Verifiable Distributed Aggregation Functions

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Table of Contents

1. Introduction

The ubiquity of the Internet makes it an ideal platform for measurement of large-scale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial overlap, however, between information that is valuable to measure and information that users consider private.¶

For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition.¶

In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy.¶

Most prior approaches to this problem fall under the rubric of "differential privacy (DP)" [Dwo06]. Roughly speaking, a data aggregation system that is differentially private ensures that the degree to which any individual measurement influences the value of the aggregate result can be precisely controlled. For example, in systems like RAPPOR [EPK14], each user samples noise from a well-known distribution and adds it to their input before submitting to the aggregation server. The aggregation server then adds up the noisy inputs, and because it knows the distribution from whence the noise was sampled, it can estimate the true sum with reasonable precision.¶

Differentially private systems like RAPPOR are easy to deploy and provide a useful guarantee. On its own, however, DP falls short of the strongest privacy property one could hope for. Specifically, depending on the "amount" of noise a client adds to its input, it may be possible for a curious aggregator to make a reasonable guess of the input's true value. Indeed, the more noise the clients add, the less reliable will be the server's estimate of the output. Thus systems employing DP techniques alone must strike a delicate balance between privacy and utility.¶

The ideal goal for a privacy-preserving measurement system is that of secure multi-party computation (MPC): No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal.¶

VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of non-colluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of input measurements.¶

In addition to these MPC-style security goals, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the aggregate result itself does not leak too much information about any one measurement.¶

• TODO(issue #94) Provide guidance for local and central DP and point to it here.¶

The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA].¶

The VDAF abstraction laid out in Section 5 represents a class of multi-party protocols for privacy-preserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:¶

1. Providing higher-level protocols like [DAP] with a simple, uniform interface for accessing privacy-preserving measurement schemes, and documenting relevant operational and security bounds for that interface:¶

   1. General patterns of communications among the various actors involved in the system (clients, aggregation servers, and the collector of the aggregate result);¶
   2. Capabilities of a malicious coalition of servers attempting to divulge information about client measurements; and¶
   3. Conditions that are necessary to ensure that malicious clients cannot corrupt the computation.¶
2. Providing cryptographers with design criteria that provide a clear deployment roadmap for new constructions.¶

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

• The aforementioned Prio system [CGB17] allows for the privacy-preserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶

  1. Each client shards its measurement into a sequence of additive shares and distributes the shares among the aggregation servers.¶
  2. Next, each server adds up its shares locally, resulting in an additive share of the aggregate.¶
  3. Finally, the aggregation servers send their aggregate shares to the data collector, who combines them to obtain the aggregate result.¶

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

  In Section 7 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶

• More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar for solving the t-heavy-hitters problem in a privacy-preserving manner. Here each client holds a bit-string of length n, and the goal of the aggregation servers is to compute the set of inputs that occur at least t times. The core primitive used in their protocol is a specialized Distributed Point Function (DPF) [GI14] that allows the servers to "query" their DPF shares on any bit-string of length shorter than or equal to n. As a result of this query, each of the servers has an additive share of a bit indicating whether the string is a prefix of the client's input. The protocol also specifies a multi-party computation for verifying that at most one string among a set of candidates is a prefix of the client's input.¶

  In Section 8 we describe a VDAF called Poplar1 that implements this functionality.¶

Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the client-generated measurements are prepared for aggregation. Because these constructions are based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this process may require multiple round trips over the network.¶

There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.¶

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

The remainder of this document is organized as follows: Section 3 gives a brief overview of DAFs and VDAFs; Section 4 defines the syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6 defines various functionalities that are common to our constructions; Section 7 describes the Prio3 construction; Section 8 describes the Poplar1 construction; and Section 9 enumerates the security considerations for VDAFs.¶

2. Conventions and Definitions

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶

Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception.¶

A variable with type Bytes is a byte string. This document defines several byte-string constants. When comprised of printable ASCII characters, they are written as Python 3 byte-string literals (e.g., b'some constant string').¶

A global constant VERSION is defined, which algorithms are free to use as desired. Its value SHALL be b'vdaf-03'.¶

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).¶
• concat(parts: Vec[Bytes]) -> Bytes returns the concatenation of the input byte strings, i.e., parts[0] || ... || parts[len(parts)-1].¶
• xor(left: Bytes, right: Bytes) -> Bytes returns the bitwise XOR of left and right. An exception is raised if the inputs are not the same length.¶
• I2OSP and OS2IP from [RFC8017], which are used, respectively, to convert a non-negative integer to a byte string and convert a byte string to a non-negative integer.¶
• next_power_of_2(n: Unsigned) -> Unsigned returns the smallest integer greater than or equal to n that is also a power of two.¶

3. Overview

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

      Input shares           Aggregate shares

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.¶

4. Definition of DAFs

By way of a gentle introduction to VDAFs, this section describes a simpler class of schemes called Distributed Aggregation Functions (DAFs). Unlike VDAFs, DAFs do not provide verifiability of the computation. Clients must therefore be trusted to compute their input shares correctly. Because of this fact, the use of a DAF is NOT RECOMMENDED for most applications. See Section 9 for additional discussion.¶

A DAF scheme is used to compute a particular "aggregation function" over a set of measurements generated by Clients. Depending on the aggregation function, the Collector might select an "aggregation parameter" and disseminates it to the Aggregators. The semantics of this parameter is specific to the aggregation function, but in general it is used to represent the set of "queries" that can be made on the measurement set. For example, the aggregation parameter is used to represent the candidate prefixes in Poplar1 Section 8.¶

Execution of a DAF has four distinct stages:¶

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

Sharding and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶

A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table.¶

These types define some of the inputs and outputs of DAF methods at various stages of the computation. Observe that only the measurements, output shares, the aggregate result, and the aggregation parameter have an explicit type. All other values --- in particular, the input shares and the aggregate shares --- have type Bytes and are treated as opaque byte strings. This is because these values must be transmitted between parties over a network.¶

• OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. This way each method parameter has a meaningful type hint. See issue#58.¶

Each DAF is identified by a unique, 32-bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 16.¶

    Client
    ======

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

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

    out_share_0_0   out_share_1_0       out_share_[SHARES-1]_0
    out_share_0_1   out_share_1_1       out_share_[SHARES-1]_1
    out_share_0_2   out_share_1_2       out_share_[SHARES-1]_2
         ...             ...                     ...
    out_share_0_B   out_share_1_B       out_share_[SHARES-1]_B
      |               |                   |
      V               V                   V
    +-----------+   +-----------+       +-----------+
    | out2agg   |   | out2agg   |   ... | out2agg   |
    +-----------+   +-----------+       +-----------+
      |               |                   |
      V               V                   V
    agg_share_0     agg_share_1         agg_share_[SHARES-1]

