InternetDraft  VDAF  August 2022 
Barnes, et al.  Expires 25 February 2023  [Page] 
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a family of multiparty protocols for computing aggregate statistics over user measurements. These protocols are designed to ensure that, as long as at least one aggregation server executes the protocol honestly, individual measurements are never seen by any server in the clear. At the same time, VDAFs allow the servers to detect if a malicious or misconfigured client submitted an input that would result in an incorrect aggregate result.¶
This note is to be removed before publishing as an RFC.¶
Discussion of this document takes place on the Crypto Forum Research Group mailing list (cfrg@ietf.org), which is archived at https://mailarchive.ietf.org/arch/search/?email_list=cfrg.¶
Source for this draft and an issue tracker can be found at https://github.com/cjpatton/vdaf.¶
This InternetDraft is submitted in full conformance with the provisions of BCP 78 and BCP 79.¶
InternetDrafts are working documents of the Internet Engineering Task Force (IETF). Note that other groups may also distribute working documents as InternetDrafts. The list of current InternetDrafts is at https://datatracker.ietf.org/drafts/current/.¶
InternetDrafts are draft documents valid for a maximum of six months and may be updated, replaced, or obsoleted by other documents at any time. It is inappropriate to use InternetDrafts as reference material or to cite them other than as "work in progress."¶
This InternetDraft will expire on 25 February 2023.¶
Copyright (c) 2022 IETF Trust and the persons identified as the document authors. All rights reserved.¶
This document is subject to BCP 78 and the IETF Trust's Legal Provisions Relating to IETF Documents (https://trustee.ietf.org/licenseinfo) in effect on the date of publication of this document. Please review these documents carefully, as they describe your rights and restrictions with respect to this document. Code Components extracted from this document must include Revised BSD License text as described in Section 4.e of the Trust Legal Provisions and are provided without warranty as described in the Revised BSD License.¶
The ubiquity of the Internet makes it an ideal platform for measurement of largescale phenomena, whether public health trends or the behavior of computer systems at scale. There is substantial overlap, however, between information that is valuable to measure and information that users consider private.¶
For example, consider an application that provides health information to users. The operator of an application might want to know which parts of their application are used most often, as a way to guide future development of the application. Specific users' patterns of usage, though, could reveal sensitive things about them, such as which users are researching a given health condition.¶
In many situations, the measurement collector is only interested in aggregate statistics, e.g., which portions of an application are most used or what fraction of people have experienced a given disease. Thus systems that provide aggregate statistics while protecting individual measurements can deliver the value of the measurements while protecting users' privacy.¶
Most prior approaches to this problem fall under the rubric of "differential privacy (DP)" [Dwo06]. Roughly speaking, a data aggregation system that is differentially private ensures that the degree to which any individual measurement influences the value of the aggregate result can be precisely controlled. For example, in systems like RAPPOR [EPK14], each user samples noise from a wellknown 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 privacypreserving measurement system is that of secure multiparty computation (MPC): No participant in the protocol should learn anything about an individual input beyond what it can deduce from the aggregate. In this document, we describe Verifiable Distributed Aggregation Functions (VDAFs) as a general class of protocols that achieve this goal.¶
VDAF schemes achieve their privacy goal by distributing the computation of the aggregate among a number of noncolluding aggregation servers. As long as a subset of the servers executes the protocol honestly, VDAFs guarantee that no input is ever accessible to any party besides the client that submitted it. At the same time, VDAFs are "verifiable" in the sense that malformed inputs that would otherwise garble the output of the computation can be detected and removed from the set of input measurements.¶
In addition to these MPCstyle security goals, VDAFs can be composed with various mechanisms for differential privacy, thereby providing the added assurance that the aggregate result itself does not leak too much information about any one measurement.¶
TODO(issue #94) Provide guidance for local and central DP and point to it here.¶
The cost of achieving these security properties is the need for multiple servers to participate in the protocol, and the need to ensure they do not collude to undermine the VDAF's privacy guarantees. Recent implementation experience has shown that practical challenges of coordinating multiple servers can be overcome. The Prio system [CGB17] (essentially a VDAF) has been deployed in systems supporting hundreds of millions of users: The Mozilla Origin Telemetry project [OriginTelemetry] and the Exposure Notification Private Analytics collaboration among the Internet Security Research Group (ISRG), Google, Apple, and others [ENPA].¶
The VDAF abstraction laid out in Section 5 represents a class of multiparty protocols for privacypreserving measurement proposed in the literature. These protocols vary in their operational and security considerations, sometimes in subtle but consequential ways. This document therefore has two important goals:¶
Providing higherlevel protocols like [DAP] with a simple, uniform interface for accessing privacypreserving measurement schemes, and documenting relevant operational and security bounds for that interface:¶
This document also specifies two concrete VDAF schemes, each based on a protocol from the literature.¶
The aforementioned Prio system [CGB17] allows for the privacypreserving computation of a variety aggregate statistics. The basic idea underlying Prio is fairly simple:¶
The difficult part of this system is ensuring that the servers hold shares of a valid input, e.g., the input is an integer in a specific range. Thus Prio specifies a multiparty protocol for accomplishing this task.¶
In Section 7 we describe Prio3, a VDAF that follows the same overall framework as the original Prio protocol, but incorporates techniques introduced in [BBCGGI19] that result in significant performance gains.¶
More recently, Boneh et al. [BBCGGI21] described a protocol called Poplar
for solving the t
heavyhitters problem in a privacypreserving manner. Here
each client holds a bitstring of length n
, and the goal of the aggregation
servers is to compute the set of inputs that occur at least t
times. The
core primitive used in their protocol is a specialized Distributed Point
Function (DPF) [GI14] that allows the servers to "query" their DPF shares on
any bitstring of length shorter than or equal to n
. As a result of this
query, each of the servers has an additive share of a bit indicating whether
the string is a prefix of the client's input. The protocol also specifies a
multiparty computation for verifying that at most one string among a set of
candidates is a prefix of the client's input.¶
In Section 8 we describe a VDAF called Poplar1 that implements this functionality.¶
Finally, perhaps the most complex aspect of schemes like Prio3 and Poplar1 is the process by which the clientgenerated measurements are prepared for aggregation. Because these constructions are based on secret sharing, the servers will be required to exchange some amount of information in order to verify the measurement is valid and can be aggregated. Depending on the construction, this process may require multiple round trips over the network.¶
There are applications in which this verification step may not be necessary, e.g., when the client's software is run a trusted execution environment. To support these applications, this document also defines Distributed Aggregation Functions (DAFs) as a simpler class of protocols that aim to provide the same privacy guarantee as VDAFs but fall short of being verifiable.¶
OPEN ISSUE Decide if we should give one or two example DAFs. There are natural variants of Prio3 and Poplar1 that might be worth describing.¶
The remainder of this document is organized as follows: Section 3 gives a brief overview of DAFs and VDAFs; Section 4 defines the syntax for DAFs; Section 5 defines the syntax for VDAFs; Section 6 defines various functionalities that are common to our constructions; Section 7 describes the Prio3 construction; Section 8 describes the Poplar1 construction; and Section 9 enumerates the security considerations for VDAFs.¶
(*) Indicates a change that breaks wire compatibility with the previous draft.¶
03:¶
02:¶
01:¶
prep_next()
to
prep_shares_to_prep()
. (*)¶
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and "OPTIONAL" in this document are to be interpreted as described in BCP 14 [RFC2119] [RFC8174] when, and only when, they appear in all capitals, as shown here.¶
Algorithms in this document are written in Python 3. Type hints are used to define input and output types. A fatal error in a program (e.g., failure to parse one of the function parameters) is usually handled by raising an exception.¶
A variable with type Bytes
is a byte string. This document defines several
bytestring constants. When comprised of printable ASCII characters, they are
written as Python 3 bytestring literals (e.g., b'some constant string'
).¶
A global constant VERSION
is defined, which algorithms are free to use as
desired. Its value SHALL be b'vdaf03'
.¶
This document describes algorithms for multiparty computations in which the parties typically communicate over a network. Wherever a quantity is defined that must be be transmitted from one party to another, this document prescribes a particular encoding of that quantity as a byte string.¶
OPEN ISSUE It might be better to not be prescriptive about how quantities are encoded on the wire. See issue #58.¶
Some common functionalities:¶
zeros(len: Unsigned) > Bytes
returns an array of zero bytes. The length of
output
MUST be len
.¶
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 nonnegative integer to a byte string and convert a byte string to a
nonnegative 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.¶
In a DAF or VDAFbased private measurement system, we distinguish three types of actors: Clients, Aggregators, and Collectors. The overall flow of the measurement process is as follows:¶
The Aggregators convert their input shares into "output shares".¶
Aggregators are a new class of actor relative to traditional measurement systems where clients submit measurements to a single server. They are critical for both the privacy properties of the system and, in the case of VDAFs, the correctness of the measurements obtained. The privacy properties of the system are assured by noncollusion among Aggregators, and Aggregators are the entities that perform validation of Client measurements. Thus clients trust Aggregators not to collude (typically it is required that at least one Aggregator is honest), and Collectors trust Aggregators to correctly run the protocol.¶
Within the bounds of the noncollusion requirements of a given (V)DAF instance, it is possible for the same entity to play more than one role. For example, the Collector could also act as an Aggregator, effectively using the other Aggregator(s) to augment a basic clientserver protocol.¶
In this document, we describe the computations performed by the actors in this system. It is up to the higherlevel protocol making use of the (V)DAF to arrange for the required information to be delivered to the proper actors in the proper sequence. In general, we assume that all communications are confidential and mutually authenticated, with the exception that Clients submitting measurements may be anonymous.¶
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 and Preparation are done once per measurement. Aggregation and Unsharding are done over a batch of measurements (more precisely, over the recovered output shares).¶
A concrete DAF specifies an algorithm for the computation needed in each of these stages. The interface of each algorithm is defined in the remainder of this section. In addition, a concrete DAF defines the associated constants and types enumerated in the following table.¶
Parameter  Description 

