draft-irtf-cfrg-vdaf-03
Internet-Draft
VDAF
August 2022
Barnes, et al.
Expires 25 February 2023
[Page]
Workgroup:
CFRG
Internet-Draft:
draft-irtf-cfrg-vdaf-03
Published:
24 August 2022
Intended Status:
Informational
Expires:
25 February 2023
Authors:
R. L. Barnes
Cisco
C. Patton
Cloudflare
P. Schoppmann
Google
Verifiable Distributed Aggregation Functions
Abstract
This document describes Verifiable Distributed Aggregation Functions (VDAFs), a
family of multi-party protocols for computing aggregate statistics over user
measurements. These protocols are designed to ensure that, as long as at least
one aggregation server executes the protocol honestly, individual measurements
are never seen by any server in the clear. At the same time, VDAFs allow the
servers to detect if a malicious or misconfigured client submitted an
input that would result in an incorrect aggregate result.
Discussion Venues
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
Source for this draft and an issue tracker can be found at
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This Internet-Draft will expire on 25 February 2023.
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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:
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:
General patterns of communications among the various actors involved in
the system (clients, aggregation servers, and the collector of the
aggregate result);
Capabilities of a malicious coalition of servers attempting to divulge
information about client measurements; and
Conditions that are necessary to ensure that malicious clients cannot
corrupt the computation.
Providing cryptographers with design criteria that provide a clear
deployment roadmap for new constructions.
This document also specifies two concrete VDAF schemes, each based on a protocol
from the literature.
The aforementioned Prio system
CGB17
allows for the privacy-preserving
computation of a variety aggregate statistics. The basic idea underlying Prio
is fairly simple:
Each client shards its measurement into a sequence of additive shares and
distributes the shares among the aggregation servers.
Next, each server adds up its shares locally, resulting in an additive
share of the aggregate.
Finally, the aggregation servers send their aggregate shares to the data
collector, who combines them to obtain the aggregate result.
The difficult part of this system is ensuring that the servers hold shares of
a valid 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
-heavy-hitters problem in a privacy-preserving manner. Here
each client holds a bit-string of length
, and the goal of the aggregation
servers is to compute the set of inputs that occur at least
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
. 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.
1.1.
Change Log
(*) Indicates a change that breaks wire compatibility with the previous draft.
03:
Define codepoints for (V)DAFs and use them for domain separation in Prio3 and
Poplar1. (*)
Prio3: Align joint randomness computation with revised paper
BBCGGI19
This change mitigates an attack on robustness. (*)
Prio3: Remove an intermediate PRG evaluation from query randomness generation.
(*)
Add additional guidance for choosing FFT-friendly fields.
02:
Complete the initial specification of Poplar1.
Extend (V)DAF syntax to include a "public share" output by the Client and
distributed to all of the Aggregators. This is to accommodate "extractable"
IDPFs as required for Poplar1. (See
BBCGGI21
, Section 4.3 for details.)
Extend (V)DAF syntax to allow the unsharding step to take into account the
number of measurements aggregated.
Extend FLP syntax by adding a method for decoding the aggregate result from a
vector of field elements. The new method takes into account the number of
measurements.
Prio3: Align aggregate result computation with updated FLP syntax.
Prg: Add a method for statefully generating a vector of field elements.
Field: Require that field elements are fully reduced before decoding. (*)
Define new field Field255.
01:
Require that VDAFs specify serialization of aggregate shares.
Define Distributed Aggregation Functions (DAFs).
Prio3: Move proof verifier check from
prep_next()
to
prep_shares_to_prep()
. (*)
Remove public parameter and replace verification parameter with a
"verification key" and "Aggregator ID".
2.
Conventions and Definitions
The key words "
MUST
", "
MUST NOT
", "
REQUIRED
", "
SHALL
", "
SHALL NOT
", "
SHOULD
", "
SHOULD NOT
", "
RECOMMENDED
", "
NOT RECOMMENDED
",
MAY
", and "
OPTIONAL
" in this document are to be interpreted as
described in BCP 14
RFC2119
RFC8174
when, and only when, they
appear in all capitals, as shown here.
Algorithms in this document are written in Python 3. Type hints are used to
define input and output types. A fatal error in a program (e.g., failure to
parse one of the function parameters) is usually handled by raising an
exception.
A variable with type
Bytes
is a byte string. This document defines several
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
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
Figure 1
Overall data flow of a (V)DAF
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.
Table 1
Constants and types defined by each concrete DAF.
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, 32-bit integer
ID
. Identifiers for each
(V)DAF specified in this document are defined in
Table 16
4.1.
Sharding
In order to protect the privacy of its measurements, a DAF Client shards its
measurements into a sequence of input shares. The
measurement_to_input_shares
method is used for this purpose.
Daf.measurement_to_input_shares(input: Measurement) -> (Bytes, Vec[Bytes])
is the randomized input-distribution algorithm run by each Client. It consumes
the measurement and produces a "public share", distributed to each of the
Aggregators, and a corresponding sequence of input shares, one for each
Aggregator. The length of the output vector
MUST
be
SHARES
Client
======
measurement
+----------------------------------------------+
| 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
Figure 2
The Client divides its measurement into input shares and distributes them to the Aggregators.
4.2.
Preparation
Once an Aggregator has received the public share and one of the input shares,
the next step is to prepare the input share for aggregation. This is
accomplished using the following algorithm:
Daf.prep(agg_id: Unsigned, agg_param: AggParam, public_share: Bytes,
input_share: Bytes) -> OutShare
is the deterministic preparation algorithm.
It takes as input the public share and one of the input shares generated by a
Client, the Aggregator's unique identifier, and the aggregation parameter
selected by the Collector and returns an output share.
The protocol in which the DAF is used
MUST
ensure that the Aggregator's
identifier is equal to the integer in range
[0, SHARES)
that matches the index
of
input_share
in the sequence of input shares output by the Client.
4.3.
