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IRTF CFRG Internet-Draft This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multi-party protocols for computing aggregate statistics over user measurements. These protocols are designed to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers to detect if a malicious or misconfigured client submitted an input that would result in an incorrect aggregate result. Discussion Venues Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at . Source for this draft and an issue tracker can be found at .

Introduction The ubiquity of the Internet makes it an ideal platform for measurement of large-scale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial overlap, however, between information that is valuable to measure and information that users consider private. For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition. In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy. Most prior approaches to this problem fall under the rubric of "differential privacy (DP)" . 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 , 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. Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically, depending on the "amount" of noise a client adds to its input, it may be possible for a curious aggregator to make a reasonable guess of the input's true value. Indeed, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility. The ideal goal for a privacy-preserving measurement system is that of secure multi-party computation: No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal. VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of non-colluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of inputs. The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others . The VDAF abstraction laid out in represents a class of multi-party protocols for privacy-preserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:

1. Providing higher-level protocols like with a simple, uniform interface for accessing privacy-preserving measurement schemes, and documenting relevant operational and security bounds for that interface:
   1. General patterns of communications among the various actors involved in the system (clients, aggregation servers, and the collector of the aggregate result);
   2. Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and
   3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.
2. Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions.

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

• The aforementioned Prio system allows for the privacy-preserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:
  1. Each client shards its measurement into a sequence of additive shares and distributes the shares among the aggregation servers.
  2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate.
  3. Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result.
  The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multi-party protocol for accomplishing this task. In we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in that result in significant performance gains.
• More recently, Boneh et al. described a protocol called Poplar for solving the t-heavy-hitters problem in a privacy-preserving manner. Here each client holds a bit-string of length n, and the goal of the aggregation servers is to compute the set of inputs that occur at least t times. The core primitive used in their protocol is a specialized Distributed Point Function (DPF) that allows the servers to "query" their DPF shares on any bit-string of length shorter than or equal to n. As a result of this query, each of the servers has an additive share of a bit indicating whether the string is a prefix of the client's input. The protocol also specifies a multi-party computation for verifying that at most one string among a set of candidates is a prefix of the client's input. In we describe a VDAF called Poplar1 that implements this functionality.

Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the client-generated measurements are prepared for aggregation. Because these constructions are based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this process may require multiple round trips over the network. There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.

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

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

Change Log (*) Indicates a change that breaks wire compatibility with the previous draft. 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 , Section 4.3 for details.)
• Extend (V)DAF syntax to allow the unsharding step to take into account the number of measurements aggregated.
• Extend FLP syntax by adding a method for decoding the aggregate result from a vector of field elements. The new method takes into account the number of measurements.
• Prio3: Align aggregate result computation with updated FLP syntax.
• Prg: Add a method for statefully generating a vector of field elements.
• Field: Require that field elements are fully reduced before decoding. (*)
• Define new field Field255.

01:

• Require that VDAFs specify serialization of aggregate shares.
• Define Distributed Aggregation Functions (DAFs).
• Prio3: Move proof verifier check from prep_next() to prep_shares_to_prep(). (*)
• Remove public parameter and replace verification parameter with a "verification key" and "Aggregator ID".

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 when, and only when, they appear in all capitals, as shown here. Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception. A variable with type Bytes is a byte string. This document defines several byte-string constants. When comprised of printable ASCII characters, they are written as Python 3 byte-string literals (e.g., b'some constant string'). A global constant VERSION is defined, which algorithms are free to use as desired. Its value SHALL be b'vdaf-02'. This document describes algorithms for multi-party computations in which the parties typically communicate over a network. Wherever a quantity is defined that must be be transmitted from one party to another, this document prescribes a particular encoding of that quantity as a byte string.

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

Some common functionalities:

• zeros(len: Unsigned) -> Bytes returns an array of zero bytes. The length of output MUST be len.
• gen_rand(len: Unsigned) -> Bytes returns an array of random bytes. The length of output MUST be len.
• byte(int: Unsigned) -> Bytes returns the representation of int as a byte string. The value of int MUST be in [0,256).
• xor(left: Bytes, right: Bytes) -> Bytes returns the bitwise XOR of left and right. An exception is raised if the inputs are not the same length.
• I2OSP and OS2IP from , which are used, respectively, to convert a non-negative integer to a byte string and convert a byte string to a non-negative integer.
• next_power_of_2(n: Unsigned) -> Unsigned returns the smallest integer greater than or equal to n that is also a power of two.

Overview

Overall data flow of a (V)DAF | Aggregator 0 |----+ | +--------------+ | | ^ | | | | | V | | +--------------+ | | +-->| Aggregator 1 |--+ | | | +--------------+ | | +--------+-+ | ^ | +->+-----------+ | Client |---+ | +--->| Collector |--> Aggregate +--------+-+ +->+-----------+ | ... | | | | | | | V | | +----------------+ | +--->| Aggregator N-1 |---+ +----------------+ Input shares Aggregate shares ]]>

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

• To submit an individual measurement, the Client shards the measurement into "input shares" and sends one input share to each Aggregator.
• The Aggregators convert their input shares into "output shares".
  • Output shares are in one-to-one correspondence with the input shares.
  • Just as each Aggregator receives one input share of each input, at the end of this process, each aggregator holds one output share.
  • In VDAFs, Aggregators will need to exchange information among themselves as part of the validation process.
• Each Aggregator combines the output shares across inputs in the batch to compute the "aggregate share" for that batch, i.e., its share of the desired aggregate result.
• The Aggregators submit their aggregate shares to the Collector, who combines them to obtain the aggregate result over the batch.

Aggregators are a new class of actor relative to traditional measurement systems where clients submit measurements to a single server. They are critical for both the privacy properties of the system and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by non-collusion among Aggregators, and Aggregators are the entities that perform validation of Client measurements. Thus clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to correctly run the protocol. Within the bounds of the non-collusion requirements of a given (V)DAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregator(s) to augment a basic client-server protocol. In this document, we describe the computations performed by the actors in this system. It is up to the higher-level protocol making use of the (V)DAF to arrange for the required information to be delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting measurements may be anonymous.

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 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 . Execution of a DAF has four distinct stages:

• Sharding - Each Client generates input shares from its measurement and distributes them among the Aggregators.
• Preparation - Each Aggregator converts each input share into an output share compatible with the aggregation function. This computation involves the aggregation parameter. In general, each aggregation parameter may result in a different an output share.
• Aggregation - Each Aggregator combines a sequence of output shares into its aggregate share and sends the aggregate share to the Collector.
• Unsharding - The Collector combines the aggregate shares into the aggregate result.

Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares). A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table. Constants and types defined by each concrete DAF.

Parameter	Description
SHARES	Number of input shares into which each measurement is sharded
Measurement	Type of each measurement
AggParam	Type of aggregation parameter
OutShare	Type of each output share
AggResult	Type of the aggregate result

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.

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) -> (Bytes, Vec[Bytes]) is the randomized input-distribution algorithm run by each Client. 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.

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

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.

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.

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

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

• OPEN ISSUE It may be worthwhile to explicitly define the "streaming" API. See issue#47.

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.

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 .) Or is this out-of-scope of this document? See https://github.com/ietf-wg-ppm/ppm-specification/issues/19.

Execution of a DAF Securely executing a DAF involves emulating the following procedure.

Execution of a DAF.

The inputs to this procedure are the same as the aggregation function computed by the DAF: An aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a "benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure point-to-point channels. In reality, these channels need to be instantiated by some "wrapper protocol", such as , 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. describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.

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 are defined by each concrete VDAF. Constants and types defined by each concrete VDAF.