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

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

    Collector
    =========

def run_daf(Daf,
            agg_param: Daf.AggParam,
            measurements: Vec[Daf.Measurement]):
    out_shares = [ [] for j in range(Daf.SHARES) ]
    for measurement in measurements:
        # Each Client shards its measurement into input shares and
        # distributes them among the Aggregators.
        (public_share, input_shares) = \
            Daf.measurement_to_input_shares(measurement)

        # Each Aggregator prepares its input share for aggregation.
        for j in range(Daf.SHARES):
            out_shares[j].append(
                Daf.prep(j, agg_param, public_share, input_shares[j]))

    # Each Aggregator aggregates its output shares into an aggregate
    # share and sends it to the Collector.
    agg_shares = []
    for j in range(Daf.SHARES):
        agg_share_j = Daf.out_shares_to_agg_share(agg_param,
                                                  out_shares[j])
        agg_shares.append(agg_share_j)

    # Collector unshards the aggregate result.
    num_measurements = len(measurements)
    agg_result = Daf.agg_shares_to_result(agg_param, agg_shares,
                                          num_measurements)
    return agg_result

5. Definition of VDAFs

Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Client-generated measurements. Evaluation of a VDAF involves the same four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the computation's correctness. Accommodating this interaction will require syntactic changes.¶

Overall execution of a VDAF comprises the following stages:¶

• Sharding - Computing input shares from an individual measurement¶
• Preparation - Conversion and verification of input shares to output shares compatible with the aggregation function being computed¶
• Aggregation - Combining a sequence of output shares into an aggregate share¶
• Unsharding - Combining a sequence of aggregate shares into an aggregate result¶

In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: Verification of the validity of the recovered output shares. This process ensures that aggregating the output shares will not lead to a garbled aggregate result.¶

The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF.¶

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

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

Each VDAF is identified by a unique, 32-bit integer ID. Identifiers for each (V)DAF specified in this document are defined in Table 16.¶

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

    input_share_0  input_share_1       input_share_[SHARES-1]
      |              |              ...  |
      V              V                   V
    +-----------+  +-----------+       +-----------+
    | prep_init |  | prep_init |       | prep_init |
    +-----------+  +------------+      +-----------+
      |              |              ...  |             \
      V              V                   V             |
    +-----------+  +-----------+       +-----------+   |
    | prep_next |  | prep_next |       | prep_next |   |
    +-----------+  +-----------+       +-----------+   |
      |              |              ...  |             |
      V              V                   V             | x ROUNDS
    +----------------------------------------------+   |
    | prep_shares_to_prep                          |   |
    +----------------------------------------------+   |
                     |                                 |
      +--------------+-------------------+             |
      |              |              ...  |             |
      V              V                   V             /
     ...            ...                 ...
      |              |                   |
      V              V                   V
    +-----------+  +-----------+       +-----------+
    | prep_next |  | prep_next |       | prep_next |
    +-----------+  +-----------+       +-----------+
      |              |              ...  |
      V              V                   V
    out_share_0    out_share_1         out_share_[SHARES-1]

def run_vdaf(Vdaf,
             agg_param: Vdaf.AggParam,
             nonces: Vec[Bytes],
             measurements: Vec[Vdaf.Measurement]):
    # Generate the long-lived verification key.
    verify_key = gen_rand(Vdaf.VERIFY_KEY_SIZE)

    out_shares = []
    for (nonce, measurement) in zip(nonces, measurements):
        # Each Client shards its measurement into input shares.
        (public_share, input_shares) = \
            Vdaf.measurement_to_input_shares(measurement)

        # Each Aggregator initializes its preparation state.
        prep_states = []
        for j in range(Vdaf.SHARES):
            state = Vdaf.prep_init(verify_key, j,
                                   agg_param,
                                   nonce,
                                   public_share,
                                   input_shares[j])
            prep_states.append(state)

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

        # The final outputs of prepare phase are the output shares.
        out_shares.append(outbound)

    # Each Aggregator aggregates its output shares into an
    # aggregate share. In a distributed VDAF computation, the
    # aggregate shares are sent over the network.
    agg_shares = []
    for j in range(Vdaf.SHARES):
        out_shares_j = [out[j] for out in out_shares]
        agg_share_j = Vdaf.out_shares_to_agg_share(agg_param,
                                                   out_shares_j)
        agg_shares.append(agg_share_j)

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

6. Preliminaries

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

def encode_vec(Field, data: Vec[Field]) -> Bytes:
    encoded = Bytes()
    for x in data:
        encoded += I2OSP(x.as_unsigned(), Field.ENCODED_SIZE)
    return encoded

def decode_vec(Field, encoded: Bytes) -> Vec[Field]:
    L = Field.ENCODED_SIZE
    if len(encoded) % L != 0:
        raise ERR_DECODE

    vec = []
    for i in range(0, len(encoded), L):
        encoded_x = encoded[i:i+L]
        x = OS2IP(encoded_x)
        if x >= Field.MODULUS:
            raise ERR_DECODE # Integer is larger than modulus
        vec.append(Field(x))
    return vec

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

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

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

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

class PrgAes128:

    SEED_SIZE: Unsigned = 16

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

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

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

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

7. Prio3

• NOTE This construction has not undergone significant security analysis.¶

This section describes Prio3, a VDAF for Prio [CGB17]. Prio is suitable for a wide variety of aggregation functions, including (but not limited to) sum, mean, standard deviation, estimation of quantiles (e.g., median), and linear regression. In fact, the scheme described in this section is compatible with any aggregation function that has the following structure:¶

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

At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multi-party computation to determine if their shares correspond to a valid input (as determined by the arithmetic circuit). This computation involves a "proof" of validity generated by the Client. Next, each Aggregator sums up its input shares locally. Finally, the Collector sums up the aggregate shares and computes the aggregate result.¶

This VDAF does not have an aggregation parameter. Instead, the output share is derived from the input share by applying a fixed map. See Section 8 for an example of a VDAF that makes meaningful use of the aggregation parameter.¶

As the name implies, Prio3 is a descendant of the original Prio construction. A second iteration was deployed in the [ENPA] system, and like the VDAF described here, the ENPA system was built from techniques introduced in [BBCGGI19] that significantly improve communication cost. That system was specialized for a particular aggregation function; the goal of Prio3 is to provide the same level of generality as the original construction.¶

The core component of Prio3 is a "Fully Linear Proof (FLP)" system. Introduced by [BBCGGI19], the FLP encapsulates the functionality required for encoding and validating inputs. Prio3 can be thought of as a transformation of a particular class of FLPs into a VDAF.¶