Aggregation
Once an Aggregator holds output shares for a batch of measurements (where
batches are defined by the application), it combines them into a share of the
desired aggregate result:
Daf.out_shares_to_agg_share(agg_param: AggParam, out_shares: Vec[OutShare])
-> agg_share: Bytes
is the deterministic aggregation algorithm. It is run by
each Aggregator a set of recovered output shares.
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]
Figure 3
Aggregation of output shares. `B` indicates the number of measurements in the batch.
For simplicity, we have written this algorithm in a "one-shot" form, where all
output shares for a batch are provided at the same time. Many DAFs may also
support a "streaming" form, where shares are processed one at a time.
OPEN ISSUE It may be worthwhile to explicitly define the "streaming" API. See
issue#47.
4.4.
Unsharding
After the Aggregators have aggregated a sufficient number of output shares, each
sends its aggregate share to the Collector, who runs the following algorithm to
recover the following output:
Daf.agg_shares_to_result(agg_param: AggParam,
agg_shares: Vec[Bytes], num_measurements: Unsigned) -> AggResult
is
run by the Collector in order to compute the aggregate result from
the Aggregators' shares. The length of
agg_shares
MUST
be
SHARES
num_measurements
is the number of measurements that contributed to
each of the aggregate shares. This algorithm is deterministic.
Aggregator 0 Aggregator 1 Aggregator SHARES-1
============ ============ ===================
agg_share_0 agg_share_1 agg_share_[SHARES-1]
| | |
V V V
+-----------------------------------------------+
| agg_shares_to_result |
+-----------------------------------------------+
agg_result
Collector
=========
Figure 4
Computation of the final aggregate result from aggregate shares.
QUESTION Maybe the aggregation algorithms should be randomized in order to
allow the Aggregators (or the Collector) to add noise for differential
privacy. (See the security considerations of
DAP
.)
Or is this out-of-scope of this document? See
4.5.
Execution of a DAF
Securely executing a DAF involves emulating the following procedure.
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
Figure 5
Execution of a DAF.
The inputs to this procedure are the same as the aggregation function computed by
the DAF: An aggregation parameter and a sequence of measurements. The procedure
prescribes how a DAF is executed in a "benign" environment in which there is no
adversary and the messages are passed among the protocol participants over
secure point-to-point channels. In reality, these channels need to be
instantiated by some "wrapper protocol", such as
DAP
that realizes these channels using suitable cryptographic mechanisms. Moreover,
some fraction of the Aggregators (or Clients) may be malicious and diverge from
their prescribed behaviors.
Section 9
describes the execution of the DAF in
various adversarial environments and what properties the wrapper protocol needs
to provide in each.
5.
Definition of VDAFs
Like DAFs described in the previous section, a VDAF scheme is used to compute a
particular aggregation function over a set of 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.
Table 2
Constants and types 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 {[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
5.1.
Sharding
Sharding is syntactically identical to DAFs (cf.
Section 4.1
):
Vdaf.measurement_to_input_shares(measurement: Measurement) -> (Bytes,
Vec[Bytes])
is the randomized input-distribution algorithm run by each
Client. It consumes the measurement and produces a public share, distributed
to each of Aggregators, and the corresponding sequence of input shares, one
for each Aggregator. Depending on the VDAF, the input shares may encode
additional information used to verify the recovered output shares (e.g., the
"proof shares" in Prio3
Section 7
). The length of the output vector
MUST
be
SHARES
5.2.
Preparation
To recover and verify output shares, the Aggregators interact with one another
over
ROUNDS
rounds. Prior to each round, each Aggregator constructs an
outbound message. Next, the sequence of outbound messages is combined into a
single message, called a "preparation message". (Each of the outbound messages
are called "preparation-message shares".) Finally, the preparation message is
distributed to the Aggregators to begin the next round.
An Aggregator begins the first round with its input share and it begins each
subsequent round with the previous preparation message. Its output in the last
round is its output share and its output in each of the preceding rounds is a
preparation-message share.
This process involves a value called the "aggregation parameter" used to map the
input shares to output shares. The Aggregators need to agree on this parameter
before they can begin preparing inputs for aggregation.
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]
Figure 6
VDAF preparation process on the input shares for a single measurement. At the end of the computation, each Aggregator holds an output share or an error.
To facilitate the preparation process, a concrete VDAF implements the following
class methods:
Vdaf.prep_init(verify_key: Bytes, agg_id: Unsigned, agg_param: AggParam,
nonce: Bytes, public_share: Bytes, input_share: Bytes) -> Prep
is the
deterministic preparation-state initialization algorithm run by each
Aggregator to begin processing its input share into an output share. Its
inputs are the shared verification key (
verify_key
), the Aggregator's unique
identifier (
agg_id
), the aggregation parameter (
agg_param
), the nonce
provided by the environment (
nonce
, see
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 preparation-state update algorithm run
by each Aggregator. It updates the Aggregator's preparation state (
prep
) and
returns either its next preparation state and its message share for the
current round or, if this is the last round, its output share. An exception is
raised if a valid output share could not be recovered. The input of this
algorithm is the inbound preparation message or, if this is the first round,
None
Vdaf.prep_shares_to_prep(agg_param: AggParam, prep_shares: Vec[Bytes]) ->
Bytes
is the deterministic preparation-message pre-processing algorithm. It
combines the preparation-message shares generated by the Aggregators in the
previous round into the preparation message consumed by each in the next
round.
In effect, each Aggregator moves through a linear state machine with
ROUNDS+1
states. The Aggregator enters the first state on using the initialization
algorithm, and the update algorithm advances the Aggregator to the next state.
Thus, in addition to defining the number of rounds (
ROUNDS
), a VDAF instance
defines the state of the Aggregator after each round.
TODO Consider how to bake this "linear state machine" condition into the
syntax. Given that Python 3 is used as our pseudocode, it's easier to specify
the preparation state using a class.
The preparation-state update accomplishes two tasks: recovery of output shares
from the input shares and ensuring that the recovered output shares are valid.