Parameter	Description
VERIFY_KEY_SIZE	Size (in bytes) of the verification key ()
ROUNDS	Number of rounds of communication during the Preparation stage ()
SHARES	Number of input shares into which each measurement is sharded ()
Measurement	Type of each measurement
AggParam	Type of aggregation parameter
Prep	State of each Aggregator during Preparation ()
OutShare	Type of each output share
AggResult	Type of the aggregate result

Similarly to DAFs (see {[sec-daf}}), any output of a VDAF method that must be transmitted from one party to another is treated as an opaque byte string. All other quantities are given a concrete type.

• OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. See issue#58.

Sharding Sharding is syntactically identical to DAFs (cf. ):

• Vdaf.measurement_to_input_shares(measurement: Measurement) -> (Bytes, Vec[Bytes]) is the randomized input-distribution algorithm run by each Client. It consumes the measurement 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 ). The length of the output vector MUST be SHARES.

Preparation To recover and verify output shares, the Aggregators interact with one another over ROUNDS rounds. Prior to each round, each Aggregator constructs an outbound message. Next, the sequence of outbound messages is combined into a single message, called a "preparation message". (Each of the outbound messages are called "preparation-message shares".) Finally, the preparation message is distributed to the Aggregators to begin the next round. An Aggregator begins the first round with its input share and it begins each subsequent round with the previous preparation message. Its output in the last round is its output share and its output in each of the preceding rounds is a preparation-message share. This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing inputs for aggregation.

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, agg_id: Unsigned, agg_param: AggParam, nonce: Bytes, public_share: Bytes, input_share: Bytes) -> Prep is the deterministic preparation-state initialization algorithm run by each Aggregator to begin processing its input share into an output share. Its inputs are the shared verification key (verify_key), the Aggregator's unique identifier (agg_id), the aggregation parameter (agg_param), the nonce provided by the environment (nonce, see ), and 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. The length of verify_key MUST be VERIFY_KEY_SIZE. It is up to the high level protocol in which the VDAF is used to arrange for the distribution of the verification key among the Aggregators prior to the start of this phase of VDAF evaluation.
  • OPEN ISSUE What security properties do we need for this key exchange? See issue#18.
  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. In addition, 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 for details.)
• Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) -> Union[Tuple[Prep, Bytes], OutShare] is the deterministic preparation-state update algorithm run by each Aggregator. It updates the Aggregator's preparation state (prep) and returns either its next preparation state and its message share for the current round or, if this is the last round, its output share. An exception is raised if a valid output share could not be recovered. The input of this algorithm is the inbound preparation message or, if this is the first round, None.
• Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) -> Bytes is the deterministic preparation-message pre-processing algorithm. It combines the preparation-message shares generated by the Aggregators in the previous round into the preparation message consumed by each in the next round.

In effect, each Aggregator moves through a linear state machine with ROUNDS+1 states. The Aggregator enters the first state on using the initialization algorithm, and the update algorithm advances the Aggregator to the next state. Thus, in addition to defining the number of rounds (ROUNDS), a VDAF instance defines the state of the Aggregator after each round.

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

The preparation-state update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that 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+ based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all. Note that it is possible for a VDAF to specify ROUNDS == 0, in which case each Aggregator runs the preparation-state update algorithm once and immediately recovers its output share without interacting with the other Aggregators. However, most, if not all, constructions will require some amount of interaction in order to ensure validity of the output shares (while also maintaining privacy).

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

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

• 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.

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

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

• 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 .

Execution of a VDAF Secure execution of a VDAF involves simulating the following procedure.

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 for details. As explained in , the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol participants.

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

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.

Derived class methods for finite fields. Bytes: encoded = Bytes() for x in data: encoded += I2OSP(x.as_unsigned(), Field.ENCODED_SIZE) return encoded def decode_vec(Field, encoded: Bytes) -> Vec[Field]: L = Field.ENCODED_SIZE if len(encoded) % L != 0: raise ERR_DECODE vec = [] for i in range(0, len(encoded), L): encoded_x = encoded[i:i+L] x = OS2IP(encoded_x) if x >= Field.MODULUS: raise ERR_DECODE # Integer is larger than modulus vec.append(Field(x)) return vec ]]>

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.

Common functions for finite fields. 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))) ]]>

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

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

FFT-friendly fields also define the following parameter:

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

Parameters The tables below define finite fields used in the remainder of this document. Field64, an FFT-friendly field.

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

Field128, an FFT-friendly field.

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

Field255.

Parameter	Value
MODULUS	2^255 - 19
ENCODED_SIZE	32

• OPEN ISSUE We currently use big-endian for encoding field elements. However, for implementations of GF(2^255-19), little endian is more common. See issue#90.

Pseudorandom Generators A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed into a long string of pseudorandom bits. A PRG suitable for this document implements the interface specified in this section. Concrete constructions are described in the subsections that follow. 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, info: Bytes) constructs an instance of Prg from the given seed and info string. The seed MUST be of length SEED_SIZE and MUST be generated securely (i.e., it is either the output of gen_rand or a previous invocation of the PRG). The info string is used for domain separation.
• 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).

Derived class methods for PRGs. bytes: prg = Prg(seed, info) 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 = OS2IP(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, info: Bytes, length: Unsigned): prg = Prg(seed, info) return prg.next_vec(Field, length) ]]>

PrgAes128

• OPEN ISSUE Phillipp points out that a fixed-key mode of AES may be more performant (https://eprint.iacr.org/2019/074.pdf). See issue#32.

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

Definition of PRG PrgAes128.

Prio3

• NOTE This construction has not undergone significant security analysis.

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

• Each measurement is encoded as a vector over some finite field.
• Input validity is determined by an arithmetic circuit evaluated over the encoded input. (An "arithmetic circuit" is a function comprised of arithmetic operations in the field.) The circuit's output is a single field element: if zero, then the input is said to be "valid"; otherwise, if the output is non-zero, then the input is said to be "invalid".
• The aggregate result is obtained by summing up the encoded input vectors and computing some function of the sum.

At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multi-party computation to determine if their shares correspond 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 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 system, and like the VDAF described here, the ENPA system was built from techniques introduced in 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 , 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 . The generic transformation of an FLP into Prio3 is specified in . Next, a concrete FLP suitable for any validity circuit is specified in . Finally, instantiations of Prio3 for various types of measurements are specified in . Test vectors can be found in .

Fully Linear Proof (FLP) Systems Conceptually, an FLP is a two-party protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid input. (See for details.) As usual, we will describe the interface implemented by a concrete FLP in terms of an abstract base class Flp that specifies the set of methods and parameters a concrete FLP must provide. The parameters provided by a concrete FLP are listed in . Constants and types defined by a concrete FLP.

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 ()
OUTPUT_LEN	Length of the aggregatable output ()
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 ()

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

• Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field]) -> Vec[Field] is the deterministic proof-generation algorithm run by the prover. Its inputs are the encoded input, the "prover randomness" prove_rand, and the "joint randomness" joint_rand. The prover randomness is used only by the prover, but the joint randomness is shared by both the prover and verifier.
• Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field], joint_rand: Vec[Field], num_shares: Unsigned) -> Vec[Field] is the query-generation algorithm run by the verifier. This is used to "query" the input and proof. The result of the query (i.e., the output of this function) is called the "verifier message". In addition to the input and proof, this algorithm takes as input the query randomness query_rand and the joint randomness joint_rand. The former is used only by the verifier. num_shares specifies how many input and proof shares were generated.
• Flp.decide(verifier: Vec[Field]) -> Bool is the deterministic decision algorithm run by the verifier. It takes as input the verifier message and outputs a boolean indicating if the input from which it was generated is valid.

