halfgates garbling (in the multiinstance setting)", CRYPTO 2020 , 2020, <https://link.springer.com/ chapter/10.1007/9783030568801_28>. [OriginTelemetry] "Origin Telemetry", 2020, <https://firefoxsource docs.mozilla.org/toolkit/components/telemetry/collection/ origin.html>. Acknowledgments The security considerations in Section 9 are based largely on the security analysis of [DPRS23]. Thanks to Hannah Davis and Mike Rosulek, who lent their time to developing definitions and security proofs. Thanks to Henry CorriganGibbs, Armando FazHernández, Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, Xiao Wang, and Christopher Wood for useful feedback on and contributions to the spec. Test Vectors NOTE 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 Barnes, et al. Expires 14 September 2023 [Page 89] InternetDraft VDAF March 2023 verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 1 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > input_share_0: > e7a225b76420dd6dd0682380363bd782c8ca9ace6e7abc559dd873bafb503e3cc8b7 9f3b2b8b0a14676172e46be2ce2f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f round_0: prep_share_0: > 56df20acd68725eab85e44c706629df555d635a5b899df767fe28effd0c45f60 prep_share_1: > ab20df532878da1518b04178e08a39d0d40fdbfe1876e479dc5e94b43afc68d5 prep_message: > out_share_0:  e7a225b76420dd6d out_share_1:  1b5dda489adf2292 agg_share_0: > e7a225b76420dd6d agg_share_1: > 1b5dda489adf2292 agg_result: 1 Prio3Sum bits: 8 verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 100 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > da6f5f000d95651e628464f5d086fde6a544464ac776c3812e2299cce1d29095 input_share_0: > 2d97ed6ebc5f6748db3017211c7a31ceeba9500b8317a7f61578b85c08a7f7d95c63 7bb14b95c062c74bd5b08625bdafbfd91fd01d22291b22b76e54a9be1f7e7015a396 ae0d6d034b3d22795b3d98db2a9aa51b4ea56c247496d380367deb786a9b34d5cb88 f0f43e5cb4ab71910c8fa20b0b518f7dd9ce11ec9bf4ce460c81d3e9db05c34e0f05 14d53f5eadea231dbc1fc575ad52e01622671eaf49a66533a6511710eeea024df0da 7e7d21451ff5fd9d38e36eee8b78bf7c0767b91e747c6b80d6ca6b726286edebeae4 ff377040d87f84288f4d0e5960e27b79387caef816c27481be166b71d38d1695e5e5 061c26f0f667f8fff3133c2ae3ac1853ac542a9af0b5e422764609abd882dedf6aa8 c835228bbc0cb99a695e81c45a2876a961af0c680662de4400ec96004c45bdc32726 68b01d7f861f2e03fc18adcc352bdeb7b06b39c2de749da45e9b48f46c45a390cb59 455dbf3e8be5938203def1c45af5eb5bf2037be005768d74faf11efd255b699f5ae9 bce8b3bf4688907d11ca89e7609b77c04abde845f63091b934017cf9a83f80eb86fd Barnes, et al. Expires 14 September 2023 [Page 90] InternetDraft VDAF March 2023 aa063630f86ee982d812062db0d1d760d84cfebbf99927cad516dd028bf2f9e9b7d2 4aa0bd9c0cb16a46c469eacfe1c48b8d7bf626e5d3eedb3f3cc4972f14a4ba605565 ec795dd9e68af4ac6185f370f1f77610c64b04d295007b34ed59ed7bceace7b9dea8 8346a2ee47dfec257ede5cd6985c34abaf5983d067786a695ae1039b631555bbe876 7baeedfe4f30df6b47479543c58a87e0aaf371234fc9094c4b2b79d025229634a3fd f68a36be0e8661a4f043504b7025f03f02e57704b4e410c60e69e88050d0c33ea71b c37f306c225f457fbab4929c65ba9401dc57a2dc31bd6d3f53099a00303132333435 363738393a3b3c3d3e3f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021 22232425262728292a2b2c2d2e2f