The abstraction boundary is drawn so that an Aggregator only recovers an output
share if it is deemed valid (at least, based on the Aggregator's view of the
protocol). Another way to draw this boundary would be to have the Aggregators
recover output shares first, then verify that they are valid. However, this
would allow the possibility of misusing the API by, say, aggregating an invalid
output share. Moreover, in protocols like Prio+
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 preparation-state update algorithm once and immediately
recovers its output share without interacting with the other Aggregators.
However, most, if not all, constructions will require some amount of interaction
in order to ensure validity of the output shares (while also maintaining
privacy).
OPEN ISSUE accommodating 0-round VDAFs may require syntax changes if, for
example, public keys are required. On the other hand, we could consider
defining this class of schemes as a different primitive. See issue#77.
5.3.
Aggregation
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 "one-shot" form, where all shares for a
batch are provided at the same time. VDAFs typically also support a "streaming"
form, where shares are processed one at a time.
5.4.
Unsharding
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.
The data flow for this stage is illustrated in
Figure 4
5.5.
Execution of a VDAF
Secure execution of a VDAF involves simulating the following procedure.
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
Figure 7
Execution of a VDAF.
The inputs to this algorithm are the aggregation parameter, a list of
measurements, and a nonce for each measurement. This document does not specify
how the nonces are chosen, but security requires that the nonces be unique. See
Section 9
for details. As explained in
Section 4.5
, the secure
execution of a VDAF requires the application to instantiate secure channels
between each of the protocol participants.
6.
Preliminaries
This section describes the primitives that are common to the VDAFs specified in
this document.
6.1.
Finite Fields
Both Prio3 and Poplar1 use finite fields of prime order. Finite field
elements are represented by a class
Field
with the following associated
parameters:
MODULUS: Unsigned
is the prime modulus that defines the field.
ENCODED_SIZE: Unsigned
is the number of bytes used to encode a field element
as a byte string.
A concrete
Field
also implements the following class methods:
Field.zeros(length: Unsigned) -> output: Vec[Field]
returns a vector of
zeros. The length of
output
MUST
be
length
Field.rand_vec(length: Unsigned) -> output: Vec[Field]
returns a vector of
random field elements. The length of
output
MUST
be
length
A field element is an instance of a concrete
Field
. The concrete class defines
the usual arithmetic operations on field elements. In addition, it defines the
following instance method for converting a field element to an unsigned integer:
elem.as_unsigned() -> Unsigned
returns the integer representation of
field element
elem
Likewise, each concrete
Field
implements a constructor for converting an
unsigned integer into a field element:
Field(integer: Unsigned)
returns
integer
represented as a field element.
The value of
integer
MUST
be less than
Field.MODULUS
Finally, each concrete
Field
has two derived class methods, one for encoding
a vector of field elements as a byte string and another for decoding a vector of
field elements.
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
Figure 8
Derived class methods for finite fields.
6.1.1.
Auxiliary Functions
The following auxiliary functions on vectors of field elements are used in the
remainder of this document. Note that an exception is raised by each function if
the operands are not the same length.
# Compute the inner product of the operands.
def inner_product(left: Vec[Field], right: Vec[Field]) -> Field:
return sum(map(lambda x: x[0] * x[1], zip(left, right)))
# Subtract the right operand from the left and return the result.
def vec_sub(left: Vec[Field], right: Vec[Field]):
return list(map(lambda x: x[0] - x[1], zip(left, right)))
# Add the right operand to the left and return the result.
def vec_add(left: Vec[Field], right: Vec[Field]):
return list(map(lambda x: x[0] + x[1], zip(left, right)))
Figure 9
Common functions for finite fields.
6.1.2.
FFT-Friendly Fields
Some VDAFs require fields that are suitable for efficient computation of the
discrete Fourier transform, as this allows for fast polynomial interpolation.
(One example is Prio3 (
Section 7
) when instantiated with the generic FLP of
Section 7.3.3
.) Specifically, a field is said to be
"FFT-friendly" 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 FFT-friendly, 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
FFT-friendly fields also define the following parameter:
GEN_ORDER: Unsigned
is the order of a multiplicative subgroup generated by
Field.gen()
6.1.3.
Parameters
The tables below define finite fields used in the remainder of this document.
Table 3
Field64, an FFT-friendly field.
Parameter
Value
MODULUS
2^32 * 4294967295 + 1
ENCODED_SIZE
Generator
7^4294967295
GEN_ORDER
2^32
Table 4
Field128, an FFT-friendly field.
Parameter
Value
MODULUS
2^66 * 4611686018427387897 + 1
ENCODED_SIZE
16
Generator
7^4611686018427387897
GEN_ORDER
2^66
Table 5
Field255.
Parameter
Value
MODULUS
2^255 - 19
ENCODED_SIZE
32
OPEN ISSUE We currently use big-endian for encoding field elements. However,
for implementations of
GF(2^255-19)
, little endian is more common. See
issue#90.
6.2.
Pseudorandom Generators
A pseudorandom generator (PRG) is used to expand a short, (pseudo)random seed
into a long string of pseudorandom bits. A PRG suitable for this document
implements the interface specified in this section. 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).
# 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)
Figure 10
Derived class methods for PRGs.
6.2.1.
PrgAes128
OPEN ISSUE Phillipp points out that a fixed-key mode of AES may be more
performant (https://eprint.iacr.org/2019/074.pdf). See issue#32.
TODO(issue #106) Decide if it's safe to model this construction as a random
oracle.
PrgAes128.derive_seed()
is used for the Fiat-Shamir heuristic in
Prio3 (
Section 7
). A fixed-key is used for this step (the all-zero string). A
reasonable starting point would be to model AES as an ideal cipher.
Our first construction,
PrgAes128
, converts a blockcipher, namely AES-128,
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 CTR-mode to produce a key stream for
generating the output. A fixed initialization vector (IV) is used.
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:]
Figure 11
Definition of PRG PrgAes128.
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"
7.1.
Fully Linear Proof (FLP) Systems
Conceptually, an FLP is a two-party protocol executed by a prover and a
verifier. In actual use, however, the prover's computation is carried out by the
Client, and the verifier's computation is distributed among the Aggregators. The
Client generates a "proof" of its input's validity and distributes shares of the
proof to the Aggregators. Each Aggregator then performs some a computation on
its input share and proof share locally and sends the result to the other
Aggregators. Combining the exchanged messages allows each Aggregator to decide
if it holds a share of a valid input. (See
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
Table 6
Constants and types defined by a concrete FLP.