Our application requires that the FLP is "fully linear" in the sense defined in . As a practical matter, what this property implies is that, when run on a share of the input and proof, the query-generation algorithm outputs a share of the verifier message. Furthermore, the "strong zero-knowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the query-generation algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm. The query-generation algorithm includes a parameter num_shares that specifies the number of shares of the input and proof that were generated. If these data are not secret shared, then num_shares == 1. This parameter is useful for certain FLP constructions. For example, the FLP in 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:

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

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 in which an integer in range [0, 2^k) is encoded as a vector of k field elements (this type is also defined in ). After consuming this vector, all that is needed is the integer it represents. Thus the FLP defines an algorithm for truncating the input to the length of the aggregated output:

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

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

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

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

Construction This section specifies Prio3, an implementation of the Vdaf interface (). It has two generic parameters: an Flp () and a Prg (). The associated constants and types required by the Vdaf interface are defined in . The methods required for sharding, preparation, aggregation, and unsharding are described in the remaining subsections. Associated parameters for the Prio3 VDAF.

Parameter	Value
VERIFY_KEY_SIZE	Prg.SEED_SIZE
ROUNDS	1
SHARES	in [2, 255)
Measurement	Flp.Measurement
AggParam	None
Prep	Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]
OutShare	Vec[Flp.Field]
AggResult	Flp.AggResult

Sharding Recall from that the FLP syntax calls for "joint randomness" shared by the prover (i.e., the Client) and the verifier (i.e., the Aggregators). VDAFs have no such notion. Instead, the Client derives the joint randomness from its input in a way that allows the Aggregators to reconstruct it from their input shares. (This idea is based on the Fiat-Shamir heuristic and is described in Section 6.2.3 of .) The input-distribution algorithm involves the following steps:

1. Encode the Client's raw measurement as an input for the FLP
2. Shard the input into a sequence of input shares
3. Derive the joint randomness from the input shares
4. Run the FLP proof-generation algorithm using the derived joint 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 code refers to a pair of auxiliary functions for encoding the shares. These are called encode_leader_share and encode_helper_share respectively and they are described in .

Input-distribution algorithm for Prio3.

Preparation This section describes the process of recovering output shares from the input shares. The high-level idea is that each Aggregator first queries its input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept. In addition, the Aggregators must ensure that they have all used the same joint randomness for the query-generation algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives an XOR secret share of this seed from its input share and the "blind" generated by the client. Thus, before running the query-generation algorithm, it must first gather the XOR secret shares derived by the other Aggregators. In order to avoid extra round of communication, the Client sends each Aggregator a "hint" equal to the XOR of the other Aggregators' shares of the joint randomness seed. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their XOR shares of the PRG seed along with their verifier shares.

• NOTE This optimization somewhat diverges from Section 6.2.3 of . Security analysis is needed.

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

Preparation state for Prio3. 0: encoded = Prio3.Flp.Field.encode_vec(input_share) k_joint_rand_share = Prio3.Prg.derive_seed( k_blind, byte(agg_id) + encoded) k_joint_rand = xor(k_hint, k_joint_rand_share) joint_rand = Prio3.Prg.expand_into_vec( Prio3.Flp.Field, k_joint_rand, dst, Prio3.Flp.JOINT_RAND_LEN ) verifier_share = Prio3.Flp.query(input_share, proof_share, query_rand, joint_rand, Prio3.SHARES) prep_msg = Prio3.encode_prep_share(verifier_share, k_joint_rand_share) return (out_share, k_joint_rand, prep_msg) def prep_next(Prio3, prep, inbound): (out_share, k_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_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) k_joint_rand_check = zeros(Prio3.Prg.SEED_SIZE) for encoded in prep_shares: (verifier_share, k_joint_rand_share) = \ Prio3.decode_prep_share(encoded) verifier = vec_add(verifier, verifier_share) if Prio3.Flp.JOINT_RAND_LEN > 0: k_joint_rand_check = xor(k_joint_rand_check, k_joint_rand_share) if not Prio3.Flp.decide(verifier): raise ERR_VERIFY # proof verifier check failed return Prio3.encode_prep_msg(k_joint_rand_check) ]]>

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

Aggregation algorithm for Prio3.

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

Computation of the aggregate result for Prio3.

Auxiliary Functions

Helper functions required for Prio3. 0: encoded += k_blind encoded += k_hint return encoded def decode_leader_share(Prio3, encoded): l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.INPUT_LEN encoded_input_share, encoded = encoded[:l], encoded[l:] input_share = Prio3.Flp.Field.decode_vec(encoded_input_share) l = Prio3.Flp.Field.ENCODED_SIZE * Prio3.Flp.PROOF_LEN encoded_proof_share, encoded = encoded[:l], encoded[l:] proof_share = Prio3.Flp.Field.decode_vec(encoded_proof_share) l = Prio3.Prg.SEED_SIZE k_blind, k_hint = None, None if Prio3.Flp.JOINT_RAND_LEN > 0: k_blind, encoded = encoded[:l], encoded[l:] k_hint, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (input_share, proof_share, k_blind, k_hint) def encode_helper_share(Prio3, k_input_share, k_proof_share, k_blind, k_hint): encoded = Bytes() encoded += k_input_share encoded += k_proof_share if Prio3.Flp.JOINT_RAND_LEN > 0: encoded += k_blind encoded += k_hint return encoded def decode_helper_share(Prio3, dst, agg_id, encoded): l = Prio3.Prg.SEED_SIZE k_input_share, encoded = encoded[:l], encoded[l:] input_share = Prio3.Prg.expand_into_vec(Prio3.Flp.Field, k_input_share, dst + 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, dst + byte(agg_id), Prio3.Flp.PROOF_LEN) k_blind, k_hint = None, None if Prio3.Flp.JOINT_RAND_LEN > 0: k_blind, encoded = encoded[:l], encoded[l:] k_hint, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (input_share, proof_share, k_blind, k_hint) def encode_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) k_joint_rand = None if Prio3.Flp.JOINT_RAND_LEN > 0: l = Prio3.Prg.SEED_SIZE k_joint_rand, encoded = encoded[:l], encoded[l:] if len(encoded) != 0: raise ERR_DECODE return (verifier, k_joint_rand) 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): k_joint_rand_check = None if Prio3.Flp.JOINT_RAND_LEN > 0: 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 ]]>

A General-Purpose FLP This section describes an FLP based on the construction from in , Section 4.2. We begin in with an overview of their proof system and the extensions to their proof system made here. The construction is specified in .

• OPEN ISSUE We're not yet sure if specifying this general-purpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the general-purpose FLP. In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the general-purpose construction. The reference implementation can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc.

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

Overview In the proof system of , validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is non-zero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (), this computation is distributed among multiple Aggregators, each of which has only a share of the input. Suppose for a moment that the validity circuit C is affine, meaning its only operations are addition and multiplication-by-constant. In particular, suppose the circuit does not contain a multiplication gate whose operands are both non-constant. Then to decide if an input x is valid, each Aggregator could evaluate C on its share of x locally, broadcast the output share to its peers, then combine the output shares locally to recover C(x). This is true because for any SHARES-way secret sharing of x it holds that (Note that, for this equality to hold, it may be necessary to scale any constants in the circuit by SHARES.) However this is not the case if C is not-affine (i.e., it contains at least one multiplication gate whose operands are non-constant). In the proof system of , the proof is designed to allow the (distributed) verifier to compute the non-affine operations using only linear operations on (its share of) the input and proof. To make this work, the proof system is restricted to validity circuits that exhibit a special structure. Specifically, an arithmetic circuit with "G-gates" (see , Definition 5.2) is composed of affine gates and any number of instances of a distinguished gate G, which may be non-affine. We will refer to this class of circuits as 'gadget circuits' and to G as the "gadget". As an illustrative example, consider a validity circuit C that recognizes the set L = set([0], [1]). That is, C takes as input a length-1 vector x and returns 0 if x[0] is in [0,2) and outputs something else otherwise. This circuit can be expressed as the following degree-2 polynomial: 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 , Theorem 4.3, the circuit needs to be rewritten in terms of a gadget that subsumes this non-affine operation. For example, the gadget might be multiplication: The validity circuit can then be rewritten in terms of Mul like so: The proof system of allows the verifier to evaluate each instance of the gadget (i.e., Mul(x[0], x[0]) in our example) using a linear function of the input and proof. The proof is constructed roughly as follows. Let C be the validity circuit and suppose the gadget is arity-L (i.e., it has L input wires.). Let wire[j-1,k-1] denote the value of the jth wire of the kth call to the gadget during the evaluation of C(x). Suppose there are M such calls and fix distinct field elements alpha[0], ..., alpha[M-1]. (We will require these points to have a special property, as we'll discuss in ; but for the moment it is only important that they are distinct.) The prover constructs from wire and alpha a polynomial that, when evaluated at alpha[k-1], produces the output of the kth call to the gadget. Let us call this the "gadget polynomial". Polynomial evaluation is linear, which means that, in the distributed setting, the Client can disseminate additive shares of the gadget polynomial that the Aggregators then use to compute additive shares of each gadget output, allowing each Aggregator to compute its share of C(x) locally. There is one more wrinkle, however: It is still possible for a malicious prover to produce a gadget polynomial that would result in C(x) being computed incorrectly, potentially resulting in an invalid 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 .