ID

Algorithm identifier for this DAF. 
SHARES

Number of input shares into which each measurement is sharded 
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.¶
Each DAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 16.¶
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 inputdistribution 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
.¶
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.¶
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.¶
For simplicity, we have written this algorithm in a "oneshot" form, where all output shares for a batch are provided at the same time. Many DAFs may also support a "streaming" form, where shares are processed one at a time.¶
OPEN ISSUE It may be worthwhile to explicitly define the "streaming" API. See issue#47.¶
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.¶
QUESTION Maybe the aggregation algorithms should be randomized in order to allow the Aggregators (or the Collector) to add noise for differential privacy. (See the security considerations of [DAP].) Or is this outofscope of this document? See https://github.com/ietfwgppm/ppmspecification/issues/19.¶
Securely executing a DAF involves emulating the following procedure.¶
The inputs to this procedure are the same as the aggregation function computed by the DAF: An aggregation parameter and a sequence of measurements. The procedure prescribes how a DAF is executed in a "benign" environment in which there is no adversary and the messages are passed among the protocol participants over secure pointtopoint channels. In reality, these channels need to be instantiated by some "wrapper protocol", such as [DAP], that realizes these channels using suitable cryptographic mechanisms. Moreover, some fraction of the Aggregators (or Clients) may be malicious and diverge from their prescribed behaviors. Section 9 describes the execution of the DAF in various adversarial environments and what properties the wrapper protocol needs to provide in each.¶
Like DAFs described in the previous section, a VDAF scheme is used to compute a particular aggregation function over a set of Clientgenerated measurements. Evaluation of a VDAF involves the same four stages as for DAFs: Sharding, Preparation, Aggregation, and Unsharding. However, the Preparation stage will require interaction among the Aggregators in order to facilitate verifiability of the computation's correctness. Accommodating this interaction will require syntactic changes.¶
Overall execution of a VDAF comprises the following stages:¶
In contrast to DAFs, the Preparation stage for VDAFs now performs an additional task: Verification of the validity of the recovered output shares. This process ensures that aggregating the output shares will not lead to a garbled aggregate result.¶
The remainder of this section defines the VDAF interface. The attributes are listed in Table 2 are defined by each concrete VDAF.¶
Parameter  Description 