The remainder of this section is structured as follows. The syntax for FLPs is described in Section 7.1. The generic transformation of an FLP into Prio3 is specified in Section 7.2. Next, a concrete FLP suitable for any validity circuit is specified in Section 7.3. Finally, instantiations of Prio3 for various types of measurements are specified in Section 7.4. Test vectors can be found in Appendix "Test Vectors".¶

def run_flp(Flp, inp: Vec[Flp.Field], num_shares: Unsigned):
    joint_rand = Flp.Field.rand_vec(Flp.JOINT_RAND_LEN)
    prove_rand = Flp.Field.rand_vec(Flp.PROVE_RAND_LEN)
    query_rand = Flp.Field.rand_vec(Flp.QUERY_RAND_LEN)

    # Prover generates the proof.
    proof = Flp.prove(inp, prove_rand, joint_rand)

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

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

def measurement_to_input_shares(Prio3, measurement):
    # Domain separation tag for PRG info string
    dst = VERSION + I2OSP(Prio3.ID, 4)
    inp = Prio3.Flp.encode(measurement)

    # Generate input shares.
    leader_input_share = inp
    k_helper_input_shares = []
    k_helper_blinds = []
    k_joint_rand_parts = []
    for j in range(Prio3.SHARES-1):
        k_blind = gen_rand(Prio3.Prg.SEED_SIZE)
        k_share = gen_rand(Prio3.Prg.SEED_SIZE)
        helper_input_share = Prio3.Prg.expand_into_vec(
            Prio3.Flp.Field,
            k_share,
            dst + byte(j+1),
            Prio3.Flp.INPUT_LEN
        )
        leader_input_share = vec_sub(leader_input_share,
                                     helper_input_share)
        encoded = Prio3.Flp.Field.encode_vec(helper_input_share)
        k_joint_rand_part = Prio3.Prg.derive_seed(
            k_blind, dst + byte(j+1) + encoded)
        k_helper_input_shares.append(k_share)
        k_helper_blinds.append(k_blind)
        k_joint_rand_parts.append(k_joint_rand_part)
    k_leader_blind = gen_rand(Prio3.Prg.SEED_SIZE)
    encoded = Prio3.Flp.Field.encode_vec(leader_input_share)
    k_leader_joint_rand_part = Prio3.Prg.derive_seed(
        k_leader_blind, dst + byte(0) + encoded)
    k_joint_rand_parts.insert(0, k_leader_joint_rand_part)

    # Compute joint randomness seed.
    k_joint_rand = Prio3.joint_rand(k_joint_rand_parts)

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

    # The Leader's "hint" consists of the joint randomness parts of the
    # other Aggregators.
    k_leader_hint = k_joint_rand_parts[1:]
    input_shares = []
    input_shares.append(Prio3.encode_leader_share(
        leader_input_share,
        leader_proof_share,
        k_leader_blind,
        k_leader_hint,
    ))
    for j in range(Prio3.SHARES-1):
        # Each Helper's hint consists of the joint randomness part of the
        # other Aggregators.
        k_helper_hint = k_joint_rand_parts[:j+1] + \
                        k_joint_rand_parts[j+2:]
        input_shares.append(Prio3.encode_helper_share(
            k_helper_input_shares[j],
            k_helper_proof_shares[j],
            k_helper_blinds[j],
            k_helper_hint,
        ))
    return (b'', input_shares)

def prep_init(Prio3, verify_key, agg_id, _agg_param,
              nonce, _public_share, input_share):
    # Domain separation tag for PRG info string
    dst = VERSION + I2OSP(Prio3.ID, 4)

    (input_share, proof_share, k_blind, k_hint) = \
        Prio3.decode_leader_share(input_share) if agg_id == 0 else \
        Prio3.decode_helper_share(dst, agg_id, input_share)
    out_share = Prio3.Flp.truncate(input_share)

    # Compute joint randomness.
    joint_rand, k_joint_rand, k_joint_rand_part = [], None, None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded = Prio3.Flp.Field.encode_vec(input_share)
        k_joint_rand_part = Prio3.Prg.derive_seed(
            k_blind, dst + byte(agg_id) + encoded)
        k_joint_rand_parts = k_hint[:agg_id] + \
                             [k_joint_rand_part] + \
                             k_hint[agg_id:]
        k_joint_rand = Prio3.joint_rand(k_joint_rand_parts)
        joint_rand = Prio3.Prg.expand_into_vec(
            Prio3.Flp.Field,
            k_joint_rand,
            dst,
            Prio3.Flp.JOINT_RAND_LEN
        )

    # Query the input and proof share.
    query_rand = Prio3.Prg.expand_into_vec(
        Prio3.Flp.Field,
        verify_key,
        dst + byte(255) + nonce,
        Prio3.Flp.QUERY_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_part)
    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):
    dst = VERSION + I2OSP(Prio3.ID, 4)
    verifier = Prio3.Flp.Field.zeros(Prio3.Flp.VERIFIER_LEN)
    k_joint_rand_parts = []
    for encoded in prep_shares:
        (verifier_share, k_joint_rand_part) = \
            Prio3.decode_prep_share(encoded)

        verifier = vec_add(verifier, verifier_share)

        if Prio3.Flp.JOINT_RAND_LEN > 0:
            k_joint_rand_parts.append(k_joint_rand_part)

    if not Prio3.Flp.decide(verifier):
        raise ERR_VERIFY # proof verifier check failed

    k_joint_rand_check = None
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        k_joint_rand_check = Prio3.joint_rand(k_joint_rand_parts)
    return Prio3.encode_prep_msg(k_joint_rand_check)

def out_shares_to_agg_share(Prio3, _agg_param, out_shares):
    agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for out_share in out_shares:
        agg_share = vec_add(agg_share, out_share)
    return Prio3.Flp.Field.encode_vec(agg_share)

def agg_shares_to_result(Prio3, _agg_param, agg_shares,
                         num_measurements):
    agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
    for agg_share in agg_shares:
        agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share))
    return Prio3.Flp.decode(agg, num_measurements)

def joint_rand(Prio3, k_joint_rand_parts):
    dst = VERSION + I2OSP(Prio3.ID, 4)
    return Prio3.Prg.derive_seed(
        zeros(Prio3.Prg.SEED_SIZE),
        dst + byte(255) + concat(k_joint_rand_parts))

def encode_leader_share(Prio3,
                        input_share,
                        proof_share,
                        k_blind,
                        k_hint):
    encoded = Bytes()
    encoded += Prio3.Flp.Field.encode_vec(input_share)
    encoded += Prio3.Flp.Field.encode_vec(proof_share)
    if Prio3.Flp.JOINT_RAND_LEN > 0:
        encoded += k_blind
        encoded += concat(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 = []
        for _ in range(Prio3.SHARES-1):
            k_joint_rand_part, encoded = encoded[:l], encoded[l:]
            k_hint.append(k_joint_rand_part)
    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 += concat(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 = []
        for _ in range(Prio3.SHARES-1):
            k_joint_rand_part, encoded = encoded[:l], encoded[l:]
            k_hint.append(k_joint_rand_part)
    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