Extensions The FLP described in the next section extends the proof system , 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 ). This allows for circuit optimizations that trade a small soundness error for a shorter proof. For example, consider a circuit that recognizes the set of length-N vectors for which each element is either one or zero. A deterministic circuit could be constructed for this language, but it would involve a large number of multiplications that would result in a large proof. (See the discussion in , Section 5.2 for details). A much shorter proof can be constructed for the following randomized circuit: (Note that this is a special case of , Theorem 5.2.) Here inp is the length-N input and r is a random field element. The gadget circuit Range2 is the "range-check" polynomial described above, i.e., Range2(x) = x^2 - x. The idea is that, if inp is valid (i.e., each inp[j] is in [0,2)), then the circuit will evaluate to 0 regardless of the value of r; but if inp[j] is not in [0,2) for some j, the output will be non-zero with high probability. The second extension implemented by our FLP allows the validity circuit to contain multiple gadget types. (This generalization was suggested in , 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): Finally, , Theorem 4.3 makes no restrictions on the choice of the fixed points alpha[0], ..., alpha[M-1], other than to require that the points are distinct. In this document, the fixed points are chosen so that the gadget polynomial can be constructed efficiently using the Cooley-Tukey FFT ("Fast Fourier Transform") algorithm. Note that this requires the field to be "FFT-friendly" as defined in .

Validity Circuits The FLP described in is defined in terms of a validity circuit Valid that implements the interface described here. A concrete Valid defines the following parameters: Validity circuit parameters.

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

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 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:

Derived methods for validity circuits.

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

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

The FLP parameters for FlpGeneric are defined in . 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)

FLP Parameters of FlpGeneric.

Parameter	Value
PROVE_RAND_LEN	Valid.prove_rand_len() (see )
QUERY_RAND_LEN	Valid.query_rand_len() (see )
JOINT_RAND_LEN	Valid.JOINT_RAND_LEN
INPUT_LEN	Valid.INPUT_LEN
OUTPUT_LEN	Valid.OUTPUT_LEN
PROOF_LEN	Valid.proof_len() (see )
VERIFIER_LEN	Valid.verifier_len() (see )
Measurement	Valid.Measurement
Field	Valid.Field

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[j-1,k-1] to the value on the jth wire into the kth call to gadget G_i.
4. Compute the "wire polynomials". That is, for every i in [H] and j in [L_i], construct poly_wire_i[j-1], the jth wire polynomial for the ith gadget, as follows:
   • Let w = [seed_i[j-1], wire_i[j-1,0], ..., wire_i[j-1,M_i-1]].
   • Let padded_w = w + Field.zeros(P_i - len(w)).
   • NOTE We pad w to the nearest power of 2 so that we can use FFT for interpolating the wire polynomials. Perhaps there is some clever math for picking wire_inp in a way that avoids having to pad.
   • Let poly_wire_i[j-1] be the lowest degree polynomial for which poly_wire_i[j-1](alpha_i^k) == padded_w[k] for all k in [P_i].
5. Compute the "gadget polynomials". That is, for every i in [H]:
   • Let poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-1]). That is, evaluate the circuit G_i on the wire polynomials for the ith gadget. (Arithmetic is in the ring of polynomials over Field.)

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

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 .
3. Evaluate Valid on input of inp and joint_rand, recording the inputs of each gadget in the corresponding table. This step is similar to the prover's step (3.) except the verifier does not evaluate the gadgets. Instead, it computes the output of the kth call to G_i by evaluating poly_gadget_i(alpha_i^k). Let v denote the output of the circuit evaluation.
4. Compute the wire polynomials just as in the prover's step (4.).
5. Compute the tests for well-formedness of the gadget polynomials. That is, for every i in [H]:
   • Let t = query_rand[i]. Check if t^(P_i) == 1: If so, then raise ERR_ABORT and halt. (This prevents the verifier from inadvertently leaking a gadget output in the verifier message.)
   • Let y_i = poly_gadget_i(t).
   • For each j in [0,L_i) let x_i[j-1] = poly_wire_i[j-1](t).

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

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

1. Parse verifier into v, x_1, y_1, ..., x_H, y_H as defined in .
2. Check for well-formedness of the gadget polynomials. For every i in [H]:
   • Let z = G_i(x_i). That is, evaluate the circuit G_i on x_i and set z to the output.
   • If z != y_i, then return False and halt.
3. Return True if v == 0 and False otherwise.

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

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

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

Prio3Aes128Count Our first instance of Prio3 is for a simple counter: Each measurement is either one or zero and the aggregate result is the sum of the measurements. This instance uses PrgAes128 () as its PRG. Its validity circuit, denoted Count, uses Field64 () as its finite field. Its gadget, denoted Mul, is the degree-2, arity-2 gadget defined as The validity circuit is defined as The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below. Parameters of validity circuit Count.

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 ()

Prio3Aes128Sum The next instance of Prio3 supports summing of integers in a pre-determined range. Each measurement is an integer in range [0, 2^bits), where bits is an associated parameter. This instance of Prio3 uses PrgAes128 () as its PRG. Its validity circuit, denoted Sum, uses Field128 () as its finite field. The measurement is encoded as a length-bits vector of field elements, where the lth element of the vector represents the lth bit of the summand: measurement or measurement >= 2^Sum.INPUT_LEN: raise ERR_INPUT encoded = [] for l in range(Sum.INPUT_LEN): encoded.append(Sum.Field((measurement >> l) & 1)) return encoded def truncate(Sum, inp): decoded = Sum.Field(0) for (l, b) in enumerate(inp): w = Sum.Field(1 << l) decoded += w * b return [decoded] def decode(Sum, output, _num_measurements): return output[0].as_unsigned() ]]> The validity circuit checks that the input consists of ones and zeros. Its gadget, denoted Range2, is the degree-2, arity-1 gadget defined as The validity circuit is defined as Parameters of validity circuit Sum.

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 ()

Prio3Aes128Histogram This instance of Prio3 allows for estimating the distribution of the measurements by computing a simple histogram. Each measurement is an arbitrary integer and the aggregate result counts the number of measurements that fall in a set of fixed buckets. This instance of Prio3 uses PrgAes128 () as its PRG. Its validity circuit, denoted Histogram, uses Field128 () as its finite field. The measurement is encoded as a one-hot vector representing the bucket into which the measurement falls (let bucket denote a sequence of monotonically increasing integers): The validity circuit uses Range2 (see ) as its single gadget. It checks for one-hotness in two steps, as follows: Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3. Parameters of validity circuit Histogram.

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 ()

Poplar1

• NOTE This construction has not undergone significant security analysis.