ID

Algorithm identifier for this VDAF. 
VERIFY_KEY_SIZE

Size (in bytes) of the verification key (Section 5.2) 
ROUNDS

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

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

Type of each measurement 
AggParam

Type of aggregation parameter 
Prep

State of each Aggregator during Preparation (Section 5.2) 
OutShare

Type of each output share 
AggResult

Type of the aggregate result 
Similarly to DAFs (see {[secdaf}}), any output of a VDAF method that must be transmitted from one party to another is treated as an opaque byte string. All other quantities are given a concrete type.¶
OPEN ISSUE It might be cleaner to define a type for each value, then have that type implement an encoding where necessary. See issue#58.¶
Each VDAF is identified by a unique, 32bit integer ID
. Identifiers for each
(V)DAF specified in this document are defined in Table 16.¶
Sharding is syntactically identical to DAFs (cf. Section 4.1):¶
Vdaf.measurement_to_input_shares(measurement: Measurement) > (Bytes,
Vec[Bytes])
is the randomized inputdistribution 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 Section 7). The length of the output vector MUST be
SHARES
.¶
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 "preparationmessage shares".) Finally, the preparation message is
distributed to the Aggregators to begin the next round.¶
An Aggregator begins the first round with its input share and it begins each subsequent round with the previous preparation message. Its output in the last round is its output share and its output in each of the preceding rounds is a preparationmessage share.¶
This process involves a value called the "aggregation parameter" used to map the input shares to output shares. The Aggregators need to agree on this parameter before they can begin preparing inputs for aggregation.¶
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 preparationstate initialization algorithm run by each
Aggregator to begin processing its input share into an output share. Its
inputs are the shared verification key (verify_key
), the Aggregator's unique
identifier (agg_id
), the aggregation parameter (agg_param
), the nonce
provided by the environment (nonce
, see Figure 7), 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 Section 8 for details.)¶
Vdaf.prep_next(prep: Prep, inbound: Optional[Bytes]) > Union[Tuple[Prep,
Bytes], OutShare]
is the deterministic preparationstate update algorithm run
by each Aggregator. It updates the Aggregator's preparation state (prep
) and
returns either its next preparation state and its message share for the
current round or, if this is the last round, its output share. An exception is
raised if a valid output share could not be recovered. The input of this
algorithm is the inbound preparation message or, if this is the first round,
None
.¶
Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) >
Bytes
is the deterministic preparationmessage preprocessing algorithm. It
combines the preparationmessage shares generated by the Aggregators in the
previous round into the preparation message consumed by each in the next
round.¶
In effect, each Aggregator moves through a linear state machine with ROUNDS+1
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (ROUNDS
), a VDAF instance
defines the state of the Aggregator after each round.¶
TODO Consider how to bake this "linear state machine" condition into the syntax. Given that Python 3 is used as our pseudocode, it's easier to specify the preparation state using a class.¶
The preparationstate update accomplishes two tasks: recovery of output shares from the input shares and ensuring that the recovered output shares are valid. The abstraction boundary is drawn so that an Aggregator only recovers an output share if it is deemed valid (at least, based on the Aggregator's view of the protocol). Another way to draw this boundary would be to have the Aggregators recover output shares first, then verify that they are valid. However, this would allow the possibility of misusing the API by, say, aggregating an invalid output share. Moreover, in protocols like Prio+ [AGJOP21] based on oblivious transfer, it is necessary for the Aggregators to interact in order to recover aggregatable output shares at all.¶
Note that it is possible for a VDAF to specify ROUNDS == 0
, in which case each
Aggregator runs the preparationstate update algorithm once and immediately
recovers its output share without interacting with the other Aggregators.
However, most, if not all, constructions will require some amount of interaction
in order to ensure validity of the output shares (while also maintaining
privacy).¶
OPEN ISSUE accommodating 0round VDAFs may require syntax changes if, for example, public keys are required. On the other hand, we could consider defining this class of schemes as a different primitive. See issue#77.¶
VDAF Aggregation is identical to DAF Aggregation (cf. Section 4.3):¶
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 Figure 3. Here again, we have the aggregation algorithm in a "oneshot" form, where all shares for a batch are provided at the same time. VDAFs typically also support a "streaming" form, where shares are processed one at a time.¶
VDAF Unsharding is identical to DAF Unsharding (cf. Section 4.4):¶
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.¶
Secure execution of a VDAF involves simulating the following procedure.¶
The inputs to this algorithm are the aggregation parameter, a list of measurements, and a nonce for each measurement. This document does not specify how the nonces are chosen, but security requires that the nonces be unique. See Section 9 for details. As explained in Section 4.5, the secure execution of a VDAF requires the application to instantiate secure channels between each of the protocol participants.¶
This section describes the primitives that are common to the VDAFs specified in this document.¶
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.¶
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.¶
Some VDAFs require fields that are suitable for efficient computation of the discrete Fourier transform, as this allows for fast polynomial interpolation. (One example is Prio3 (Section 7) when instantiated with the generic FLP of Section 7.3.3.) Specifically, a field is said to be "FFTfriendly" if, in addition to satisfying the interface described in Section 6.1, it implements the following method:¶
Field.gen() > Field
returns the generator of a large subgroup of the
multiplicative group. To be FFTfriendly, the order of this subgroup NUST be a
power of 2. In addition, the size of the subgroup dictates how large
interpolated polynomials can be. It is RECOMMENDED that a generator is chosen
with order at least 2^20
.¶
FFTfriendly fields also define the following parameter:¶
GEN_ORDER: Unsigned
is the order of a multiplicative subgroup generated by
Field.gen()
.¶
The tables below define finite fields used in the remainder of this document.¶
Parameter  Value 