C(x_shares[0] + ... + x_shares[SHARES-1]) =
    C(x_shares[0]) + ... + C(x_shares[SHARES-1])

C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]

Mul(left, right) = left * right

C(x[0]) = Mul(x[0], x[0]) - x[0]

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

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

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

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

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

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

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

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

def encode(Sum, measurement: Integer):
    if 0 > measurement or measurement >= 2^Sum.INPUT_LEN:
        raise ERR_INPUT

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

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

def decode(Sum, output, _num_measurements):
    return output[0].as_unsigned()

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

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

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

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

def decode(Histogram, output: Vec[Field128], _num_measurements):
    return [bucket_count.as_unsigned() for bucket_count in output]

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

    # Check that the buckets sum to 1.
    sum_check = -Field128(1) * Field128(num_shares).inv()
    for b in inp:
        sum_check += b

    out = joint_rand[1]   * range_check + \
          joint_rand[1]^2 * sum_check
    return out

8. 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 [BBCGGI21]. 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 [BBCGGI21] that generalizes the notion of a Distributed Point Function (DPF) [GI14]. Briefly, a DPF is used to distribute the computation of a "point function", a function that evaluates to zero on every input except at a programmable "point". The computation is distributed in such a way that no one party knows either the point or what it evaluates to.¶

An IDPF generalizes this "point" to a path on a full binary tree from the root to one of the leaves. It is evaluated on an "index" representing a unique node of the tree. If the node is on the programmed path, then 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 [BBCGGI21]. This protocol ensures that evaluating a set of input shares on a unique set of candidate prefixes results in shares of a "one-hot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one.¶

The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors".¶

def current_field(Idpf, level):
    return Idpf.FieldInner if level < Idpf.BITS-1 \
                else Idpf.FieldLeaf

    A = -2*a + k
    B = a^2 + b - k*a + c

def measurement_to_input_shares(Poplar1, measurement):
    dst = VERSION + I2OSP(Poplar1.ID, 4)
    prg = Poplar1.Idpf.Prg(
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE), dst + byte(255))

    # Construct the IDPF values for each level of the IDPF tree.
    # Each "data" value is 1; in addition, the Client generates
    # a random "authenticator" value used by the Aggregators to
    # compute the sketch during preparation. This sketch is used
    # to verify the one-hotness of their output shares.
    beta_inner = [
        [Poplar1.Idpf.FieldInner(1), k] \
            for k in prg.next_vec(Poplar1.Idpf.FieldInner,
                                  Poplar1.Idpf.BITS - 1) ]
    beta_leaf = [Poplar1.Idpf.FieldLeaf(1)] + \
        prg.next_vec(Poplar1.Idpf.FieldLeaf, 1)

    # Generate the IDPF keys.
    (public_share, keys) = \
        Poplar1.Idpf.gen(measurement, beta_inner, beta_leaf)

    # Generate correlated randomness used by the Aggregators to
    # compute a sketch over their output shares. PRG seeds are
    # used to encode shares of the `(a, b, c)` triples.
    # (See [BBCGGI21, Appendix C.4].)
    corr_seed = [
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE),
        gen_rand(Poplar1.Idpf.Prg.SEED_SIZE),
    ]
    corr_prg = [
        Poplar1.Idpf.Prg(corr_seed[0], dst + byte(0)),
        Poplar1.Idpf.Prg(corr_seed[1], dst + byte(1)),
    ]

    # For each level of the IDPF tree, shares of the `(A, B)`
    # pairs are computed from the corresponding `(a, b, c)`
    # triple and authenticator value `k`.
    corr_inner = [[], []]
    for level in range(Poplar1.Idpf.BITS):
        Field = Poplar1.Idpf.current_field(level)
        k = beta_inner[level][1] if level < Poplar1.Idpf.BITS - 1 \
            else beta_leaf[1]
        (a, b, c) = vec_add(corr_prg[0].next_vec(Field, 3),
                            corr_prg[1].next_vec(Field, 3))
        A = -Field(2) * a + k
        B = a^2 + b - a * k + c
        corr1 = prg.next_vec(Field, 2)
        corr0 = vec_sub([A, B], corr1)
        if level < Poplar1.Idpf.BITS - 1:
            corr_inner[0] += corr0
            corr_inner[1] += corr1
        else:
            corr_leaf = [corr0, corr1]

    # Each input share consists of the Aggregator's IDPF key
    # and a share of the correlated randomness.
    return (public_share,
            Poplar1.encode_input_shares(
                keys, corr_seed, corr_inner, corr_leaf))

def prep_init(Poplar1, verify_key, agg_id, agg_param,
              nonce, public_share, input_share):
    dst = VERSION + I2OSP(Poplar1.ID, 4)
    (level, prefixes) = agg_param
    (key, corr_seed, corr_inner, corr_leaf) = \
        Poplar1.decode_input_share(input_share)

    # Evaluate the IDPF key at the given set of prefixes.
    value = Poplar1.Idpf.eval(
        agg_id, public_share, key, level, prefixes)

    # Get correlation shares for the given level of the IDPF tree.
    #
    # Implementation note: Computing the shares of `(a, b, c)`
    # requires expanding PRG seeds into a vector of field elements
    # of length proportional to the level of the tree. Typically
    # the IDPF will be evaluated incrementally beginning with
    # `level == 0`. Implementations can save computation by
    # storing the intermediate PRG state between evaluations.
    corr_prg = Poplar1.Idpf.Prg(corr_seed, dst + byte(agg_id))
    for current_level in range(level+1):
        Field = Poplar1.Idpf.current_field(current_level)
        (a_share, b_share, c_share) = corr_prg.next_vec(Field, 3)
    (A_share, B_share) = corr_inner[2*level:2*(level+1)] \
        if level < Poplar1.Idpf.BITS - 1 else corr_leaf