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

1. Each Client splits its input string into input shares and sends one share to each Aggregator.
2. The Aggregators agree on an initial set of candidate prefixes, say 0 and 1.
3. The Aggregators evaluate the VDAF on each set of input shares and aggregate the recovered output shares. The aggregation parameter is the set of candidate prefixes.
4. The Aggregators send their aggregate shares to the Collector, who combines them to recover the counts of each candidate prefix.
5. Let H denote the set of prefixes that occurred at least t times. If the prefixes all have length BITS, then H is the set of t-heavy-hitters. Otherwise compute the next set of candidate prefixes as follows. For each p in H, add add p || 0 and p || 1 to the set. Repeat step 3 with the new set of candidate prefixes.

Poplar1 is constructed from an "Incremental Distributed Point Function (IDPF)", a primitive described by that generalizes the notion of a Distributed Point Function (DPF) . 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 function evaluates to to a non-zero value; otherwise it evaluates to zero. This structure allows an IDPF to provide the functionality required for the above protocol, while at the same time ensuring the same degree of privacy as a DPF. Poplar1 composes an IDPF with the "secure sketching" protocol of . This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "one-hot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one. The remainder of this section is structured as follows. IDPFs are defined in ; a concrete instantiation is given . The Poplar1 VDAF is defined in in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in ; test vectors can be found in .

Incremental Distributed Point Functions (IDPFs) An IDPF is defined over a domain of size 2^BITS, where BITS is constant defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range [0, 2^BITS). The Client specifies an index alpha and a vector of values beta, one for each "level" L in range [0, BITS). The key generation algorithm generates one IDPF "key" for each Aggregator. When evaluated at level L and index 0 <= prefix < 2^L, each IDPF key returns an additive share of beta[L] if prefix is the L-bit prefix of alpha and shares of zero otherwise. An index x is defined to be a prefix of another index y as follows. Let LSB(x, N) denote the least significant N bits of positive integer x. By definition, a positive integer 0 <= x < 2^L is said to be the length-L prefix of positive integer 0 <= y < 2^BITS if LSB(x, L) is equal to the most significant L bits of LSB(y, BITS), For example, 6 (110 in binary) is the length-3 prefix of 25 (11001), but 7 (111) is not. Each of the programmed points beta is a vector of elements of some finite field. We distinguish two types of fields: One for inner nodes (denoted Idpf.FieldInner), and one for leaf nodes (Idpf.FieldLeaf). (Our instantiation of Poplar1 () will use a much larger field for leaf nodes than for inner nodes. This is to ensure the IDPF is "extractable" as defined in , Definition 1.) A concrete IDPF defines the types and constants enumerated in . 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]) -> (Bytes, Vec[Bytes]) is the randomized IDPF-key generation algorithm. Its inputs are the index alpha and the values beta. 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]) -> Idpf.Vec is the deterministic, stateless IDPF-key evaluation algorithm run by each Aggregator. Its inputs are the Aggregator's unique identifier, the public share distributed to all of the Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the IDPF, and the sequence of candidate prefixes. It returns the share of the value corresponding to each candidate prefix. The output type depends on the value of level: If level < Idpf.BITS-1, the output is the value for an inner node, which has type Vec[Vec[Idpf.FieldInner]]; otherwise, if level == Idpf.BITS-1, then the output is the value for a leaf node, which has type Vec[Vec[Idpf.FieldLeaf]]. The value of level MUST be in range [0, BITS). The indexes in prefixes MUST all be distinct and in range [0, 2^level). Applications MUST ensure that the Aggregator's identifier is equal to the integer in range [0, SHARES) that matches the index of key in the sequence of IDPF keys output by the Client.

In addition, the following method is derived for each concrete Idpf: 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 for details. Constants and types defined by a concrete IDPF.

Parameter	Description
SHARES	Number of IDPF keys output by IDPF-key generator
BITS	Length in bits of each input string
VALUE_LEN	Number of field elements of each output value
KEY_SIZE	Size in bytes of each IDPF key
FieldInner	Implementation of Field () used for values of inner nodes
FieldLeaf	Implementation of Field used for values of leaf nodes
Prg	Implementation of Prg ()

Construction This section specifies Poplar1, an implementation of the Vdaf interface (). It is defined in terms of any Idpf () for which Idpf.SHARES == 2 and Idpf.VALUE_LEN == 2. The associated constants and types required by the Vdaf interface are defined in . The methods required for sharding, preparation, aggregation, and unsharding are described in the remaining subsections. Associated parameters for the Poplar1 VDAF.

Parameter	Value
VERIFY_KEY_SIZE	Idpf.Prg.SEED_SIZE
ROUNDS	2
SHARES	2
Measurement	Unsigned
AggParam	Tuple[Unsigned, Vec[Unsigned]]
Prep	Tuple[Bytes, Unsigned, Idpf.Vec]
OutShare	Idpf.Vec
AggResult	Vec[Unsigned]

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 , Appendix C.4. After evaluating their IDPF key shares on a given sequence of candidate prefixes, the sketching protocol is used by the Aggregators to verify that they hold shares of a one-hot vector. In addition, for each level of the tree, the prover generates random elements a, b, and c and computes and sends additive shares of a, b, c, A and B to the Aggregators. Putting everything together, the input-distribution algorithm is defined as follows. Function encode_input_shares is defined in .

The input-distribution algorithm for Poplar1.

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

Preparation state for Poplar1.

Aggregation Aggregation involves simply adding up the output shares.

Aggregation algorithm for Poplar1.

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

Computation of the aggregate result for Poplar1.

Auxiliary Functions

Helper functions for Poplar1. 2^16 - 1: raise ERR_INPUT # level too deep if len(prefixes) > 2^16 - 1: raise ERR_INPUT # too many prefixes encoded = Bytes() encoded += I2OSP(level, 2) encoded += I2OSP(len(prefixes), 2) packed = 0 for (i, prefix) in enumerate(prefixes): packed |= prefix << ((level+1) * i) l = floor(((level+1) * len(prefixes) + 7) / 8) encoded += I2OSP(packed, l) return encoded def verify_context(Poplar1, nonce, level, prefixes): if len(nonce) > 255: raise ERR_INPUT # nonce too long context = Bytes() context += byte(254) context += byte(len(nonce)) context += nonce context += Poplar1.encode_agg_param(level, prefixes) return context ]]>

The IDPF scheme of 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 .

• TODO We should consider giving IdpfPoplar a more distinctive name.

The constant and type definitions required by the Idpf interface are given in . Constants and type definitions for IdpfPoplar.

Parameter	Value
SHARES	2
BITS	any positive integer
VALUE_LEN	any positive integer
KEY_SIZE	Prg.SEED_SIZE
FieldInner	Field64 ()
FieldLeaf	Field255 ()
Prg	any implementation of Prg ()

Key Generation

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

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

IDPF-key generation algorithm of IdpfPoplar. = 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 init_seed = [ gen_rand(IdpfPoplar.Prg.SEED_SIZE), gen_rand(IdpfPoplar.Prg.SEED_SIZE), ] seed = init_seed.copy() ctrl = [Field2(0), Field2(1)] correction_words = [] for level in range(IdpfPoplar.BITS): keep = (alpha >> (IdpfPoplar.BITS - level - 1)) & 1 lose = 1 - keep bit = Field2(keep) (s0, t0) = IdpfPoplar.extend(seed[0]) (s1, t1) = IdpfPoplar.extend(seed[1]) 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], seed_cw) if ctrl[0] == Field2(1) \ else s0[keep] x1 = xor(s1[keep], seed_cw) if ctrl[1] == Field2(1) \ else s1[keep] (seed[0], w0) = IdpfPoplar.convert(level, x0) (seed[1], w1) = IdpfPoplar.convert(level, x1) ctrl[0] = t0[keep] + ctrl[0] * ctrl_cw[keep] ctrl[1] = t1[keep] + ctrl[1] * ctrl_cw[keep] b = beta_inner[level] if level < IdpfPoplar.BITS-1 \ else beta_leaf if len(b) != IdpfPoplar.VALUE_LEN: raise ERR_INPUT # beta too long or too short w_cw = vec_add(vec_sub(b, w0), w1) if ctrl[1] == Field2(1): w_cw = vec_neg(w_cw) correction_words.append((seed_cw, ctrl_cw, w_cw)) public_share = IdpfPoplar.encode_public_share(correction_words) return (public_share, init_seed) ]]>