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

MODULUS  2^66 * 4611686018427387897 + 1 
ENCODED_SIZE  16 
Generator  7^4611686018427387897 
GEN_ORDER  2^66 
OPEN ISSUE We currently use bigendian for encoding field elements. However,
for implementations of GF(2^25519)
, little endian is more common. See
issue#90.¶
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).¶
OPEN ISSUE Phillipp points out that a fixedkey mode of AES may be more performant (https://eprint.iacr.org/2019/074.pdf). See issue#32.¶
TODO(issue #106) Decide if it's safe to model this construction as a random
oracle. PrgAes128.derive_seed()
is used for the FiatShamir heuristic in
Prio3 (Section 7). A fixedkey is used for this step (the allzero string). A
reasonable starting point would be to model AES as an ideal cipher.¶
Our first construction, PrgAes128
, converts a blockcipher, namely AES128,
into a PRG. Seed expansion involves two steps. In the first step, CMAC
[RFC4493] is applied to the seed and info string to get a fresh key. In the
second step, the fresh key is used in CTRmode to produce a key stream for
generating the output. A fixed initialization vector (IV) is used.¶
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:¶
At a high level, Prio3 distributes this computation as follows. Each Client first shards its measurement by first encoding it, then splitting the vector into secret shares and sending a share to each Aggregator. Next, in the preparation phase, the Aggregators carry out a multiparty computation to determine if their shares correspond to a valid 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".¶
Conceptually, an FLP is a twoparty protocol executed by a prover and a verifier. In actual use, however, the prover's computation is carried out by the Client, and the verifier's computation is distributed among the Aggregators. The Client generates a "proof" of its input's validity and distributes shares of the proof to the Aggregators. Each Aggregator then performs some a computation on its input share and proof share locally and sends the result to the other Aggregators. Combining the exchanged messages allows each Aggregator to decide if it holds a share of a valid input. (See Section 7.2 for details.)¶
As usual, we will describe the interface implemented by a concrete FLP in terms
of an abstract base class Flp
that specifies the set of methods and parameters
a concrete FLP must provide.¶
The parameters provided by a concrete FLP are listed in Table 6.¶
Parameter  Description 

PROVE_RAND_LEN

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

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

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

Length of the encoded measurement (Section 7.1.1) 
OUTPUT_LEN

Length of the aggregatable output (Section 7.1.1) 
PROOF_LEN

Length of the proof 
VERIFIER_LEN

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

Type of the measurement 
AggResult

Type of the aggregate result 
Field

As defined in (Section 6.1) 
An FLP specifies the following algorithms for generating and verifying proofs of validity (encoding is described below in Section 7.1.1):¶
Flp.prove(input: Vec[Field], prove_rand: Vec[Field], joint_rand: Vec[Field])
> Vec[Field]
is the deterministic proofgeneration algorithm run by the
prover. Its inputs are the encoded input, the "prover randomness"
prove_rand
, and the "joint randomness" joint_rand
. The prover randomness is
used only by the prover, but the joint randomness is shared by both the prover
and verifier.¶
Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field],
joint_rand: Vec[Field], num_shares: Unsigned) > Vec[Field]
is the
querygeneration algorithm run by the verifier. This is used to "query" the
input and proof. The result of the query (i.e., the output of this function)
is called the "verifier message". In addition to the input and proof, this
algorithm takes as input the query randomness query_rand
and the joint
randomness joint_rand
. The former is used only by the verifier. num_shares
specifies how many input and proof shares were generated.¶
Flp.decide(verifier: Vec[Field]) > Bool
is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the input from which it was generated is
valid.¶
Our application requires that the FLP is "fully linear" in the sense defined in [BBCGGI19]. As a practical matter, what this property implies is that, when run on a share of the input and proof, the querygeneration algorithm outputs a share of the verifier message. Furthermore, the "strong zeroknowledge" property of the FLP system ensures that the verifier message reveals nothing about the input's validity. Therefore, to decide if an input is valid, the Aggregators will run the querygeneration algorithm locally, exchange verifier shares, combine them to recover the verifier message, and run the decision algorithm.¶
The querygeneration algorithm includes a parameter num_shares
that specifies
the number of shares of the input and proof that were generated. If these data
are not secret shared, then num_shares == 1
. This parameter is useful for
certain FLP constructions. For example, the FLP in Section 7.3 is defined in
terms of an arithmetic circuit; when the circuit contains constants, it is
sometimes necessary to normalize those constants to ensure that the circuit's
output, when run on a valid input, is the same regardless of the number of
shares.¶
An FLP is executed by the prover and verifier as follows:¶
The proof system is constructed so that, if input
is a valid input, then
run_flp(Flp, input, 1)
always returns True
. On the other hand, if input
is
invalid, then as long as joint_rand
and query_rand
are generated uniform
randomly, the output is False
with overwhelming probability.¶
We remark that [BBCGGI19] defines a much larger class of fully linear proof systems than we consider here. In particular, what is called an "FLP" here is called a 1.5round, publiccoin, interactive oracle proof system in their paper.¶
The type of measurement being aggregated is defined by the FLP. Hence, the FLP also specifies a method of encoding raw measurements as a vector of field elements:¶
Flp.encode(measurement: Measurement) > Vec[Field]
encodes a raw measurement
as a vector of field elements. The return value MUST be of length INPUT_LEN
.¶
For some FLPs, the encoded input also includes redundant field elements that are
useful for checking the proof, but which are not needed after the proof has been
checked. An example is the "integer sum" data type from [CGB17] in which an
integer in range [0, 2^k)
is encoded as a vector of k
field elements (this
type is also defined in Section 7.4.2). After consuming this vector,
all that is needed is the integer it represents. Thus the FLP defines an
algorithm for truncating the input to the length of the aggregated output:¶
Flp.truncate(input: Vec[Field]) > Vec[Field]
maps an encoded input to an
aggregatable output. The length of the input MUST be INPUT_LEN
and the length
of the output MUST be OUTPUT_LEN
.¶
Once the aggregate shares have been computed and combined together, their sum can be converted into the aggregate result. This could be a projection from the FLP's field to the integers, or it could include additional postprocessing.¶
Flp.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult
maps a sum of aggregate shares to an aggregate result. The length of the
input MUST be OUTPUT_LEN
. num_measurements
is the number of measurements
that contributed to the aggregated output.¶
We remark that, taken together, these three functionalities correspond roughly to the notion of "Affineaggregatable encodings (AFEs)" from [CGB17].¶
This section specifies Prio3
, an implementation of the Vdaf
interface
(Section 5). It has two generic parameters: an Flp
(Section 7.1) and a Prg
(Section 6.2). The associated constants and types required by the Vdaf
interface
are defined in Table 7. The methods required for sharding, preparation,
aggregation, and unsharding are described in the remaining subsections.¶
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