round_0: prep_share_0: > 7ae72bff1c063181bfea222dd77bcd40c5d15f62174d4af984d728effb82edd302 74c927a9ffa51b90c167c60b40c3cdda6f5f000d95651e628464f5d086fde6 prep_share_1: > 8718d400e3f9ce7e2415ddd2288432bf85623812cb8a7cdfed57f5a417c422e25e 84a67ab22c0b8ef7b0500c366feb95a544464ac776c3812e2299cce1d29095 prep_message: > 4f351791f4d706a31c54ce699ef1af97 out_share_0:  a18d0c7e2a74470727d4f628ff934e52 out_share_1:  c472f381d58bb8f8bc2b09d7006cb1ad agg_share_0: > a18d0c7e2a74470727d4f628ff934e52 agg_share_1: > c472f381d58bb8f8bc2b09d7006cb1ad agg_result: 100 Prio3Histogram Barnes, et al. Expires 14 September 2023 [Page 91] InternetDraft VDAF March 2023 buckets: [1, 10, 100] verify_key: "000102030405060708090a0b0c0d0e0f" upload_0: measurement: 50 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > 5e015517900cfc204138c24f808ddf4ee85eca87ba246cd715d116195172e500 input_share_0: > c269f2189db431ca2ebe7230bacd692ac02a7790818681b6ee5ecbd509587b2976c7 897326840bb70083ee4df8ffe3dc65dbd71f56735a7dcc3a150da8e77171e6d380f2 f6886845e310dabec4382a037ffc3520737af8874f5b2aec2fae1405cce12c257ef2 9d22b3e511c30e8f9545251286b3ef3e3728256fc0cc21f5a8ea1d095187cc29132e 52d24ad778920c740af125833e9a8d2f6b0255740e8d2a4e7b718446610a0650ba40 61e6a3ee805c1dcffec90d00a3ea5817c9956459538c99efbb56b38eaa6aed93f5a0 0f11ab0210af030b91c56e6b2b47bb30fcdc5f44a28b5d733e4661cbd736ae071e78 4f99ba86ff2688f6751d2d16e80ddd99c9aceeb39ca660215f0fc1178f828e295840 f941fa9a3217afd52b32ec6c8ee3ec58291f2c88e8150374ccfd503f07bc0021d6a9 f1c81050df96ec5ce5c7d4f41255303132333435363738393a3b3c3d3e3f input_share_1: > 000102030405060708090a0b0c0d0e0f101112131415161718191a1b1c1d1e1f2021 22232425262728292a2b2c2d2e2f round_0: prep_share_0: > ac872edaaaa532de61bcd63b168e522230b35064b09b6b35c1352bc354560fdeac 684da645d3e509c6db8ba31f58dabb5e015517900cfc204138c24f808ddf4e prep_share_1: > 5578d125555acd21824329c4e971adddc07187f0f25a30ceba8407513f60f8e3d1 5c17270c5631a718b3b4f5702aff54e85eca87ba246cd715d116195172e500 prep_message: > 890bfbdf0c619d5c853a92e28bce41b3 out_share_0:  c269f2189db431ca2ebe7230bacd692a  c02a7790818681b6ee5ecbd509587b29  76c7897326840bb70083ee4df8ffe3dc  65dbd71f56735a7dcc3a150da8e77171 out_share_1:  3f960de7624bce35b5418dcf453296d5  41d5886f7e797e49f5a0342af6a784d6  8c38768cd97bf448e37c11b207001c23  9c2428e0a98ca58217c5eaf257188e8e agg_share_0: > c269f2189db431ca2ebe7230bacd692ac02a7790818681b6ee5ecbd509587b2976c789 7326840bb70083ee4df8ffe3dc65dbd71f56735a7dcc3a150da8e77171 agg_share_1: > 3f960de7624bce35b5418dcf453296d541d5886f7e797e49f5a0342af6a784d68c3876 8cd97bf448e37c11b207001c239c2428e0a98ca58217c5eaf257188e8e agg_result: [0, 0, 1, 0] Barnes, et al. Expires 14 September 2023 [Page 92] InternetDraft VDAF March 2023 Poplar1 Sharding bits: 4 upload_0: measurement: 13 nonce: "000102030405060708090a0b0c0d0e0f" public_share: > 8b332c8b09b84f5801ad1cfd30b0df3fba69d11ddf6065a9cdb87cb31579ba0443a4 82f4ed21ac079b96f5944d1669a1159406148c00878ac9953bcfbadc0273042653fd 0fae8c43ce746fd31f248d1cbf5d1bc01fe02a883571cf38a1a4c10f54d41979b6fc 35ef1fb051f968ba4397b8e61a81d835fa7ee54b949a35abf76c13e0a9daf90f1528 