Key Evaluation

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

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

IDPF-evaluation generation algorithm of IdpfPoplar. = IdpfPoplar.SHARES: raise ERR_INPUT # invalid aggregator ID if level >= IdpfPoplar.BITS: raise ERR_INPUT # level too deep if len(set(prefixes)) != len(prefixes): raise ERR_INPUT # candidate prefixes are non-unique correction_words = IdpfPoplar.decode_public_share(public_share) out_share = [] for prefix in prefixes: if prefix >= 2^(level+1): raise ERR_INPUT # prefix too long # The Aggregator's output share is the value of a node of # the IDPF tree at the given `level`. The node's value is # computed by traversing the path defined by the candidate # `prefix`. Each node in the tree is represented by a seed # (`seed`) and a set of control bits (`ctrl`). seed = init_seed ctrl = Field2(agg_id) for current_level in range(level+1): bit = (prefix >> (level - current_level)) & 1 # Implementation note: Typically the current round of # candidate prefixes would have been derived from # aggregate results computed during previous rounds. For # example, when using `IdpfPoplar` to compute heavy # hitters, a string whose hit count exceeded the given # threshold in the last round would be the prefix of each # `prefix` in the current round. (See [BBCGGI21, # Section 5.1].) In this case, part of the path would # have already been traversed. # # Re-computing nodes along previously traversed paths is # wasteful. Implementations can eliminate this added # complexity by caching nodes (i.e., `(seed, ctrl)` # pairs) output by previous calls to `eval_next()`. (seed, ctrl, y) = IdpfPoplar.eval_next(seed, ctrl, correction_words[current_level], current_level, bit) out_share.append(y if agg_id == 0 else vec_neg(y)) return out_share # 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): (seed_cw, ctrl_cw, w_cw) = correction_word (s, t) = IdpfPoplar.extend(prev_seed) if prev_ctrl == Field2(1): s[0] = xor(s[0], seed_cw) s[1] = xor(s[1], seed_cw) t[0] = t[0] + ctrl_cw[0] t[1] = t[1] + ctrl_cw[1] next_ctrl = t[bit] (next_seed, y) = IdpfPoplar.convert(level, s[bit]) if next_ctrl == Field2(1): y = vec_add(y, w_cw) return (next_seed, next_ctrl, y) ]]>

Auxiliary Functions

Helper functions for IdpfPoplar. > 1) & 1)] return (s, t) def convert(IdpfPoplar, level, seed): dst = VERSION + b' idpf poplar convert' prg = IdpfPoplar.Prg(seed, dst) next_seed = prg.next(IdpfPoplar.Prg.SEED_SIZE) Field = IdpfPoplar.current_field(level) w = prg.next_vec(Field, IdpfPoplar.VALUE_LEN) return (next_seed, w) def encode_public_share(IdpfPoplar, correction_words): encoded = Bytes() packed_ctrl = 0 for (level, (_, ctrl_cw, _)) \ in enumerate(correction_words): packed_ctrl |= ctrl_cw[0].as_unsigned() << (2*level) packed_ctrl |= ctrl_cw[1].as_unsigned() << (2*level+1) l = floor((2*IdpfPoplar.BITS + 7) / 8) encoded += I2OSP(packed_ctrl, l) 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:] packed_ctrl = OS2IP(encoded_ctrl) correction_words = [] for level in range(IdpfPoplar.BITS): Field = IdpfPoplar.current_field(level) ctrl_cw = (Field2(packed_ctrl & 1), Field2((packed_ctrl >> 1) & 1)) packed_ctrl >>= 2 l = IdpfPoplar.Prg.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 ]]>

Poplar1Aes128 We refer to Poplar1 instantiated with IdpfPoplar (VALUE_LEN == 2) and PrgAes128 () as Poplar1Aes128. This VDAF is suitable for any positive value of BITS. Test vectors can be found in .

Security Considerations

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

VDAFs have two essential security goals:

1. Privacy: An attacker that controls the network, the Collector, and a subset of Clients and Aggregators learns nothing about the measurements of honest Clients beyond what it can deduce from the aggregate result.
2. Robustness: An attacker that controls the network and a subset of Clients cannot cause the Collector to compute anything other than the aggregate of the measurements of honest Clients.

Note that, to achieve robustness, it is important to ensure that the verification key distributed to the Aggregators (verify_key, see ) is never revealed to the Clients. It is also possible to consider a stronger form of robustness, where the attacker also controls a subset of Aggregators (see , Section 6.3). To satisfy this stronger notion of robustness, it is necessary to prevent the attacker from sharing the verification key with the Clients. It is therefore RECOMMENDED that the Aggregators generate verify_key only after a set of Client inputs has been collected for verification, and re-generate them for each such set of inputs. In order to achieve robustness, the Aggregators MUST ensure that the nonces used to process the measurements in a batch are all unique. A VDAF is the core cryptographic primitive of a protocol that achieves the above privacy and robustness goals. It is not sufficient on its own, however. The application will need to assure a few security properties, for example:

• Securely distributing the long-lived parameters.
• Establishing secure channels:
  • Confidential and authentic channels among Aggregators, and between the Aggregators and the Collector; and
  • Confidential and Aggregator-authenticated channels between Clients and Aggregators.
• Enforcing the non-collusion properties required of the specific VDAF in use.

In such an environment, a VDAF provides the high-level privacy property described above: The Collector learns only the aggregate measurement, and nothing about individual measurements aside from what can be inferred from the aggregate result. The Aggregators learn neither individual measurements nor the aggregate result. The Collector is assured that the aggregate statistic accurately reflects the inputs as long as the Aggregators correctly executed their role in the VDAF. On their own, VDAFs do not mitigate Sybil attacks . In this attack, the adversary observes a subset of input shares transmitted by a Client it is interested in. It allows the input shares to be processed, but corrupts and picks bogus inputs for the remaining Clients. Applications can guard against these risks by adding additional controls on measurement submission, such as client authentication and rate limits. VDAFs do not inherently provide differential privacy . The VDAF approach to private measurement can be viewed as complementary to differential privacy, relying on non-collusion instead of statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and removing it after unsharding.

IANA Considerations This document makes no request of IANA.

References Normative References In many standards track documents several words are used to signify the requirements in the specification. These words are often capitalized. This document defines these words as they should be interpreted in IETF documents. This document specifies an Internet Best Current Practices for the Internet Community, and requests discussion and suggestions for improvements. RFC 2119 specifies common key words that may be used in protocol specifications. This document aims to reduce the ambiguity by clarifying that only UPPERCASE usage of the key words have the defined special meanings. This document provides recommendations for the implementation of public-key cryptography based on the RSA algorithm, covering cryptographic primitives, encryption schemes, signature schemes with appendix, and ASN.1 syntax for representing keys and for identifying the schemes. This document represents a republication of PKCS #1 v2.2 from RSA Laboratories' Public-Key Cryptography Standards (PKCS) series. By publishing this RFC, change control is transferred to the IETF. This document also obsoletes RFC 3447. The National Institute of Standards and Technology (NIST) has recently specified the Cipher-based Message Authentication Code (CMAC), which is equivalent to the One-Key CBC MAC1 (OMAC1) submitted by Iwata and Kurosawa. This memo specifies an authentication algorithm based on CMAC with the 128-bit Advanced Encryption Standard (AES). This new authentication algorithm is named AES-CMAC. The purpose of this document is to make the AES-CMAC algorithm conveniently available to the Internet Community. This memo provides information for the Internet community. Informative References ISRG Cloudflare Mozilla Cloudflare There are many situations in which it is desirable to take measurements of data which people consider sensitive. In these cases, the entity taking the measurement is usually not interested in people's individual responses but rather in aggregated data. Conventional methods require collecting individual responses and then aggregating them, thus representing a threat to user privacy and rendering many such measurements difficult and impractical. This document describes a multi-party distributed aggregation protocol (DAP) for privacy preserving measurement (PPM) which can be used to collect aggregate data without revealing any individual user's data.