This section describes the process of recovering output shares from the input shares. The highlevel idea is that each Aggregator first queries its input and proof share locally, then exchanges its verifier share with the other Aggregators. The verifier shares are then combined into the verifier message, which is used to decide whether to accept.¶
In addition, the Aggregators must ensure that they have all used the same joint randomness for the querygeneration algorithm. The joint randomness is generated by a PRG seed. Each Aggregator derives a "part" of this seed from its input share and the "blind" generated by the client. The seed is derived by hashing together the parts, so before running the querygeneration algorithm, it must first gather the parts derived by the other Aggregators.¶
In order to avoid extra round of communication, the Client sends each Aggregator a "hint" consisting of the other Aggregators' parts of the joint randomness seed. This leaves open the possibility that the Client cheated by, say, forcing the Aggregators to use joint randomness that biases the proof check procedure some way in its favor. To mitigate this, the Aggregators also check that they have all computed the same joint randomness seed before accepting their output shares. To do so, they exchange their parts of the joint randomness along with their verifier shares.¶
The algorithms required for preparation are defined as follows. These algorithms make use of methods defined in Section 7.2.5.¶
Aggregating a set of output shares is simply a matter of adding up the vectors elementwise.¶
To unshard a set of aggregate shares, the Collector first adds up the vectors elementwise. It then converts each element of the vector into an integer.¶
This section describes an FLP based on the construction from in [BBCGGI19], Section 4.2. We begin in Section 7.3.1 with an overview of their proof system and the extensions to their proof system made here. The construction is specified in Section 7.3.3.¶
OPEN ISSUE We're not yet sure if specifying this generalpurpose FLP is desirable. It might be preferable to specify specialized FLPs for each data type that we want to standardize, for two reasons. First, clear and concise specifications are likely easier to write for specialized FLPs rather than the general one. Second, we may end up tailoring each FLP to the measurement type in a way that improves performance, but breaks compatibility with the generalpurpose FLP.¶
In any case, we can't make this decision until we know which data types to standardize, so for now, we'll stick with the generalpurpose construction. The reference implementation can be found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc.¶
OPEN ISSUE Chris Wood points out that the this section reads more like a paper than a standard. Eventually we'll want to work this into something that is readily consumable by the CFRG.¶
In the proof system of [BBCGGI19], validity is defined via an arithmetic circuit evaluated over the input: If the circuit output is zero, then the input is deemed valid; otherwise, if the circuit output is nonzero, then the input is deemed invalid. Thus the goal of the proof system is merely to allow the verifier to evaluate the validity circuit over the input. For our application (Section 7), this computation is distributed among multiple Aggregators, each of which has only a share of the input.¶
Suppose for a moment that the validity circuit C
is affine, meaning its only
operations are addition and multiplicationbyconstant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
nonconstant. Then to decide if an input x
is valid, each Aggregator could
evaluate C
on its share of x
locally, broadcast the output share to its
peers, then combine the output shares locally to recover C(x)
. This is true
because for any SHARES
way secret sharing of x
it holds that¶
C(x_shares[0] + ... + x_shares[SHARES1]) = C(x_shares[0]) + ... + C(x_shares[SHARES1])¶
(Note that, for this equality to hold, it may be necessary to scale any
constants in the circuit by SHARES
.) However this is not the case if C
is
notaffine (i.e., it contains at least one multiplication gate whose operands
are nonconstant). In the proof system of [BBCGGI19], the proof is designed to
allow the (distributed) verifier to compute the nonaffine operations using only
linear operations on (its share of) the input and proof.¶
To make this work, the proof system is restricted to validity circuits that
exhibit a special structure. Specifically, an arithmetic circuit with "Ggates"
(see [BBCGGI19], Definition 5.2) is composed of affine gates and any number of
instances of a distinguished gate G
, which may be nonaffine. We will refer to
this class of circuits as 'gadget circuits' and to G
as the "gadget".¶
As an illustrative example, consider a validity circuit C
that recognizes the
set L = set([0], [1])
. That is, C
takes as input a length1 vector x
and
returns 0 if x[0]
is in [0,2)
and outputs something else otherwise. This
circuit can be expressed as the following degree2 polynomial:¶
C(x) = (x[0]  1) * x[0] = x[0]^2  x[0]¶
This polynomial recognizes L
because x[0]^2 = x[0]
is only true if x[0] ==
0
or x[0] == 1
. Notice that the polynomial involves a nonaffine operation,
x[0]^2
. In order to apply [BBCGGI19], Theorem 4.3, the circuit needs to be
rewritten in terms of a gadget that subsumes this nonaffine operation. For
example, the gadget might be multiplication:¶
Mul(left, right) = left * right¶
The validity circuit can then be rewritten in terms of Mul
like so:¶
C(x[0]) = Mul(x[0], x[0])  x[0]¶
The proof system of [BBCGGI19] allows the verifier to evaluate each instance
of the gadget (i.e., Mul(x[0], x[0])
in our example) using a linear function
of the input and proof. The proof is constructed roughly as follows. Let C
be
the validity circuit and suppose the gadget is arityL
(i.e., it has L
input
wires.). Let wire[j1,k1]
denote the value of the j
th wire of the k
th
call to the gadget during the evaluation of C(x)
. Suppose there are M
such
calls and fix distinct field elements alpha[0], ..., alpha[M1]
. (We will
require these points to have a special property, as we'll discuss in
Section 7.3.1.1; but for the moment it is only important
that they are distinct.)¶
The prover constructs from wire
and alpha
a polynomial that, when evaluated
at alpha[k1]
, produces the output of the k
th call to the gadget. Let us
call this the "gadget polynomial". Polynomial evaluation is linear, which means
that, in the distributed setting, the Client can disseminate additive shares of
the gadget polynomial that the Aggregators then use to compute additive shares
of each gadget output, allowing each Aggregator to compute its share of C(x)
locally.¶
There is one more wrinkle, however: It is still possible for a malicious prover
to produce a gadget polynomial that would result in C(x)
being computed
incorrectly, potentially resulting in an invalid input being accepted. To
prevent this, the verifier performs a probabilistic test to check that the
gadget polynomial is wellformed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in Section 7.3.3.¶
The FLP described in the next section extends the proof system [BBCGGI19], Section 4.2 in three ways.¶
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the input shares in Prio3;
see Section 7.2). This allows for circuit optimizations that trade a
small soundness error for a shorter proof. For example, consider a circuit that
recognizes the set of lengthN
vectors for which each element is either one or
zero. A deterministic circuit could be constructed for this language, but it
would involve a large number of multiplications that would result in a large
proof. (See the discussion in [BBCGGI19], Section 5.2 for details). A much
shorter proof can be constructed for the following randomized circuit:¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1])¶
(Note that this is a special case of [BBCGGI19], Theorem 5.2.) Here inp
is
the lengthN
input and r
is a random field element. The gadget circuit
Range2
is the "rangecheck" polynomial described above, i.e., Range2(x) = x^2 
x
. The idea is that, if inp
is valid (i.e., each inp[j]
is in [0,2)
),
then the circuit will evaluate to 0 regardless of the value of r
; but if
inp[j]
is not in [0,2)
for some j
, the output will be nonzero with high
probability.¶
The second extension implemented by our FLP allows the validity circuit to
contain multiple gadget types. (This generalization was suggested in
[BBCGGI19], Remark 4.5.) For example, the following circuit is allowed, where
Mul
and Range2
are the gadgets defined above (the input has length N+1
):¶
C(inp, r) = r * Range2(inp[0]) + ... + r^N * Range2(inp[N1]) + \ 2^0 * inp[0] + ... + 2^(N1) * inp[N1]  \ Mul(inp[N], inp[N])¶
Finally, [BBCGGI19], Theorem 4.3 makes no restrictions on the choice of the
fixed points alpha[0], ..., alpha[M1]
, other than to require that the points
are distinct. In this document, the fixed points are chosen so that the gadget
polynomial can be constructed efficiently using the CooleyTukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFTfriendly" as defined in Section 6.1.2.¶
The FLP described in Section 7.3.3 is defined in terms of a
validity circuit Valid
that implements the interface described here.¶
A concrete Valid
defines the following parameters:¶
Parameter  Description 