    # Compute the Aggregator's first round of the sketch. These are
    # called the "masked input values" [BBCGGI21, Appendix C.4].
    Field = Poplar1.Idpf.current_field(level)
    verify_rand_prg = Poplar1.Idpf.Prg(verify_key,
        dst + Poplar1.verify_context(nonce, level, prefixes))
    verify_rand = verify_rand_prg.next_vec(Field, len(prefixes))
    sketch_share = [a_share, b_share, c_share]
    out_share = []
    for (i, r) in enumerate(verify_rand):
        (data_share, auth_share) = value[i]
        sketch_share[0] += data_share * r
        sketch_share[1] += data_share * r^2
        sketch_share[2] += auth_share * r
        out_share.append(data_share)

    prep_mem = sketch_share \
                + [A_share, B_share, Field(agg_id)] \
                + out_share
    return (b'ready', level, prep_mem)

def prep_next(Poplar1, prep_state, opt_sketch):
    (step, level, prep_mem) = prep_state
    Field = Poplar1.Idpf.current_field(level)

    # Aggregators exchange masked input values (step (3.)
    # of [BBCGGI21, Appendix C.4]).
    if step == b'ready' and opt_sketch == None:
        sketch_share, prep_mem = prep_mem[:3], prep_mem[3:]
        return ((b'sketch round 1', level, prep_mem),
                Field.encode_vec(sketch_share))

    # Aggregators exchange evaluated shares (step (4.)).
    elif step == b'sketch round 1' and opt_sketch != None:
        prev_sketch = Field.decode_vec(opt_sketch)
        if len(prev_sketch) == 0:
            prev_sketch = Field.zeros(3)
        elif len(prev_sketch) != 3:
            raise ERR_INPUT # prep message malformed
        (A_share, B_share, agg_id), prep_mem = \
            prep_mem[:3], prep_mem[3:]
        sketch_share = [
            agg_id * (prev_sketch[0]^2 \
                        - prev_sketch[1]
                        - prev_sketch[2]) \
                + A_share * prev_sketch[0] \
                + B_share
        ]
        return ((b'sketch round 2', level, prep_mem),
                Field.encode_vec(sketch_share))

    elif step == b'sketch round 2' and opt_sketch != None:
        prev_sketch = Field.decode_vec(opt_sketch)
        if len(prev_sketch) == 0:
            prev_sketch = Field.zeros(1)
        elif len(prev_sketch) != 1:
            raise ERR_INPUT # prep message malformed
        if prev_sketch[0] != Field(0):
            raise ERR_VERIFY
        return prep_mem # Output shares

    raise ERR_INPUT # unexpected input

def prep_shares_to_prep(Poplar1, agg_param, prep_shares):
    if len(prep_shares) != 2:
        raise ERR_INPUT # unexpected number of prep shares
    (level, _) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    sketch = vec_add(Field.decode_vec(prep_shares[0]),
                     Field.decode_vec(prep_shares[1]))
    if sketch == Field.zeros(len(sketch)):
        # In order to reduce communication overhead, let the
        # empty string denote the zero vector of the required
        # length.
        return b''
    return Field.encode_vec(sketch)

def out_shares_to_agg_share(Poplar1, agg_param, out_shares):
    (level, prefixes) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    agg_share = Field.zeros(len(prefixes))
    for out_share in out_shares:
        agg_share = vec_add(agg_share, out_share)
    return Field.encode_vec(agg_share)

def agg_shares_to_result(Poplar1, agg_param,
                         agg_shares, _num_measurements):
    (level, prefixes) = agg_param
    Field = Poplar1.Idpf.current_field(level)
    agg = Field.zeros(len(prefixes))
    for agg_share in agg_shares:
        agg = vec_add(agg, Field.decode_vec(agg_share))
    return list(map(lambda x: x.as_unsigned(), agg))

def encode_input_shares(Poplar1, keys,
                        corr_seed, corr_inner, corr_leaf):
    input_shares = []
    for (key, seed, inner, leaf) in zip(keys,
                                        corr_seed,
                                        corr_inner,
                                        corr_leaf):
        encoded = Bytes()
        encoded += key
        encoded += seed
        encoded += Poplar1.Idpf.FieldInner.encode_vec(inner)
        encoded += Poplar1.Idpf.FieldLeaf.encode_vec(leaf)
        input_shares.append(encoded)
    return input_shares

def decode_input_share(Poplar1, encoded):
    l = Poplar1.Idpf.KEY_SIZE
    key, encoded = encoded[:l], encoded[l:]
    l = Poplar1.Idpf.Prg.SEED_SIZE
    corr_seed, encoded = encoded[:l], encoded[l:]
    l = Poplar1.Idpf.FieldInner.ENCODED_SIZE \
        * 2 * (Poplar1.Idpf.BITS - 1)
    encoded_corr_inner, encoded = encoded[:l], encoded[l:]
    corr_inner = Poplar1.Idpf.FieldInner.decode_vec(
        encoded_corr_inner)
    l = Poplar1.Idpf.FieldLeaf.ENCODED_SIZE * 2
    encoded_corr_leaf, encoded = encoded[:l], encoded[l:]
    corr_leaf = Poplar1.Idpf.FieldLeaf.decode_vec(
        encoded_corr_leaf)
    if len(encoded) != 0:
        raise ERR_INPUT
    return (key, corr_seed, corr_inner, corr_leaf)

def encode_agg_param(Poplar1, level, prefixes):
    if level > 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

def gen(IdpfPoplar, alpha, beta_inner, beta_leaf):
    if alpha >= 2^IdpfPoplar.BITS:
        raise ERR_INPUT # alpha too long
    if len(beta_inner) != IdpfPoplar.BITS - 1:
        raise ERR_INPUT # beta_inner vector is the wrong size

    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)

def eval(IdpfPoplar, agg_id, public_share, init_seed,
         level, prefixes):
    if agg_id >= IdpfPoplar.SHARES:
        raise ERR_INPUT # invalid aggregator ID
    if level >= IdpfPoplar.BITS:
        raise ERR_INPUT # level too deep
    if len(set(prefixes)) != len(prefixes):
        raise ERR_INPUT # candidate prefixes are non-unique

    correction_words = IdpfPoplar.decode_public_share(public_share)
    out_share = []
    for prefix in prefixes:
        if prefix >= 2^(level+1):
            raise ERR_INPUT # prefix too long

        # The Aggregator's output share is the value of a node of
        # the IDPF tree at the given `level`. The node's value is
        # computed by traversing the path defined by the candidate
        # `prefix`. Each node in the tree is represented by a seed
        # (`seed`) and a set of control bits (`ctrl`).
        seed = init_seed
        ctrl = Field2(agg_id)
        for current_level in range(level+1):
            bit = (prefix >> (level - current_level)) & 1