Parameter
Description
PROVE_RAND_LEN
Length of the prover randomness, the number of random field elements consumed by the prover when generating a proof
QUERY_RAND_LEN
Length of the query randomness, the number of random field elements consumed by the verifier
JOINT_RAND_LEN
Length of the joint randomness, the number of random field elements consumed by both the prover and verifier
INPUT_LEN
Length of the encoded measurement (
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 proof-generation algorithm run by the
prover. Its inputs are the encoded input, the "prover randomness"
prove_rand
, and the "joint randomness"
joint_rand
. The prover randomness is
used only by the prover, but the joint randomness is shared by both the prover
and verifier.
Flp.query(input: Vec[Field], proof: Vec[Field], query_rand: Vec[Field],
joint_rand: Vec[Field], num_shares: Unsigned) -> Vec[Field]
is the
query-generation algorithm run by the verifier. This is used to "query" the
input and proof. The result of the query (i.e., the output of this function)
is called the "verifier message". In addition to the input and proof, this
algorithm takes as input the query randomness
query_rand
and the joint
randomness
joint_rand
. The former is used only by the verifier.
num_shares
specifies how many input and proof shares were generated.
Flp.decide(verifier: Vec[Field]) -> Bool
is the deterministic decision
algorithm run by the verifier. It takes as input the verifier message and
outputs a boolean indicating if the input from which it was generated is
valid.
Our application requires that the FLP is "fully linear" in the sense defined in
BBCGGI19
. As a practical matter, what this property implies is that, when
run on a share of the input and proof, the query-generation algorithm outputs a
share of the verifier message. Furthermore, the "strong zero-knowledge" property
of the FLP system ensures that the verifier message reveals nothing about the
input's validity. Therefore, to decide if an input is valid, the Aggregators
will run the query-generation algorithm locally, exchange verifier shares,
combine them to recover the verifier message, and run the decision algorithm.
The query-generation algorithm includes a parameter
num_shares
that specifies
the number of shares of the input and proof that were generated. If these data
are not secret shared, then
num_shares == 1
. This parameter is useful for
certain FLP constructions. For example, the FLP in
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:
def run_flp(Flp, inp: Vec[Flp.Field], num_shares: Unsigned):
joint_rand = Flp.Field.rand_vec(Flp.JOINT_RAND_LEN)
prove_rand = Flp.Field.rand_vec(Flp.PROVE_RAND_LEN)
query_rand = Flp.Field.rand_vec(Flp.QUERY_RAND_LEN)
# Prover generates the proof.
proof = Flp.prove(inp, prove_rand, joint_rand)
# Verifier queries the input and proof.
verifier = Flp.query(inp, proof, query_rand, joint_rand, num_shares)
# Verifier decides if the input is valid.
return Flp.decide(verifier)
Figure 12
Execution of an FLP.
The proof system is constructed so that, if
input
is a valid input, then
run_flp(Flp, input, 1)
always returns
True
. On the other hand, if
input
is
invalid, then as long as
joint_rand
and
query_rand
are generated uniform
randomly, the output is
False
with overwhelming probability.
We remark that
BBCGGI19
defines a much larger class of fully linear proof
systems than we consider here. In particular, what is called an "FLP" here is
called a 1.5-round, public-coin, interactive oracle proof system in their paper.
7.1.1.
Encoding the Input
The type of measurement being aggregated is defined by the FLP. Hence, the FLP
also specifies a method of encoding raw measurements as a vector of field
elements:
Flp.encode(measurement: Measurement) -> Vec[Field]
encodes a raw measurement
as a vector of field elements. The return value
MUST
be of length
INPUT_LEN
For some FLPs, the encoded input also includes redundant field elements that are
useful for checking the proof, but which are not needed after the proof has been
checked. An example is the "integer sum" data type from
CGB17
in which an
integer in range
[0, 2^k)
is encoded as a vector of
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
post-processing.
Flp.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult
maps a sum of aggregate shares to an aggregate result. The length of the
input
MUST
be
OUTPUT_LEN
num_measurements
is the number of measurements
that contributed to the aggregated output.
We remark that, taken together, these three functionalities correspond roughly
to the notion of "Affine-aggregatable encodings (AFEs)" from
CGB17
7.2.
Construction
This section specifies
Prio3
, an implementation of the
Vdaf
interface
Section 5
). It has two generic parameters: an
Flp
Section 7.1
) and a
Prg
Section 6.2
). The associated constants and types required by the
Vdaf
interface
are defined in
Table 7
. The methods required for sharding, preparation,
aggregation, and unsharding are described in the remaining subsections.
Table 7
Associated parameters for the Prio3 VDAF.
Parameter
Value
VERIFY_KEY_SIZE
Prg.SEED_SIZE
ROUNDS
SHARES
in
[2, 255)
Measurement
Flp.Measurement
AggParam
None
Prep
Tuple[Vec[Flp.Field], Optional[Bytes], Bytes]
OutShare
Vec[Flp.Field]
AggResult
Flp.AggResult
7.2.1.
Sharding
Recall from
Section 7.1
that the FLP syntax calls for "joint randomness" shared by
the prover (i.e., the Client) and the verifier (i.e., the Aggregators). VDAFs
have no such notion. Instead, the Client derives the joint randomness from its
input in a way that allows the Aggregators to reconstruct it from their input
shares. (This idea is based on the Fiat-Shamir heuristic and is described in
Section 6.2.3 of
BBCGGI19
.)
The input-distribution algorithm involves the following steps:
Encode the Client's raw measurement as an input for the FLP
Shard the input into a sequence of input shares
Derive the joint randomness from the input shares
Run the FLP proof-generation algorithm using the derived joint randomness
Shard the proof into a sequence of proof shares
The algorithm is specified below. Notice that only one set of input and proof
shares (called the "leader" shares below) are vectors of field elements. The
other shares (called the "helper" shares) are represented instead by PRG seeds,
which are expanded into vectors of field elements.