GADGETS

A list of gadgets 
GADGET_CALLS

Number of times each gadget is called 
INPUT_LEN

Length of the input 
OUTPUT_LEN

Length of the aggregatable output 
JOINT_RAND_LEN

Length of the random input 
Measurement

The type of measurement 
AggResult

Type of the aggregate result 
Field

An FFTfriendly finite field as defined in Section 6.1.2 
Each gadget G
in GADGETS
defines a constant DEGREE
that specifies the
circuit's "arithmetic degree". This is defined to be the degree of the
polynomial that computes it. For example, the Mul
circuit in
Section 7.3.1 is defined by the polynomial Mul(x) = x * x
, which
has degree 2
. Hence, the arithmetic degree of this gadget is 2
.¶
Each gadget also defines a parameter ARITY
that specifies the circuit's arity
(i.e., the number of input wires).¶
A concrete Valid
provides the following methods for encoding a measurement as
an input vector, truncating an input vector to the length of an aggregatable
output, and converting an aggregated output to an aggregate result:¶
Valid.encode(measurement: Measurement) > Vec[Field]
returns a vector of
length INPUT_LEN
representing a measurement.¶
Valid.truncate(input: Vec[Field]) > Vec[Field]
returns a vector of length
OUTPUT_LEN
representing an aggregatable output.¶
Valid.decode(output: Vec[Field], num_measurements: Unsigned) > AggResult
returns an aggregate result.¶
Finally, the following class methods are derived for each concrete Valid
:¶
This section specifies FlpGeneric
, an implementation of the Flp
interface
(Section 7.1). It has as a generic parameter a validity circuit Valid
implementing
the interface defined in Section 7.3.2.¶
NOTE A reference implementation can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/flp_generic.sage.¶
The FLP parameters for FlpGeneric
are defined in Table 9. The
required methods for generating the proof, generating the verifier, and deciding
validity are specified in the remaining subsections.¶
In the remainder, we let [n]
denote the set {1, ..., n}
for positive integer
n
. We also define the following constants:¶
Parameter  Value 

PROVE_RAND_LEN

Valid.prove_rand_len() (see Section 7.3.2) 
QUERY_RAND_LEN

Valid.query_rand_len() (see Section 7.3.2) 
JOINT_RAND_LEN

Valid.JOINT_RAND_LEN

INPUT_LEN

Valid.INPUT_LEN

OUTPUT_LEN

Valid.OUTPUT_LEN

PROOF_LEN

Valid.proof_len() (see Section 7.3.2) 
VERIFIER_LEN

Valid.verifier_len() (see Section 7.3.2) 
Measurement

Valid.Measurement

Field

Valid.Field

On input inp
, prove_rand
, and joint_rand
, the proof is computed as
follows:¶
i
in [H]
create an empty table wire_i
.¶
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.¶
Valid
on input of inp
and joint_rand
, recording the inputs of
each gadget in the corresponding table. Specifically, for every i
in [H]
,
set wire_i[j1,k1]
to the value on the j
th wire into the k
th call to
gadget G_i
.¶
Compute the "wire polynomials". That is, for every i
in [H]
and j
in
[L_i]
, construct poly_wire_i[j1]
, the j
th wire polynomial for the
i
th gadget, as follows:¶
w = [seed_i[j1], wire_i[j1,0], ..., wire_i[j1,M_i1]]
.¶
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.¶
poly_wire_i[j1]
be the lowest degree polynomial for which
poly_wire_i[j1](alpha_i^k) == padded_w[k]
for all k
in [P_i]
.¶
Compute the "gadget polynomials". That is, for every i
in [H]
:¶
poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i1])
. That
is, evaluate the circuit G_i
on the wire polynomials for the i
th
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]
.¶
On input of inp
, proof
, query_rand
, and joint_rand
, the verifier message
is generated as follows:¶
i
in [H]
create an empty table wire_i
.¶
proof
into the subvectors seed_1
, coeff_1
, ..., seed_H
,
coeff_H
defined in Section 7.3.3.1.¶
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 k
th call to G_i
by evaluating
poly_gadget_i(alpha_i^k)
. Let v
denote the output of the circuit
evaluation.¶
Compute the tests for wellformedness of the gadget polynomials. That is, for
every i
in [H]
:¶
The verifier message is the vector verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H]
.¶
This section specifies instantiations of Prio3 for various measurement types.
Each uses FlpGeneric
as the FLP (Section 7.3) and is determined by a
validity circuit (Section 7.3.2) and a PRG (Section 6.2). Test vectors for
each can be found in Appendix "Test Vectors".¶
NOTE Reference implementations of each of these VDAFs can be found in https://github.com/cfrg/draftirtfcfrgvdaf/blob/main/poc/vdaf_prio3.sage.¶
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
(Section 6.2.1) as its PRG. Its validity
circuit, denoted Count
, uses Field64
(Table 3) as its finite field. Its
gadget, denoted Mul
, is the degree2, arity2 gadget defined as¶
def Mul(x, y): return x * y¶
The validity circuit is defined as¶
def Count(inp: Vec[Field64]): return Mul(inp[0], inp[0])  inp[0]¶
The measurement is encoded and decoded as a singleton vector in the natural way. The parameters for this circuit are summarized below.¶
Parameter  Value 