            # Implementation note: Typically the current round of
            # candidate prefixes would have been derived from
            # aggregate results computed during previous rounds. For
            # example, when using `IdpfPoplar` to compute heavy
            # hitters, a string whose hit count exceeded the given
            # threshold in the last round would be the prefix of each
            # `prefix` in the current round. (See [BBCGGI21,
            # Section 5.1].) In this case, part of the path would
            # have already been traversed.
            #
            # Re-computing nodes along previously traversed paths is
            # wasteful. Implementations can eliminate this added
            # complexity by caching nodes (i.e., `(seed, ctrl)`
            # pairs) output by previous calls to `eval_next()`.
            (seed, ctrl, y) = IdpfPoplar.eval_next(seed, ctrl,
                correction_words[current_level], current_level, bit)
        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)

def extend(IdpfPoplar, seed):
    dst = VERSION + b' idpf poplar extend'
    prg = IdpfPoplar.Prg(seed, dst)
    s = [
        prg.next(IdpfPoplar.Prg.SEED_SIZE),
        prg.next(IdpfPoplar.Prg.SEED_SIZE),
    ]
    b = OS2IP(prg.next(1))
    t = [Field2(b & 1), Field2((b >> 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

9. 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 Section 5.2) is never revealed to the Clients.¶

It is also possible to consider a stronger form of robustness, where the attacker also controls a subset of Aggregators (see [BBCGGI19], Section 6.3). To satisfy this stronger notion of robustness, it is necessary to prevent the attacker from sharing the verification 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 [Dou02]. In this attack, the adversary observes a subset of input shares transmitted by a Client it is interested in. It allows the input shares to be processed, but corrupts and picks bogus inputs for the remaining Clients. Applications can guard against these risks by adding additional controls on measurement submission, such as client authentication and rate limits.¶

VDAFs do not inherently provide differential privacy [Dwo06]. The VDAF approach to private measurement can be viewed as complementary to differential privacy, relying on non-collusion instead of statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and removing it after unsharding.¶

10. IANA Considerations

A codepoint for each (V)DAF in this document is defined in the table below. Note that 0xFFFF0000 through 0xFFFFFFFF are reserved for private use.¶

• TODO Add IANA considerations for the codepoints summarized in Table 16.¶

• OPEN ISSUE Currently the scheme includes the PRG. This means that we need bits of the codepoint to differentiate between PRGs. We could instead make the PRG generic (e.g., Prio3Count(Aes128) instead of Prio3Aes128Count) and define a separate codepoint.¶

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 Prio3Aes128Sum, 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.¶

verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 1
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    ad8bb894e3222b47b70eb67d4f70cb78644826d67d31129e422b910cf0aab70c0b78
    fa57b4a7b3aaafae57bd1012e813
  input_share_1: >-
    0101010101010101010101010101010101010101010101010101010101010101
  round_0:
    prep_share_0: >-
      38d535dd68f3c02ed6681f7ff24239d46fde93c8402d24ebbafa25c77ca3535d
    prep_share_1: >-
      c72aca21970c3fd35274476a1cddd4bb4efa24ee0d71473e4a0a23713a347d78
    prep_message: >-
  out_share_0:
    - 12505291739929652039
  out_share_1:
    - 5941452329484932283
agg_share_0: >-
  ad8bb894e3222b47
agg_share_1: >-
  5274476a1cddd4bb
agg_result: 1

bits: 8
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 100
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    fc1e42a024e3d8b45f63f485ebe2dc8a356c5e5780a0bd751184e6a02a96c0767f51
    8e87282ebdc039590aef02e40e5492c9eb69dd22b6b4f1d630e7ca8612b7a7e090b3
    9460bc4036345f5ef537d691fd585bc05a2ea580c7e354680afd0fd49f3d083d5e38
    3b97755a842cf5e69870a970b14a10595c0c639ad2e7bda42c7146c4b69fd79e7403
    d89dac5816d0dc6f2bb987fccca4c4aee64444b7f46431433c59c6e7f2839fe2b7ad
    9316d31a52dcc0df07f1da14aa38e0cd88de380fda29b33704e8c3439376762739aa
    5b5cff9e925939773d24ca0e75bcf87149c9bcc2f8462afa6b50513ab003ac00c9ae
    3685ea52bdee3c814ffd5afc8357d93454b3ffaf0b5e9fd351730f0d55aed54a9cfa
    86f9119601ce9857ee0af3f579251bcc7ffe51b8393adc36ab6142eb0e0d07c9b2d5
    ab71d8d5639f32c61f7d59b45a95129cbc76d7e30c02a1329454f843553413d4e84b
    cab2c3ba1a0150292026dfa37488da5dd639c53edd51bf4eb5aa54d5b165fcd55d10
    f3496008f4e3b6d3eb200c19c5b9c42ad4f12977a857d02f787b14ced27fc5eefb05
    722b372a7d48c1891d30a32d84ec8d1f9a783a38bfac2793f0da6796cff90521e1d7
    3f497f7d2c910b7fbbea2ba4b906d437a53bcbed16986f5646fd238e736f1c3e9d3a
    910218ce7f48dea3e9a1a848c580a1c506a80edb0c0a973a269667475ce88f442467
    4b14a3a8f2b71ef529d2ca96a3c5e4da384545749a55188d4de0074ad601695e934c
    9fe71d27c139b7678ead7f904cd2ae2a3aafa96d8211579e391507df96bf42c383f2
    ac71d7a558ebf1e3d5ab086b65422415bd24be9c979ca5b4f381d51b06ec4f6740b1
    a084999cd95fe63fec4a019f635640ba18d42312de7d1994947502b9010101010101
    010101010101010101015f0721f50826593dc3908dad39353846
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    0101010101010101010101010101094240ceae2d63ba1bdda997fa0bcbd8
  round_0:
    prep_share_0: >-
      0a85b5e51cacf514ee9e9bbe5d3ac023795e910b765411e5cea8ff187973640694
      bd740cc15bc9cad60bc85785206062094240ceae2d63ba1bdda997fa0bcbd8
    prep_share_1: >-
      f57a4a1ae3530acf11616441a2c53fde804d262dc42e15e556ee02c588c3ca9d92
      4eefa735a95f6e420f2c5161706e025f0721f50826593dc3908dad39353846
    prep_message: >-
      60af733578d766f2305c1d53c840b4b5
  out_share_0:
    - 227608477929192160221239678567201956832
  out_share_1:
    - 112673888991746302725626094800698809477
agg_share_0: >-
  ab3bcc5ef693737a7a3e76cd9face3e0
agg_share_1: >-
  54c433a1096c8c6985c1893260531c85
agg_result: 100

buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
  measurement: 50
  nonce: "01010101010101010101010101010101"
  public_share: >-
  input_share_0: >-
    ee1076c1ebc2d48a557a71031bc9dd5c9cd5e91180bbb51f4ac366946bcbfa93b908
    792bd15d402f4ac8da264e24a20f645ef68472180c5894bac704ae0675d7f16776df
    4f93852a40b514593a73be51ad64d8c28322a47af92c92223dd489998a3c6687861c
    dc2e4d834885d03d8d3273af0bf742c47985ae8fec6d16c31216792bb0cdca0d1d1f
    a2287414cd069f8caa42dc08f78dd43e14c4095e2ef9d9609937caebcd534e813136
    e79a4233e873397a6c7fd164928d43673b32e061139dc6650152d8433e2342f59514
    9418929b74c9e23f1469ed1eebdaa57d0b5c62f90cb5a53dc68c8e030448bb2d9c07
    aeed50d82c93e1afe8febd68918933ed9b2dd36b9d8a35fd6c57cd76707011fca775
    26437aeb8392a2013f829c1e395f7f8ddef030f5bc869833f528ae2137a2e667aa64
    8d8643f6c13e8d76e8832ab9ef7d0101010101010101010101010101010194c3f0f1
    061c8f440b51f806ad822510
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010101
    01010101010101010101010101016195ec204fd5d65c14fac36b73723cde
  round_0:
    prep_share_0: >-
      f2dc9e823b867d760b2169644633804eabec10e5869fe8f3030c5da6dc0fce03a4
      33572cb8aaa7ca3559959f7bad68306195ec204fd5d65c14fac36b73723cde
    prep_share_1: >-
      0d23617dc479826df4de969bb9cc7fb3f5e934542e987db0271aee33551b28a4c1
      6f7ad00127c43df9c433a1c224594d94c3f0f1061c8f440b51f806ad822510
    prep_message: >-
      7912f1157c2ce3a4dca6456224aeaeea
  out_share_0:
    - 316441748434879643753815489063091297628
    - 208470253761472213750543248431791209107
    - 245951175238245845331446316072865931791
    - 133415875449384174923011884997795018199
  out_share_1:
    - 23840618486058819193050284304809468581
    - 131812113159466249196322524936109557102
    - 94331191682692617615419457295034834419
    - 206866491471554288023853888370105748010
agg_share_0: >-
  ee1076c1ebc2d48a557a71031bc9dd5c9cd5e91180bbb51f4ac366946bcbfa93b90879
  2bd15d402f4ac8da264e24a20f645ef68472180c5894bac704ae0675d7
agg_share_1: >-
  11ef893e143d2b59aa858efce43622a5632a16ee7f444ac4b53c996b9434056e46f786
  d42ea2bfb4b53725d9b1db5df39ba1097b8de7f38b6b4538fb51f98a2a
agg_result: [0, 0, 1, 0]

bits: 4
verify_key: "01010101010101010101010101010101"
agg_param: (0, [0, 1])
upload_0:
  measurement: 13
  nonce: "01010101010101010101010101010101"
  public_share: >-
    9a000000000000000000000000000000000000000000000001eb3a1bd6b5fa4a4500
    000000000000000000000000000000ffffffff0000000022522c3fd5a33cac000000
    00000000000000000000000000ffffffff0000000069f41eee46542b690000000000
    00000000000000000000000000000000000000000000000000000000000000000000
    0000000000000000017d1fd6df94280145a0dcc933ceb706e9219d50e7c4f92fd8ca
    9a0ffb7d819646
  input_share_0: >-
    0101010101010101010101010101010101010101010101010101010101010101c226
    c4542c79eafa04ece4b493baadd00114dd7a8f91b9a25b767f43733c467d2cbb7fb5
    fe9782d0a51919f2bc2b6aa57442f7b923b9a700c7ab7a6c0aaff428ee671e1ed22f
    12dbb4714b53b11f3e0354cd5709f04c6bb9b0563499a7bd94c11d6d48c24ddc68a4
    b6477c2847d8218a
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010e6b
    27437d25adc6b565aa093dcc0d27507481df2f19d80abebe750f4b0ab4b6eb20a7de
    2aebb5f610e65647225d4a1851709e3b8c95c38eccf85ffebf320fae04f6826999af
    0782036a9671411b0676717cc6a0b8fd87177836f1fd980d49e4ad23f31e83a99cea
    313c614e7106ec80

verify_key: "01010101010101010101010101010101"
agg_param: (0, [0, 1])
upload_0:
  round_0:
    prep_share_0: >-
      d15f37fd8d2de10c3eec340265f8bfaa6ea7b79536b4d12a
    prep_share_1: >-
      7e02697276b506ca402eca93b7dcf770877dff591d6fd9c5
    prep_message: >-
      4f61a17103e2e7d57f1afe961dd5b71af625b6ee5424aaef
  round_1:
    prep_share_0: >-
      cba373c9458c26ee
    prep_share_1: >-
      345c8c35ba73d913
    prep_message: >-
  out_share_0:
    - 18188801410092473065
    - 309938393258542164
  out_share_1:
    - 257942659322111256
    - 18136805676156042158
agg_share_0: >-
  fc6b9a839aaf9ae9044d1f53986e4454
agg_share_1: >-
  0394657b65506518fbb2e0ab6791bbae
agg_result: [0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (1, [0, 1, 2, 3])
upload_0:
  round_0:
    prep_share_0: >-
      9b0474605d77d4791a43cb86dc332a15b8b5f67496aa4b8d
    prep_share_1: >-
      92b13576b734a956e93851d81193d503bae64c3cda971dfb
    prep_message: >-
      2db5a9d814ac7dce037c1d5fedc6ff17739c42b271416987
  round_1:
    prep_share_0: >-
      aecb3ecaafff0dbe
    prep_share_1: >-
      5134c1345000f243
    prep_message: >-
  out_share_0:
    - 16639637265869957457
    - 1807457878431656707
    - 14875784335609201424
    - 8515527061577716649
  out_share_1:
    - 1807106803544626864
    - 16639286190982927614
    - 3570959733805382897
    - 9931217007836867673
agg_share_0: >-
  e6ebddeec787e9511915615d36666b03ce716709b74e8310762d3abecc90d3a9
agg_share_1: >-
  19142210387816b0e6ea9ea1c99994fe318e98f548b17cf189d2c540336f2c59
agg_result: [0, 0, 0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (2, [0, 2, 4, 6])
upload_0:
  round_0:
    prep_share_0: >-
      93df236f6e786bca97233e32224a1941f6dd3a9511e01acc
    prep_share_1: >-
      2ab413234f2bec549b6546e4d96fee4bad6957e433768506
    prep_message: >-
      be933692bda4581e32888517fbba078ba446927a45569fd1
  round_1:
    prep_share_0: >-
      c36e1270495f5aa6
    prep_share_1: >-
      3c91ed8eb6a0a55b
    prep_message: >-
  out_share_0:
    - 3024081010823632913
    - 6306522542811675542
    - 2996392641000911092
    - 7890625214403888205
  out_share_1:
    - 15422663058590951408
    - 12140221526602908779
    - 15450351428413673229
    - 10556118855010696117
agg_share_0: >-
  29f7b1a836259811578544d6dc487b962995533b3e7430f46d8121d38048c44d
agg_share_1: >-
  d6084e56c9da67f0a87abb2823b7846bd66aacc3c18bcf0d927ede2b7fb73bb5
agg_result: [0, 0, 0, 1]