The code refers to auxiliary functions for encoding the shares and deriving the
joint randomness. In particular,
encode_leader_share
and
encode_helper_share
are used for encoding and
joint_rand
is used for joint
randomness. These are defined in
Section 7.2.5
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)
Figure 13
Input-distribution algorithm for Prio3.
7.2.2.
Preparation
This section describes the process of recovering output shares from the input
shares. The high-level idea is that each Aggregator first queries its input and
proof share locally, then exchanges its verifier share with the other
Aggregators. The verifier shares are then combined into the verifier message,
which is used to decide whether to accept.
In addition, the Aggregators must ensure that they have all used the same joint
randomness for the query-generation algorithm. The joint randomness is generated
by a PRG seed. Each Aggregator derives 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 query-generation 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
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)
Figure 14
Preparation state for Prio3.
7.2.3.
Aggregation
Aggregating a set of output shares is simply a matter of adding up the vectors
element-wise.
def out_shares_to_agg_share(Prio3, _agg_param, out_shares):
agg_share = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return Prio3.Flp.Field.encode_vec(agg_share)
Figure 15
Aggregation algorithm for Prio3.
7.2.4.
Unsharding
To unshard a set of aggregate shares, the Collector first adds up the vectors
element-wise. It then converts each element of the vector into an integer.
def agg_shares_to_result(Prio3, _agg_param, agg_shares,
num_measurements):
agg = Prio3.Flp.Field.zeros(Prio3.Flp.OUTPUT_LEN)
for agg_share in agg_shares:
agg = vec_add(agg, Prio3.Flp.Field.decode_vec(agg_share))
return Prio3.Flp.decode(agg, num_measurements)
Figure 16
Computation of the aggregate result for Prio3.
7.2.5.
Auxiliary Functions
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
Figure 17
Helper functions required for Prio3.
7.3.
A General-Purpose FLP
This section describes an FLP based on the construction from in
BBCGGI19
Section 4.2. We begin in
Section 7.3.1
with an overview of their proof
system and the extensions to their proof system made here. The construction is
specified in
Section 7.3.3
OPEN ISSUE We're not yet sure if specifying this general-purpose FLP is
desirable. It might be preferable to specify specialized FLPs for each data
type that we want to standardize, for two reasons. First, clear and concise
specifications are likely easier to write for specialized FLPs rather than the
general one. Second, we may end up tailoring each FLP to the measurement type
in a way that improves performance, but breaks compatibility with the
general-purpose FLP.
In any case, we can't make this decision until we know which data types to
standardize, so for now, we'll stick with the general-purpose construction.
The reference implementation can be found at
OPEN ISSUE Chris Wood points out that the this section reads more like a paper
than a standard. Eventually we'll want to work this into something that is
readily consumable by the CFRG.
7.3.1.
Overview
In the proof system of
BBCGGI19
, validity is defined via an arithmetic
circuit evaluated over the input: If the circuit output is zero, then the input
is deemed valid; otherwise, if the circuit output is non-zero, then the input is
deemed invalid. Thus the goal of the proof system is merely to allow the
verifier to evaluate the validity circuit over the input. For our application
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
is affine, meaning its only
operations are addition and multiplication-by-constant. In particular, suppose
the circuit does not contain a multiplication gate whose operands are both
non-constant. Then to decide if an input
is valid, each Aggregator could
evaluate
on its share of
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
it holds that
C(x_shares[0] + ... + x_shares[SHARES-1]) =
C(x_shares[0]) + ... + C(x_shares[SHARES-1])
(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
is
not-affine (i.e., it contains at least one multiplication gate whose operands
are non-constant). In the proof system of
BBCGGI19
, the proof is designed to
allow the (distributed) verifier to compute the non-affine operations using only
linear operations on (its share of) the input and proof.
To make this work, the proof system is restricted to validity circuits that
exhibit a special structure. Specifically, an arithmetic circuit with "G-gates"
(see
BBCGGI19
, Definition 5.2) is composed of affine gates and any number of
instances of a distinguished gate
, which may be non-affine. We will refer to
this class of circuits as 'gadget circuits' and to
as the "gadget".
As an illustrative example, consider a validity circuit
that recognizes the
set
L = set([0], [1])
. That is,
takes as input a length-1 vector
and
returns 0 if
x[0]
is in
[0,2)
and outputs something else otherwise. This
circuit can be expressed as the following degree-2 polynomial:
C(x) = (x[0] - 1) * x[0] = x[0]^2 - x[0]
This polynomial recognizes
because
x[0]^2 = x[0]
is only true if
x[0] ==
or
x[0] == 1
. Notice that the polynomial involves a non-affine operation,
x[0]^2
. In order to apply
BBCGGI19
, Theorem 4.3, the circuit needs to be
rewritten in terms of a gadget that subsumes this non-affine 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
be
the validity circuit and suppose the gadget is arity-
(i.e., it has
input
wires.). Let
wire[j-1,k-1]
denote the value of the
th wire of the
th
call to the gadget during the evaluation of
C(x)
. Suppose there are
such
calls and fix distinct field elements
alpha[0], ..., alpha[M-1]
. (We will
require these points to have a special property, as we'll discuss in
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[k-1]
, produces the output of the
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 well-formed. This test, and the procedure for constructing
the gadget polynomial, are described in detail in
Section 7.3.3
7.3.1.1.
Extensions
The FLP described in the next section extends the proof system
BBCGGI19
Section 4.2 in three ways.
First, the validity circuit in our construction includes an additional, random
input (this is the "joint randomness" derived from the input shares in Prio3;
see
Section 7.2
). This allows for circuit optimizations that trade a
small soundness error for a shorter proof. For example, consider a circuit that
recognizes the set of length-
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[N-1])
(Note that this is a special case of
BBCGGI19
, Theorem 5.2.) Here
inp
is
the length-
input and
is a random field element. The gadget circuit
Range2
is the "range-check" polynomial described above, i.e.,
Range2(x) = x^2 -
. 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
; but if
inp[j]
is not in
[0,2)
for some
, the output will be non-zero with high
probability.