GADGETS

[Mul]

GADGET_CALLS

[1]

INPUT_LEN

1

OUTPUT_LEN

1

JOINT_RAND_LEN

0

Measurement

Unsigned , in range [0,2)

AggResult

Unsigned

Field

Field64 (Table 3) 
The next instance of Prio3 supports summing of integers in a predetermined
range. Each measurement is an integer in range [0, 2^bits)
, where bits
is an
associated parameter.¶
This instance of Prio3 uses PrgAes128
(Section 6.2.1) as its PRG.
Its validity circuit, denoted Sum
, uses Field128
(Table 4) as its
finite field. The measurement is encoded as a lengthbits
vector of field
elements, where the l
th element of the vector represents the l
th bit of the
summand:¶
def encode(Sum, measurement: Integer): if 0 > measurement or measurement >= 2^Sum.INPUT_LEN: raise ERR_INPUT encoded = [] for l in range(Sum.INPUT_LEN): encoded.append(Sum.Field((measurement >> l) & 1)) return encoded def truncate(Sum, inp): decoded = Sum.Field(0) for (l, b) in enumerate(inp): w = Sum.Field(1 << l) decoded += w * b return [decoded] def decode(Sum, output, _num_measurements): return output[0].as_unsigned()¶
The validity circuit checks that the input consists of ones and zeros. Its
gadget, denoted Range2
, is the degree2, arity1 gadget defined as¶
def Range2(x): return x^2  x¶
The validity circuit is defined as¶
def Sum(inp: Vec[Field128], joint_rand: Vec[Field128]): out = Field128(0) r = joint_rand[0] for x in inp: out += r * Range2(x) r *= joint_rand[0] return out¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[bits]

INPUT_LEN

bits

OUTPUT_LEN

1

JOINT_RAND_LEN

1

Measurement

Unsigned , in range [0, 2^bits)

AggResult

Unsigned

Field

Field128 (Table 4) 
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
(Section 6.2.1) as its PRG. Its
validity circuit, denoted Histogram
, uses Field128
(Table 4) as its
finite field. The measurement is encoded as a onehot vector representing the
bucket into which the measurement falls (let bucket
denote a sequence of
monotonically increasing integers):¶
def encode(Histogram, measurement: Integer): boundaries = buckets + [Infinity] encoded = [Field128(0) for _ in range(len(boundaries))] for i in range(len(boundaries)): if measurement <= boundaries[i]: encoded[i] = Field128(1) return encoded def truncate(Histogram, inp: Vec[Field128]): return inp def decode(Histogram, output: Vec[Field128], _num_measurements): return [bucket_count.as_unsigned() for bucket_count in output]¶
The validity circuit uses Range2
(see Section 7.4.2) as its single gadget. It
checks for onehotness in two steps, as follows:¶
def Histogram(inp: Vec[Field128], joint_rand: Vec[Field128], num_shares: Unsigned): # Check that each bucket is one or zero. range_check = Field128(0) r = joint_rand[0] for x in inp: range_check += r * Range2(x) r *= joint_rand[0] # Check that the buckets sum to 1. sum_check = Field128(1) * Field128(num_shares).inv() for b in inp: sum_check += b out = joint_rand[1] * range_check + \ joint_rand[1]^2 * sum_check return out¶
Note that this circuit depends on the number of shares into which the input is sharded. This is provided to the FLP by Prio3.¶
Parameter  Value 

GADGETS

[Range2]

GADGET_CALLS

[buckets + 1]

INPUT_LEN

buckets + 1

OUTPUT_LEN

buckets + 1

JOINT_RAND_LEN

2

Measurement

Integer

AggResult

Vec[Unsigned]

Field

Field128 (Table 4) 
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.¶
0
and
1
.¶
H
denote the set of prefixes that occurred at least t
times. If the
prefixes all have length BITS
, then H
is the set of t
heavyhitters.
Otherwise compute the next set of candidate prefixes 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 nonzero 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 "onehot" vector, i.e., a vector that is zero everywhere except for one element, which is equal to one.¶
The remainder of this section is structured as follows. IDPFs are defined in Section 8.1; a concrete instantiation is given Section 8.3. The Poplar1 VDAF is defined in Section 8.2 in terms of a generic IDPF. Finally, a concrete instantiation of Poplar1 is specified in Section 8.4; test vectors can be found in Appendix "Test Vectors".¶
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 lengthL
prefix of positive integer 0 <= y < 2^BITS
if LSB(x, L)
is equal to the most
significant L
bits of LSB(y, BITS)
, For example, 6 (110 in binary) is the
length3 prefix of 25 (11001), but 7 (111) is not.¶
Each of the programmed points beta
is a vector of elements of some finite
field. We distinguish two types of fields: One for inner nodes (denoted
Idpf.FieldInner
), and one for leaf nodes (Idpf.FieldLeaf
). (Our
instantiation of Poplar1 (Section 8.4) will use a much larger
field for leaf nodes than for inner nodes. This is to ensure the IDPF is
"extractable" as defined in [BBCGGI21], Definition 1.)¶
A concrete IDPF defines the types and constants enumerated in Table 13. 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 IDPFkey
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
IDPFkey evaluation algorithm run by each Aggregator. Its inputs are the
Aggregator's unique identifier, the public share distributed to all of the
Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the
IDPF, 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.BITS1
, the
output is the value for an inner node, which has type
Vec[Vec[Idpf.FieldInner]]
; otherwise, if level == Idpf.BITS1
, 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
:¶
def current_field(Idpf, level): return Idpf.FieldInner if level < Idpf.BITS1 \ else Idpf.FieldLeaf¶
Finally, an implementation note. The interface for IDPFs specified here is stateless, in the sense that there is no state carried between IDPF evaluations. This is to align the IDPF syntax with the VDAF abstraction boundary, which does not include shared state across across VDAF evaluations. In practice, of course, it will often be beneficial to expose a stateful API for IDPFs and carry the state across evaluations. See Section 8.3 for details.¶
Parameter  Description 