verify_key: "01010101010101010101010101010101"
agg_param: (3, [1, 3, 5, 7, 9, 13, 15])
upload_0:
upload_0:
  measurement: 13
  nonce: "01010101010101010101010101010101"
  public_share: >-
    9a000000000000000000000000000000000000000000000001eb3a1bd6b5fa4a4500
    000000000000000000000000000000ffffffff0000000022522c3fd5a33cac000000
    00000000000000000000000000ffffffff0000000069f41eee46542b690000000000
    00000000000000000000000000000000000000000000000000000000000000000000
    0000000000000000017d1fd6df94280145a0dcc933ceb706e9219d50e7c4f92fd8ca
    9a0ffb7d819646
  input_share_0: >-
    0101010101010101010101010101010101010101010101010101010101010101c226
    c4542c79eafa04ece4b493baadd00114dd7a8f91b9a25b767f43733c467d2cbb7fb5
    fe9782d0a51919f2bc2b6aa57442f7b923b9a700c7ab7a6c0aaff428ee671e1ed22f
    12dbb4714b53b11f3e0354cd5709f04c6bb9b0563499a7bd94c11d6d48c24ddc68a4
    b6477c2847d8218a
  input_share_1: >-
    01010101010101010101010101010101010101010101010101010101010101010e6b
    27437d25adc6b565aa093dcc0d27507481df2f19d80abebe750f4b0ab4b6eb20a7de
    2aebb5f610e65647225d4a1851709e3b8c95c38eccf85ffebf320fae04f6826999af
    0782036a9671411b0676717cc6a0b8fd87177836f1fd980d49e4ad23f31e83a99cea
    313c614e7106ec80
  round_0:
    prep_share_0: >-
      0940b06b287f0232397df2414c00cd0ccfb08f955bf4089651f019458cd7da2204
      c997d78134341745669b19da58cbef280f712d12900475ac2c4de486e9870774ba
      3c9d024245e7866528e37b9a931e61b794ab12a4bf74b41de50cec5f1214
    prep_share_1: >-
      605b876c915cb4b2d94c5999ab0642e81eefe4b06ac936b2cd9608ffeb9a6c8a45
      a9faa5b4940cdc375043075990f115770cbb762f2baf1bf9024c36685cf3051607
      a3331e036072b0202bb89be78d7ad8f8c07fbf2761884dcb4f414e966562
    prep_message: >-
      699c37d7b9dbb6e512ca4bdaf7070ff4eea07445c6bd3f491f862245787246ac4a
      73927d35c840f37cb6de2133e9bd049f1c2ca341bbb391a52e9a1aef467a0c0ac1
      dfd02045a65a3685549c178220993ab0552ad1cc20fd01e9344e3af57789
  round_1:
    prep_share_0: >-
      69ac4b4219b647d08579ebc1d61e15a87684c817b8e9e2f599dec39dd42c60e0
    prep_share_1: >-
      1653b4bde649b82f7a86143e29e1ea57897b37e847161d0a66213c622bd39f0d
    prep_message: >-
  out_share_0:
    - 31266787981623073345438631450097917412029050287225674733758559830914037991840
    - 26393661814069127500656687560668964619575440551355825110935075749282733430112
    - 29174096716912605985872100661347047626133976434476959088535854022406688824676
    - 37302603208117590782775952507020589026342905631492315271787751917771005617295
    - 55309420117672678173715719521410144937198418175813304907316032334662803449360
    - 29174390317706805594521496894406843472816744180092642437922410411508000504883
    - 21223279146307403682071881696710510174726527474394063793607036387613816968277
  out_share_1:
    - 26629256637035024366346861054246036514605942045594607285970232173042526828109
    - 31502382804588970211128804943674989307059551781464456908793716254673831389837
    - 28721947901745491725913391842996906300501015898343322931192937981549875995273
    - 20593441410540506929009539997323364900292086701327966747941040086185559202654
    - 2586624500985419538069772982933808989436574157006977112412759669293761370589
    - 28721654300951292117263995609937110453818248152727639581806381592448564315067
    - 36672765472350694029713610807633443751908464858426218226121755616342747851672
agg_share_0: >-
  45205ff6efcf67961cc100e880f3c58d5ce283585caf7fb58e09c24ba9bc49a03a5a48
  7f662ab7b240b41b6c3ac345c3754b43cbb1a5d8158e8859d1994d1560407ff41dd4ca
  cc470b3c428a61fa974d6cf9b1d14c6db9401f0e174ffce55d64527886748fe09c86af
  d58b7e7717c47ee97309fefeda64830940348d97e3e48f7a4805bcea0d023dffa0d3cb
  1bda898326421778e72467244acb17f472a4e61040801ea8170611e914421029a53aa0
  1b494ff834cf2e01239409a37b99623c332eebf34778eee2356d44b95a1681d1f394f7
  fe66cad88d59fcb458283e84d455
agg_share_1: >-
  3adfa00910309869e33eff177f0c3a72a31d7ca7a350804a71f63db45643b64d45a5b7
  8099d5484dbf4be493c53cba3c8ab4bc344e5a27ea7177a62e66b2ea8d3f800be22b35
  33b8f4c3bd759e0568b293064e2eb39246bfe0f1e8b0031aa2892d87798b701f637950
  2a748188e83b81168cf60101259b7cf6bfcb72681c1b5e05b7fa4315f2fdc2005f2c34
  e425767cd9bde88718db98dbb534e80b8d5b19dd3f7fe157e8f9ee16ebbdefd65ac55f
  e4b6b007cb30d1fedc6bf65c84669dc3bb51140cb887111dca92bb46a5e97e2e0c6b08
  0199352772a6034ba7d7c17b2b98
agg_result: [0, 0, 0, 0, 0, 1, 0]

Authors' Addresses