The second extension implemented by our FLP allows the validity circuit to
contain multiple gadget types. (This generalization was suggested in
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[N-1]) + \
2^0 * inp[0] + ... + 2^(N-1) * inp[N-1] - \
Mul(inp[N], inp[N])
Finally,
BBCGGI19
, Theorem 4.3 makes no restrictions on the choice of the
fixed points
alpha[0], ..., alpha[M-1]
, other than to require that the points
are distinct. In this document, the fixed points are chosen so that the gadget
polynomial can be constructed efficiently using the Cooley-Tukey FFT ("Fast
Fourier Transform") algorithm. Note that this requires the field to be
"FFT-friendly" as defined in
Section 6.1.2
7.3.2.
Validity Circuits
The FLP described in
Section 7.3.3
is defined in terms of a
validity circuit
Valid
that implements the interface described here.
A concrete
Valid
defines the following parameters:
Table 8
Validity circuit parameters.
Parameter
Description
GADGETS
A list of gadgets
GADGET_CALLS
Number of times each gadget is called
INPUT_LEN
Length of the input
OUTPUT_LEN
Length of the aggregatable output
JOINT_RAND_LEN
Length of the random input
Measurement
The type of measurement
AggResult
Type of the aggregate result
Field
An FFT-friendly finite field as defined in
Section 6.1.2
Each gadget
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
. Hence, the arithmetic degree of this gadget is
Each gadget also defines a parameter
ARITY
that specifies the circuit's arity
(i.e., the number of input wires).
A concrete
Valid
provides the following methods for encoding a measurement as
an input vector, truncating an input vector to the length of an aggregatable
output, and converting an aggregated output to an aggregate result:
Valid.encode(measurement: Measurement) -> Vec[Field]
returns a vector of
length
INPUT_LEN
representing a measurement.
Valid.truncate(input: Vec[Field]) -> Vec[Field]
returns a vector of length
OUTPUT_LEN
representing an aggregatable output.
Valid.decode(output: Vec[Field], num_measurements: Unsigned) -> AggResult
returns an aggregate result.
Finally, the following class methods are derived for each concrete
Valid
# Length of the prover randomness.
def prove_rand_len(Valid):
return sum(map(lambda g: g.ARITY, Valid.GADGETS))
# Length of the query randomness.
def query_rand_len(Valid):
return len(Valid.GADGETS)
# Length of the proof.
def proof_len(Valid):
length = 0
for (g, g_calls) in zip(Valid.GADGETS, Valid.GADGET_CALLS):
P = next_power_of_2(1 + g_calls)
length += g.ARITY + g.DEGREE * (P - 1) + 1
return length
# Length of the verifier message.
def verifier_len(Valid):
length = 1
for g in Valid.GADGETS:
length += g.ARITY + 1
return length
Figure 18
Derived methods for validity circuits.
7.3.3.
Construction
This section specifies
FlpGeneric
, an implementation of the
Flp
interface
Section 7.1
). It has as a generic parameter a validity circuit
Valid
implementing
the interface defined in
Section 7.3.2
NOTE A reference implementation can be found in
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
. We also define the following constants:
Let
H = len(Valid.GADGETS)
For each
in
[H]
Let
G_i = Valid.GADGETS[i]
Let
L_i = Valid.GADGETS[i].ARITY
Let
M_i = Valid.GADGET_CALLS[i]
Let
P_i = next_power_of_2(M_i+1)
Let
alpha_i = Field.gen()^(Field.GEN_ORDER / P_i)
Table 9
FLP Parameters of FlpGeneric.
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
7.3.3.1.
Proof Generation
On input
inp
prove_rand
, and
joint_rand
, the proof is computed as
follows:
For each
in
[H]
create an empty table
wire_i
Partition the prover randomness
prove_rand
into subvectors
seed_1, ...,
seed_H
where
len(seed_i) == L_i
for all
in
[H]
. Let us call these
the "wire seeds" of each gadget.
Evaluate
Valid
on input of
inp
and
joint_rand
, recording the inputs of
each gadget in the corresponding table. Specifically, for every
in
[H]
set
wire_i[j-1,k-1]
to the value on the
th wire into the
th call to
gadget
G_i
Compute the "wire polynomials". That is, for every
in
[H]
and
in
[L_i]
, construct
poly_wire_i[j-1]
, the
th wire polynomial for the
th gadget, as follows:
Let
w = [seed_i[j-1], wire_i[j-1,0], ..., wire_i[j-1,M_i-1]]
Let
padded_w = w + Field.zeros(P_i - len(w))
NOTE We pad
to the nearest power of 2 so that we can use FFT for
interpolating the wire polynomials. Perhaps there is some clever math for
picking
wire_inp
in a way that avoids having to pad.
Let
poly_wire_i[j-1]
be the lowest degree polynomial for which
poly_wire_i[j-1](alpha_i^k) == padded_w[k]
for all
in
[P_i]
Compute the "gadget polynomials". That is, for every
in
[H]
Let
poly_gadget_i = G_i(poly_wire_i[0], ..., poly_wire_i[L_i-1])
. That
is, evaluate the circuit
G_i
on the wire polynomials for the
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
in
[H]
7.3.3.2.
Query Generation
On input of
inp
proof
query_rand
, and
joint_rand
, the verifier message
is generated as follows:
For every
in
[H]
create an empty table
wire_i
Partition
proof
into the subvectors
seed_1
coeff_1
, ...,
seed_H
coeff_H
defined in
Section 7.3.3.1
Evaluate
Valid
on input of
inp
and
joint_rand
, recording the inputs of
each gadget in the corresponding table. This step is similar to the prover's
step (3.) except the verifier does not evaluate the gadgets. Instead, it
computes the output of the
th call to
G_i
by evaluating
poly_gadget_i(alpha_i^k)
. Let
denote the output of the circuit
evaluation.
Compute the wire polynomials just as in the prover's step (4.).