SHARES  Number of IDPF keys output by IDPFkey generator 
BITS  Length in bits of each input string 
VALUE_LEN  Number of field elements of each output value 
KEY_SIZE  Size in bytes of each IDPF key 
FieldInner  Implementation of Field (Section 6.1) used for values of inner nodes 
FieldLeaf  Implementation of Field used for values of leaf nodes 
Prg  Implementation of Prg (Section 6.2) 
This section specifies Poplar1
, an implementation of the Vdaf
interface
(Section 5). It is defined in terms of any Idpf
(Section 8.1) for which
Idpf.SHARES == 2
and Idpf.VALUE_LEN == 2
. The associated constants and types
required by the Vdaf
interface are defined in Table 14. The methods
required for sharding, preparation, aggregation, and unsharding are described in
the remaining subsections.¶
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]

The client's input is an IDPF index, denoted alpha
. The programmed IDPF values
are pairs of field elements (1, k)
where each k
is chosen at random. This
random value is used as part of the secure sketching protocol of [BBCGGI21],
Appendix C.4. After evaluating their IDPF key shares on a given sequence of
candidate prefixes, the sketching protocol is used by the Aggregators to verify
that they hold shares of a onehot vector. In addition, for each level of the
tree, the prover generates random elements a
, b
, and c
and computes¶
A = 2*a + k B = a^2 + b  k*a + c¶
and sends additive shares of a
, b
, c
, A
and B
to the Aggregators.
Putting everything together, the inputdistribution algorithm is defined as
follows. Function encode_input_shares
is defined in Section 8.2.5.¶
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a data_share
and auth_share
for each. The Aggregators use these and the correlation shares provided by the
Client to verify that the sequence of data_share
values are additive shares of
a onehot vector.¶
The algorithms below make use of auxiliary functions verify_context()
and
decode_input_share()
defined in Section 8.2.5.¶
Aggregation involves simply adding up the output shares.¶
Finally, the Collector unshards the aggregate result by adding up the aggregate shares.¶
In this section we specify a concrete IDPF, called IdpfPoplar, suitable for instantiating Poplar1. The scheme gets its name from the name of the protocol of [BBCGGI21].¶
TODO We should consider giving IdpfPoplar
a more distinctive name.¶
The constant and type definitions required by the Idpf
interface are given in
Table 15.¶
Parameter  Value 

SHARES 
2

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

FieldInner 
Field64 (Table 3) 
FieldLeaf 
Field255 (Table 5) 
Prg  any implementation of Prg (Section 6.2) 
TODO Describe the construction in prose, beginning with a gentle introduction to the high level idea.¶
The description of the IDPFkey generation algorithm makes use of auxiliary
functions extend()
, convert()
, and encode_public_share()
defined in
Section 8.3.3. In the following, we let Field2
denote the
field GF(2)
.¶
TODO Describe in prose how IDPFkey evaluation algorithm works.¶
The description of the IDPFevaluation algorithm makes use of auxiliary
functions extend()
, convert()
, and decode_public_share()
defined in
Section 8.3.3.¶
We refer to Poplar1 instantiated with IdpfPoplar (VALUE_LEN == 2
)
and PrgAes128 (Section 6.2.1) as Poplar1Aes128. This VDAF is suitable
for any positive value of BITS
. Test vectors can be found in
Appendix "Test Vectors".¶
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:¶
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 regenerate 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:¶
Establishing secure channels:¶
In such an environment, a VDAF provides the highlevel privacy property described above: The Collector learns only the aggregate measurement, and nothing about individual measurements aside from what can be inferred from the aggregate result. The Aggregators learn neither individual measurements nor the aggregate result. The Collector is assured that the aggregate statistic accurately reflects the inputs as long as the Aggregators correctly executed their role in the VDAF.¶
On their own, VDAFs do not 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 noncollusion instead of statistical noise to protect the privacy of the inputs. It is possible that a future VDAF could incorporate differential privacy features, e.g., by injecting noise before the sharding stage and removing it after unsharding.¶
A codepoint for each (V)DAF in this document is defined in the table below. Note
that 0xFFFF0000
through 0xFFFFFFFF
are reserved for private use.¶
Value  Scheme  Type  Reference 

0x00000000

Prio3Aes128Count  VDAF  Section 7.4.1 
0x00000001

Prio3Aes128Sum  VDAF  Section 7.4.2 
0x00000002

Prio3Aes128Histogram  VDAF  Section 7.4.3 
0x00000003 to 0x00000FFF

reserved for Prio3  VDAF  n/a 
0x00001000

Poplar1Aes128  VDAF  Section 8.4 
0xFFFF0000 to 0xFFFFFFFF

reserved  n/a  n/a 
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.¶
Thanks to David Cook, Henry CorriganGibbs, Armando FazHernandez, Simon Friedberger, Tim Geoghegan, Mariana Raykova, Jacob Rothstein, and Christopher Wood for useful feedback on and contributions to the spec.¶
NOTE Machinereadable test vectors can be found at https://github.com/cfrg/draftirtfcfrgvdaf/tree/main/poc/test_vec.¶
Test vectors cover the generation of input shares and the conversion of input
shares into output shares. Vectors specify the verification key, measurements,
aggregation parameter, and any parameters needed to construct the VDAF. (For
example, for 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]¶
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]¶