Acknowledgments Thanks to David Cook, Henry Corrigan-Gibbs, Armando Faz-Hernandez, Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, and Christopher Wood for useful feedback on and contributions to the spec.

Test Vectors

• NOTE Machine-readable test vectors can be found at https://github.com/cfrg/draft-irtf-cfrg-vdaf/tree/main/poc/test_vec.

Test vectors cover the generation of input shares and the conversion of input shares into output shares. Vectors specify the verification key, measurements, aggregation parameter, and any parameters needed to construct the VDAF. (For example, for Prio3AesSum, the user specifies the number of bits for representing each summand.) Byte strings are encoded in hexadecimal To make the tests deterministic, gen_rand() was replaced with a function that returns the requested number of 0x01 octets.

Prio3Aes128Count - input_share_0: >- a46091d07a930ace79bb11d1d03596bf4f921f7e70b853cf6b8d4969287dfb504a1d ea6b0eb01a5f50ad24225186c104 input_share_1: >- 0101010101010101010101010101010101010101010101010101010101010101 round_0: prep_share_0: >- cdbbf14ef0c19728578ce1e50411e1e26eb49eea4dadfada9ed3d13c71d846b8 prep_share_1: >- 32440eb00f3e68d95b9f6e2e856cf534948881bf63c24d315e4954159611ebb5 prep_message: >- out_share_0: - 11844627344580086478 out_share_1: - 6602116724834497844 agg_share_0: >- a46091d07a930ace agg_share_1: >- 5b9f6e2e856cf534 agg_result: 1 ]]>

Prio3Aes128Sum - input_share_0: >- a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa222 0ff24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6eb940251d 14855b16689ab3cfd9100b568729aadfb5558a3bd0277d3ae7c5d967e3525839f26f 5cb210e4fb0b0aeb0394ce0791de9c4c07f8840dcdd1ce09bb2179bb11d1d03596be 4f921f7f70b853ce99c2bd80daf08029b7bb7820b7a7632e3b9200ff98adde77a833 6469ba7fa61ebf135e7f303568ddf63e39978954c40a3fcfa08412f5f03f61699643 85f167c68c3ae5205e1c3dd075fc76224bf9407ef3fd54334bec36d453460381ee54 4027bb520f55394873f5dff846794007cc7d77e903b1474095c61140d1e488a98e00 f433c22491825191261e982affc44e20756db024e0dcdda9746aba73d2ec9a083953 4a1d0178802fc0ee9393160b34dca94999c91fa788b8308b15880ea3146aabf38e2f a4d715c0be7e183fe7f158f47438eaa5a55406da5be56d724afd86e579bc076c669c 334c93fb724891e0b72c8737a9f65c2d3d56bae20dce052f43293712a603f8a8b9e8 bcdafad8420ae14201f9a8b4891e9b60dd418d0f71ee2395b9026cb4978f901765e0 155b127a4fa95a0e0751b95d9ada245c78df820e4bc7f0c9329a5bbad2ee85688cea 47336da7a798705b6c465678363574e844d1974f6f369e9ef13e920d383f580aa382 8b7641a77f352b73134886c58e822040747311a9b094137c5f46d6e9ed05bb17d9c7 0f01b5491805ac83c447035560b00b8e7049c334fcfa003941fa0e307cf52884e461 8f794848936bfcd8d7f895781d27b006673a6349bdfa04c8c3506b2b64efff5335f7 6723bce0f1092571cae3d125dcabbb6da4867963ffd29172ef699ecf010101010101 0101010101010101010132dd9fa5604fadc81ff1636918a6db69 input_share_1: >- 01010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101646430cb3212da0c23bff5618647609c round_0: prep_share_0: >- ede97ab0d3d2dc256a1f82261078cc06b5a423c5f221e9fa029ba023bc71e70fd1 9e2db541507b4171409bc06c3ac292646430cb3212da0c23bff5618647609c prep_share_1: >- 1216854f2c2d23be95e07dd9ef8733fb779dce023a0e4f7147b74c75c5fd1723a1 1b7a116c0a89c6b97471597eb2a83c32dd9fa5604fadc81ff1636918a6db69 prep_message: >- 56b9af6e525d77c43c4e96089ee1bbf5 out_share_0: - 231691668211413349799981210015670190003 out_share_1: - 108590698709525113146884563352230576306 agg_share_0: >- ae4e3131e0f5bea01ba67704684fbfb3 agg_share_1: >- 51b1cece1f0a4143e45988fb97b040b2 agg_result: 100 ]]>

Prio3Aes128Histogram - input_share_0: >- a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa222 0ff24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6e79bb11d1 d03596be4f921f7f70b853cec138b5a4f1a798810bde46a7265fd6bed9bb8cc8d058 bb224cc75b17f2ebc7f7b6c7a708b11022329fab97ed7d23309116a6fb53d1d9ff45 9a67eafcddbd190e69069ef82f361a6d4c9616ec3396776834a7d3d5848409cdd89f ec7920a858074db9c2e7adfc4b3397339532f80e327f25f31975e3f5d149ba8c8d5b f101488f4882ab2b38ddedc84f814435c2b227ffeee7affc5076d78819d7ce68f247 355af67c52fc7ef9bbcbf8bc910af7c047a28dfeb4f92158934ce93dac747c1b55a8 f5efb18e944e0579602c918e670f8d543b530d8ba3730d0a43e8bcd2d54f736c4db1 ac64c9c49dc5ab20c0ac0cf6530b01010101010101010101010101010101d451ca17 ca18344ac4925510c71e5ac8 input_share_1: >- 01010101010101010101010101010101010101010101010101010101010101010101 0101010101010101010101010101ef377f05b86ffa85ed011213230bb1d5 round_0: prep_share_0: >- a1a7093768a9f8a3c01e31f7e9c6dc7925407d03831c510799fcf30a03c61bcd35 78329d502e06cd01e994c227d3d1afef377f05b86ffa85ed011213230bb1d5 prep_share_1: >- 5e58f6c8975607403fe1ce081639238809c0227ef8c371e897e7950d763d8a37f7 ee6acb826a97f3ef37fd59e6a95511d451ca17ca18344ac4925510c71e5ac8 prep_message: >- 3b66b5127277cecf29934703e415eb1d out_share_0: - 218494809353158975333103031759951894435 - 277877721980181983205633439801202130410 - 215511796844427285058159435311010143348 - 151150421348124479376036084370309246062 out_share_1: - 121787557567779487613762741607948871774 - 62404644940756479741232333566698635799 - 124770570076511177888706338056890622862 - 189131945572813983570829688997591520147 agg_share_0: >- a46091d17a930aaf4d02cff822f817a3d10d4d74d526daf00112da46b9b33deaa2220f f24fe104b34007f47c4531dc7471b6832358f5d6859fb3b2112e79ec6e agg_share_1: >- 5b9f6e2e856cf534b2fd3007dd07e85e2ef2b28b2ad924f3feed25b9464cc2175dddf0 0db01efb30bff80b83bace238e8e497cdca70a295e604c4deed1861393 agg_result: [0, 0, 1, 0] ]]>