Compute the tests for well-formedness of the gadget polynomials. That is, for
every
in
[H]
Let
t = query_rand[i]
. Check if
t^(P_i) == 1
: If so, then raise
ERR_ABORT and halt. (This prevents the verifier from inadvertently leaking
a gadget output in the verifier message.)
Let
y_i = poly_gadget_i(t)
For each
in
[0,L_i)
let
x_i[j-1] = poly_wire_i[j-1](t)
The verifier message is the vector
verifier = [v] + x_1 + [y_1] + ... + x_H +
[y_H]
7.3.3.3.
Decision
On input of vector
verifier
, the verifier decides if the input is valid as
follows:
Parse
verifier
into
x_1
y_1
, ...,
x_H
y_H
as defined in
Section 7.3.3.2
Check for well-formedness of the gadget polynomials. For every
in
[H]
Let
z = G_i(x_i)
. That is, evaluate the circuit
G_i
on
x_i
and set
to the output.
If
z != y_i
, then return
False
and halt.
Return
True
if
v == 0
and
False
otherwise.
7.3.3.4.
Encoding
The FLP encoding and truncation methods invoke
Valid.encode
Valid.truncate
, and
Valid.decode
in the natural way.
7.4.
Instantiations
This section specifies instantiations of Prio3 for various measurement types.
Each uses
FlpGeneric
as the FLP (
Section 7.3
) and is determined by a
validity circuit (
Section 7.3.2
) and a PRG (
Section 6.2
). Test vectors for
each can be found in
Appendix "Test Vectors"
NOTE Reference implementations of each of these VDAFs can be found in
7.4.1.
Prio3Aes128Count
Our first instance of Prio3 is for a simple counter: Each measurement is either
one or zero and the aggregate result is the sum of the measurements.
This instance uses
PrgAes128
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 degree-2, arity-2 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.
Table 10
Parameters of validity circuit Count.
Parameter
Value
GADGETS
[Mul]
GADGET_CALLS
[1]
INPUT_LEN
OUTPUT_LEN
JOINT_RAND_LEN
Measurement
Unsigned
, in range
[0,2)
AggResult
Unsigned
Field
Field64
Table 3
7.4.2.
Prio3Aes128Sum
The next instance of Prio3 supports summing of integers in a pre-determined
range. Each measurement is an integer in range
[0, 2^bits)
, where
bits
is an
associated parameter.
This instance of Prio3 uses
PrgAes128
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 length-
bits
vector of field
elements, where the
th element of the vector represents the
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 degree-2, arity-1 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
Table 11
Parameters of validity circuit Sum.
Parameter
Value
GADGETS
[Range2]
GADGET_CALLS
[bits]
INPUT_LEN
bits
OUTPUT_LEN
JOINT_RAND_LEN
Measurement
Unsigned
, in range
[0, 2^bits)
AggResult
Unsigned
Field
Field128
Table 4
7.4.3.
Prio3Aes128Histogram
This instance of Prio3 allows for estimating the distribution of the
measurements by computing a simple histogram. Each measurement is an arbitrary
integer and the aggregate result counts the number of measurements that fall in
a set of fixed buckets.
This instance of Prio3 uses
PrgAes128
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 one-hot 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 one-hotness 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.
Table 12
Parameters of validity circuit Histogram.
Parameter
Value
GADGETS
[Range2]
GADGET_CALLS
[buckets + 1]
INPUT_LEN
buckets + 1
OUTPUT_LEN
buckets + 1
JOINT_RAND_LEN
Measurement
Integer
AggResult
Vec[Unsigned]
Field
Field128
Table 4
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
-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.
Each Client splits its input string into input shares and sends one share to
each Aggregator.
The Aggregators agree on an initial set of candidate prefixes, say
and
The Aggregators evaluate the VDAF on each set of input shares and aggregate
the recovered output shares. The aggregation parameter is the set of
candidate prefixes.
The Aggregators send their aggregate shares to the Collector, who combines
them to recover the counts of each candidate prefix.
Let
denote the set of prefixes that occurred at least
times. If the
prefixes all have length
BITS
, then
is the set of
-heavy-hitters.
Otherwise compute the next set of candidate prefixes as follows. For each
in
, 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"
8.1.
Incremental Distributed Point Functions (IDPFs)
An IDPF is defined over a domain of size
2^BITS
, where
BITS
is constant
defined by the IDPF. Indexes into the IDPF tree are encoded as integers in range
[0, 2^BITS)
. The Client specifies an index
alpha
and a vector of
values
beta
, one for each "level"
in range
[0, BITS)
. The key generation
algorithm generates one IDPF "key" for each Aggregator. When evaluated at level
and index
0 <= prefix < 2^L
, each IDPF key returns an additive share of
beta[L]
if
prefix
is the
-bit prefix of
alpha
and shares of zero
otherwise.
An index
is defined to be a prefix of another index
as follows. Let
LSB(x, N)
denote the least significant
bits of positive integer
. By
definition, a positive integer
0 <= x < 2^L
is said to be the length-
prefix of positive integer
0 <= y < 2^BITS
if
LSB(x, L)
is equal to the most
significant
bits of
LSB(y, BITS)
, For example, 6 (110 in binary) is the
length-3 prefix of 25 (11001), but 7 (111) is not.
Each of the programmed points
beta
is a vector of elements of some finite
field. We distinguish two types of fields: One for inner nodes (denoted
Idpf.FieldInner
), and one for leaf nodes (
Idpf.FieldLeaf
). (Our
instantiation of Poplar1 (
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 IDPF-key
generation algorithm. Its inputs are the index
alpha
and the values
beta
The value of
alpha
MUST
be in range
[0, 2^BITS)
. The output is a public
part that is sent to all Aggregators and a vector of private IDPF keys, one
for each aggregator.
Idpf.eval(agg_id: Unsigned, public_share: Bytes, key: Bytes, level: Unsigned,
prefixes: Vec[Unsigned]) -> Idpf.Vec
is the deterministic, stateless
IDPF-key evaluation algorithm run by each Aggregator. Its inputs are the
Aggregator's unique identifier, the public share distributed to all of the
Aggregators, the Aggregator's IDPF key, the "level" at which to evaluate the
IDPF, and the sequence of candidate prefixes. It returns the share of the
value corresponding to each candidate prefix.