c40db5904b6e24d70116538b28c16d9dbfee45000fe14db45cafbb76768b28250e74 6af691f4bb1619 input_share_0: > 000102030405060708090a0b0c0d0e0f202122232425262728292a2b2c2d2e2f0734 d55dcb0dec9599b96f38ca694d46056dbe5bb24ab41ca10faaa8249839632c6503c9 795300a160e8c378762f3b329e9ebe6a081eaed2d72aaff0aeb52415211c314f609a 0377e66b7be10a3d8d4e75890da4c93f62172bc07e54be31c060c5970b52af8e8633 d63560f59fa0bb36 input_share_1: > 101112131415161718191a1b1c1d1e1f303132333435363738393a3b3c3d3e3f9558 af2229b50110d88e17b5b3a81902194a5eab5f8ad69f5c061f16ac1e92deadc3c6ed 32714a3756fa68e580d523d6c0e4b92ba2277033a7adb060e7a2ee3b2e44591d7c99 b1cbcbfc58e4ae9ef7613a1ff24e53201e7264940bed2cd318e8e80ab7aae630f342 911d75c85980691d Preparation, Aggregation, and Unsharding Barnes, et al. Expires 14 September 2023 [Page 93] InternetDraft VDAF March 2023 verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (0, [0, 1]) upload_0: round_0: prep_share_0: > a2d369b7ee44e5f8b903e889b37ea7ef4d29ee36d1abdb13 prep_share_1: > 31182ff6e075b925a06c802eac3b14c1ad66ce991ca2ffc2 prep_message: > d2eb98add0ba9e1e587068b860babbb0fa8fbcd0ed4ddbd6 round_1: prep_share_0: > f4445265dc801ab5 prep_share_1: > 0dbbad9a227fe54a prep_message: > out_share_0:  d47170d4f804449e  e16add75c1d3f0c4 out_share_1:  2d8e8f2b06fbbb61  2195228a3d2c0f3b agg_share_0: > d47170d4f804449ee16add75c1d3f0c4 agg_share_1: > 2d8e8f2b06fbbb612195228a3d2c0f3b agg_result: [0, 1] Barnes, et al. Expires 14 September 2023 [Page 94] InternetDraft VDAF March 2023 verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (1, [0, 1, 2, 3]) upload_0: round_0: prep_share_0: > 94352b588fbd978be7ef2cacb83b1b64c1d11af699253242 prep_share_1: > 4a3529a2b34159be1666064ba0a3a9cc4e72223ef6d0ac79 prep_message: > dd6a54fa43fff049fc5533f759dfc4300f443d3490f6debb round_1: prep_share_0: > 3dc2833b1f9010ee prep_share_1: > c43d7cc4df6fef11 prep_message: > out_share_0:  c03ede9261e7f441  8ef9fde3f50eb967  de3bb4acf0b9dff8  24687334644cc809 out_share_1:  41c1216d9d180bbe  7306021c09f14698  23c44b530e462007  de978ccb9ab337f6 agg_share_0: > c03ede9261e7f4418ef9fde3f50eb967de3bb4acf0b9dff824687334644cc809 agg_share_1: > 41c1216d9d180bbe7306021c09f1469823c44b530e462007de978ccb9ab337f6 agg_result: [0, 0, 0, 1] Barnes, et al. Expires 14 September 2023 [Page 95] InternetDraft VDAF March 2023 verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (2, [0, 2, 4, 6]) upload_0: round_0: prep_share_0: > b640798e11bc37ac1f4981c99f0de15569b2bcc187d2f0f2 prep_share_1: > 37a720e1739caca1b39b8fe97ce96e09277eee9cc694c2cf prep_message: > ece7996f8658e44dd2e410b31cf74f5f8f30ab5e4f67b3c2 round_1: prep_share_0: > 52a1848f964385f8 prep_share_1: > af5e7b7068bc7a07 prep_message: > out_share_0:  352395a6c5e5804e  f9b61274892e76dd  b27e3eb4accbaa0a  3b1c4a7f78d59935 out_share_1:  ccdc6a59391a7fb1  0849ed8b75d18922  4f81c14b523455f5  c7e3b580862a66ca agg_share_0: > 352395a6c5e5804ef9b61274892e76ddb27e3eb4accbaa0a3b1c4a7f78d59935 agg_share_1: > ccdc6a59391a7fb10849ed8b75d189224f81c14b523455f5c7e3b580862a66ca agg_result: [0, 0, 0, 1] verify_key: "000102030405060708090a0b0c0d0e0f" agg_param: (3, [1, 3, 5, 7, 9, 13, 15]) upload_0: round_0: prep_share_0: > 