Poplar1Aes128

Sharding - 9a00000000000000000000000000000000ffffffff00000000bd5fef66a8181fb000 000000000000000000000000000000ffffffff00000000dbf324f2b4351711000000 00000000000000000000000000ffffffff00000000984542b43f4008f40000000000 00000000000000000000000000000000000000000000000000000000000000000000 00000000000000000113aec130616497e38eaafb95841bad26f773979c71d5422d1c 875c6c007575ce input_share_0: >- 01010101010101010101010101010101010101010101010101010101010101011926 0bd7b833c3b313d7ba6d15a9f16fec202c82ddf2848909acc634a461a6b8b7758b10 e0adb885083645e2bbe0c575137f41a015e6cc2ce8fed760ca08589f957eef3d1667 07d9044d22411142e73568bba4e2f7150dab3316e30966a1d73f660ec47316af0198 e8682372a232bcef input_share_1: >- 01010101010101010101010101010101010101010101010101010101010101013422 59973b9807dbbb662f6a81067220a977e09ad54ba5048edb2118c95bb8f3ce11a0dd 6f0bda66dc2dc274f26ec69e091747a0b26b57ad629d905fae98204c5171179db567 12e5dd66f792eddc6da955476f1471eb5c12aac55b35c1d80a2022d30da1ac00dfc5 8c581e8304ab59f1 ]]>

Preparation, Aggregation, and Unsharding - 48c3581799a109398b1943c10957b8c2536f292900de6436 prep_share_1: >- 8377f3fcd4d2d1f33233948d19fe2c385f97c52d8e007f76 prep_message: >- cc3b4c146e73db2cbd4cd84e2355e4fab306ee568edee3ac round_1: prep_share_0: >- 64ce1f88d77b2a65 prep_share_1: >- 9b31e0762884d59c prep_message: >- out_share_0: - 18352387916526274078 - 14559030685141423577 out_share_1: - 94356152888310243 - 3887713384273160745 agg_share_0: >- feb0c79330b8461eca0c114565ddd5d9 agg_share_1: >- 014f386bcf47b9e335f3eeb99a222a29 agg_result: [0, 1] ]]> - 37c72ca578a032eaeffbd9e63d29e6eaa205d9d67628d055 prep_share_1: >- ab956611d15cfcef0773a4e435fdcb83909785aa50c17513 prep_message: >- e35c92b749fd2fd9f76f7eca7327b26d329d5f81c6ea4567 round_1: prep_share_0: >- ee63dfb515814411 prep_share_1: >- 119c2049ea7ebbf0 prep_message: >- out_share_0: - 7623875273889432259 - 4753532626118505607 - 9999595461072379379 - 1320535832873937471 out_share_1: - 10822868795525152062 - 13693211443296078714 - 8447148608342204942 - 17126208236540646851 agg_share_0: >- 69cd722b281042c341f7f0bcf805f4878ac5b3177ad3ddf312537c57efa02a3f agg_share_1: >- 96328dd3d7efbd3ebe080f4207fa0b7a753a4ce7852c220eedac83a7105fd5c3 agg_result: [0, 0, 0, 1] ]]> - 7eb8dbda5ea6807e53f22d3f74a43f59ed83ef578b070654 prep_share_1: >- 567632dc63ec2294c6f030077726f87ec11e2fc53a38f640 prep_message: >- d52f0eb6c292a3121ae25d47ebcb37d6aea21f1dc53ffc93 round_1: prep_share_0: >- 9e92c793980229fd prep_share_1: >- 616d386b67fdd604 prep_message: >- out_share_0: - 1646250657834468215 - 16923505979406688621 - 18330814903122515861 - 12155602082232726549 out_share_1: - 16800493411580116106 - 1523238090007895700 - 115929166292068460 - 6291141987181857773 agg_share_0: >- 16d8a840477a0377eadc5f0610bc216dfe642308980dd795a8b15fb0ce7af415 agg_share_1: >- e92757beb885fc8a1523a0f8ef43de94019bdcf667f2286c574ea04e31850bed agg_result: [0, 0, 0, 1] ]]> - 246777985de3e9fd0ea9e8ea429ae3c7255c0b2aa84cff6bd5c0079a122be38b53 4e75cbfe8abb2c54b5b277089bb2708b274a17ce26df0ea9881487ac852eee30c1 8ba6dc40beb7f33e5003fdf065eec844e320d0b9ff49b0b29f5bbbcbcc8d prep_share_1: >- 3dea0b9e72389ea0d62521e91214c654ebf9b2a3d855a32219755216f4cb8eca0d 864838efc99f1a5d2adecf87e4ff0b839fb7191d4c58c5fd6753c0c6a2fb155071 efff9719f0e9645d70b398ca24ecb2fa8fbb56a6a90ce6d95b675e1e28f2 prep_message: >- 62518336d01c889de4cf0ad354afaa1c1155bdce80a2a28def3559b106f7725560 d4be04ee545a46b1e091469080b17c0ec70130eb7337d4a6ef684873282a030133 7ba6735aafa1579bc0b796ba8adb7b3f72dc2760a856978bfac319e9f592 round_1: prep_share_0: >- 40521213baba54c83b4ddc0cb1e3e4c16897bb8831cc12ea91a82927dd30ae58 prep_share_1: >- 3fadedec4545ab37c4b223f34e1c1b3e97684477ce33ed156e57d6d822cf5195 prep_message: >- out_share_0: - 27022242238524926041754024638978288940369802983157338113702795433392718442891 - 8433446103891669428510589487794410319530091467635619512122120272876905203470 - 31927171851194534364500488903327032909992419490289770138187660233081601268509 - 13674389066590417186009193737398514883303633790292716630453797389670854762435 - 49245655988555337792760994399532100867010365510706353464574012401106060986059 - 38418164104898327452350816690091778787588836972103923142413897241001677348897 - 17473652484574293745985786122460988541722879368489070298827855759184581179349 out_share_1: - 30873802380133171670031467865365664986265189349662943906025996570563846377058 - 49462598514766428283274903016549543607104900865184662507606671731079659616479 - 25968872767463563347285003601016921016642572842530511881541131770874963551440 - 44221655552067680525776298766945439043331358542527565389274994614285710057514 - 8650388630102759919024498104811853059624626822113928555154779602850503833890 - 19477880513759770259434675814252175139046155360716358877314894762954887471053 - 40422392134083803965799706381882965384912112964331211720900936244771983640600 agg_share_0: >- 3bbe0c0f2a3ecf406419e5a8636f13c1102df2df86b4dec82daa4669b24f718b12a529 4d99b7a604a6944dbe1b81cfcc98e4a5a1fa0d08ac727f620ee0b4870e4696238e0b88 71bb6d93ab6c8fc1680da244dfc59bd0b6f70b8dae1f4c1a1b1d1e3b6e1053afb851cc ae5d9ddabb8bff59afea5f6f905e0b2dd1b8a9cc911fc36ce00db4dfb57d404c6bddc9 0a93a35fa5a476634aab5c2c46b467e2a6ce46cb54efe909f5c78891ac58d782afb5a4 4dbc5b2f0c190d4ebe8ca7eb1e4af8e02126a1bca0b523a4c67c7176bebb04afed7b50 6fc5c343e3e7727d3027d68937d5 agg_share_1: >- 4441f3f0d5c130bf9be61a579c90ec3eefd20d20794b2137d255b9964db08e626d5ad6 b2664859fb596bb241e47e3033671b5a5e05f2f7538d809df11f4b78df3969dc71f477 8e44926c5493703e97f25dbb203a642f4908f47251e0b3e5e4d061c491efac5047ae33 51a26225447400a65015a0906fa1f4d22e4756336ee02a131ff24b204a82bfb3942236 f56c5ca05a5b899cb554a3d3b94b981d5931b9222b1016f60a38776e53a7287d504a5b b243a4d0f3e6f2b141735814e1b5071fcd595e435f4adc5b39838e894144fb501284af 903a3cbc1c188d82cfd82976c818 agg_result: [0, 0, 0, 0, 0, 1, 0] ]]>