08efd12285f6e61599b2f9d941577fd843597d3b15f747d15b0b02daf6147752c6 a7a72723fa0cad98c7c38a0716ff8833dd7f97d40ed374f963dd2bced921614ccd 4dca993e6c4f5d1ab0ce9bf37df76b41e2c111868322dfc39b3145dfc832 prep_share_1: > 04eb21e562d79e4255b2c5001ed5949fc01841f40d921fdf5c1100b5c993f34e9d a143558cc507fb9c3e3832d85b751575b8f5a102257b141b94027bc0a849013f46 00d8257361b9d17e43f8cddc106b512f69c78b63bcde177e2d3c0cfb353d prep_message: > 1fdaf307e8cd8558ee64bfda5f2c14780472be2f238967b0b81c028fc0a86a2163 49eb7cafbf14a83506fcbcdf71749ea8957539d7334e8914f8dfa68e826b628b13 4ea2bfb1cd082f99f3c669d08e62bd704b899de93f01f741c96d51dafe6f Barnes, et al. Expires 14 September 2023 [Page 96] InternetDraft VDAF March 2023 round_1: prep_share_0: > 3e1ded312b78a4a2003d9365472d481c0adb6411d7049869693fb0b331750301 prep_share_1: > afe212ced4875b5dffc26c9ab8d2b7e3f5249bee28fb679696c04f4cce8afc7e prep_message: > out_share_0:  034a3226150cf3aa5e7f6e76b0c6a6b2aa9b750cdafdc6aae0ff2f4076894e2a  1413b483f800a4faa67b1e1c6bb1be2a1c9cc590761edb85f77692b67f9c696a  85591b91ee55008c7e7a58176d5bce3ca9974482b0c1aa2b7c69e0ab2e2e212c  03d9e6cde627cc30ceac7a6b5e3192a382d641e761c05c01eea4bc543c7fc41d  cd8891db6ac77a7fbba2c401c1488b8d31ee2cbc65cb26d02cb037cc99d32543  732e3e133c6d8c9a0f6647fc24be3c5d8a960b63c3163979fd4d42fd554a7812  8a2964a89cd70d7ba65bb52f0f15fb2db7dc1c534d743e37f223f74b8489c85d out_share_1:  eab5cdd9eaf30c55a18091894f39594d55648af3250239551f00d0bf8976b155  d9ec4b7c07ff5b055984e1e3944e41d5e3633a6f89e1247a08896d4980639615  68a6e46e11aaff738185a7e892a431c35668bb7d4f3e55d483961f54d1d1de53  ea26193219d833cf31538594a1ce6d5c7d29be189e3fa3fe115b43abc3803b62  20776e2495388580445d3bfe3eb77472ce11d3439a34d92fd34fc833662cda3c  7bd1c1ecc3927365f099b803db41c3a27569f49c3ce9c68602b2bd02aab5876d  63d69b576328f28459a44ad0f0ea04d24823e3acb28bc1c80ddc08b47b763722 agg_share_0: > 034a3226150cf3aa5e7f6e76b0c6a6b2aa9b750cdafdc6aae0ff2f4076894e2a1413b4 83f800a4faa67b1e1c6bb1be2a1c9cc590761edb85f77692b67f9c696a85591b91ee55 008c7e7a58176d5bce3ca9974482b0c1aa2b7c69e0ab2e2e212c03d9e6cde627cc30ce ac7a6b5e3192a382d641e761c05c01eea4bc543c7fc41dcd8891db6ac77a7fbba2c401 c1488b8d31ee2cbc65cb26d02cb037cc99d32543732e3e133c6d8c9a0f6647fc24be3c 5d8a960b63c3163979fd4d42fd554a78128a2964a89cd70d7ba65bb52f0f15fb2db7dc 1c534d743e37f223f74b8489c85d agg_share_1: > eab5cdd9eaf30c55a18091894f39594d55648af3250239551f00d0bf8976b155d9ec4b 7c07ff5b055984e1e3944e41d5e3633a6f89e1247a08896d498063961568a6e46e11aa ff738185a7e892a431c35668bb7d4f3e55d483961f54d1d1de53ea26193219d833cf31 538594a1ce6d5c7d29be189e3fa3fe115b43abc3803b6220776e2495388580445d3bfe 3eb77472ce11d3439a34d92fd34fc833662cda3c7bd1c1ecc3927365f099b803db41c3 a27569f49c3ce9c68602b2bd02aab5876d63d69b576328f28459a44ad0f0ea04d24823 e3acb28bc1c80ddc08b47b763722 agg_result: [0, 0, 0, 0, 0, 1, 0] Authors' Addresses Richard L. Barnes Cisco Email: rlb@ipv.sx Barnes, et al. Expires 14 September 2023 [Page 97] InternetDraft VDAF March 2023 David Cook ISRG Email: divergentdave@gmail.com Christopher Patton Cloudflare Email: chrispatton+ietf@gmail.com Phillipp Schoppmann Google Email: schoppmann@google.com Barnes, et al. Expires 14 September 2023 [Page 98]