The output type depends on the value of
level
: If
level < Idpf.BITS-1
, the
output is the value for an inner node, which has type
Vec[Vec[Idpf.FieldInner]]
; otherwise, if
level == Idpf.BITS-1
, then the
output is the value for a leaf node, which has type
Vec[Vec[Idpf.FieldLeaf]]
The value of
level
MUST
be in range
[0, BITS)
. The indexes in
prefixes
MUST
all be distinct and in range
[0, 2^level)
Applications
MUST
ensure that the Aggregator's identifier is equal to the
integer in range
[0, SHARES)
that matches the index of
key
in the sequence
of IDPF keys output by the Client.
In addition, the following method is derived for each concrete
Idpf
def current_field(Idpf, level):
return Idpf.FieldInner if level < Idpf.BITS-1 \
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.
Table 13
Constants and types defined by a concrete IDPF.
Parameter
Description
SHARES
Number of IDPF keys output by IDPF-key generator
BITS
Length in bits of each input string
VALUE_LEN
Number of field elements of each output value
KEY_SIZE
Size in bytes of each IDPF key
FieldInner
Implementation of
Field
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
8.2.
Construction
This section specifies
Poplar1
, an implementation of the
Vdaf
interface
Section 5
). It is defined in terms of any
Idpf
Section 8.1
) for which
Idpf.SHARES == 2
and
Idpf.VALUE_LEN == 2
. 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.
Table 14
Associated parameters for the Poplar1 VDAF.
Parameter
Value
VERIFY_KEY_SIZE
Idpf.Prg.SEED_SIZE
ROUNDS
SHARES
Measurement
Unsigned
AggParam
Tuple[Unsigned, Vec[Unsigned]]
Prep
Tuple[Bytes, Unsigned, Idpf.Vec]
OutShare
Idpf.Vec
AggResult
Vec[Unsigned]
8.2.1.
Client
The client's input is an IDPF index, denoted
alpha
. The programmed IDPF values
are pairs of field elements
(1, k)
where each
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 one-hot vector. In addition, for each level of the
tree, the prover generates random elements
, and
and computes
A = -2*a + k
B = a^2 + b - k*a + c
and sends additive shares of
and
to the Aggregators.
Putting everything together, the input-distribution algorithm is defined as
follows. Function
encode_input_shares
is defined in
Section 8.2.5
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))
Figure 19
The input-distribution algorithm for Poplar1.
8.2.2.
Preparation
The aggregation parameter encodes a sequence of candidate prefixes. When an
Aggregator receives an input share from the Client, it begins by evaluating its
IDPF share on each candidate prefix, recovering a
data_share
and
auth_share
for each. The Aggregators use these and the correlation shares provided by the
Client to verify that the sequence of
data_share
values are additive shares of
a one-hot vector.
The algorithms below make use of auxiliary functions
verify_context()
and
decode_input_share()
defined in
Section 8.2.5
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)
Figure 20
Preparation state for Poplar1.
8.2.3.
Aggregation
Aggregation involves simply adding up the output shares.
def out_shares_to_agg_share(Poplar1, agg_param, out_shares):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg_share = Field.zeros(len(prefixes))
for out_share in out_shares:
agg_share = vec_add(agg_share, out_share)
return Field.encode_vec(agg_share)
Figure 21
Aggregation algorithm for Poplar1.
8.2.4.
Unsharding
Finally, the Collector unshards the aggregate result by adding up the aggregate
shares.
def agg_shares_to_result(Poplar1, agg_param,
agg_shares, _num_measurements):
(level, prefixes) = agg_param
Field = Poplar1.Idpf.current_field(level)
agg = Field.zeros(len(prefixes))
for agg_share in agg_shares:
agg = vec_add(agg, Field.decode_vec(agg_share))
return list(map(lambda x: x.as_unsigned(), agg))
Figure 22
Computation of the aggregate result for Poplar1.
8.2.5.
Auxiliary Functions
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
Figure 23
Helper functions for Poplar1.
8.3.
The IDPF scheme of
[BBCGGI21]
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
Table 15
Constants and type definitions for IdpfPoplar.
Parameter
Value
SHARES
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
8.3.1.
Key Generation
TODO Describe the construction in prose, beginning with a gentle introduction
to the high level idea.
The description of the IDPF-key generation algorithm makes use of auxiliary
functions
extend()
convert()
, and
encode_public_share()
defined in
Section 8.3.3
. In the following, we let
Field2
denote the
field
GF(2)
def gen(IdpfPoplar, alpha, beta_inner, beta_leaf):
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)
Figure 24
IDPF-key generation algorithm of IdpfPoplar.
8.3.2.
Key Evaluation
TODO Describe in prose how IDPF-key evaluation algorithm works.
The description of the IDPF-evaluation algorithm makes use of auxiliary
functions
extend()
convert()
, and
decode_public_share()
defined in
Section 8.3.3
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)
Figure 25
IDPF-evaluation generation algorithm of IdpfPoplar.
8.3.3.
Auxiliary Functions
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
Figure 26
Helper functions for IdpfPoplar.
8.4.
Poplar1Aes128
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"
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:
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.
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.
Table 16
Unique identifiers for (V)DAFs.
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
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.
11.
References
11.1.
Normative References
[RFC2119]
Bradner, S.
"Key words for use in RFCs to Indicate Requirement Levels"
BCP 14
RFC 2119
DOI 10.17487/RFC2119
March 1997
[RFC4493]
Song, JH.
Poovendran, R.
Lee, J.
, and
T. Iwata
"The AES-CMAC Algorithm"
RFC 4493
DOI 10.17487/RFC4493
June 2006
[RFC8017]
Moriarty, K., Ed.
Kaliski, B.
Jonsson, J.
, and
A. Rusch
"PKCS #1: RSA Cryptography Specifications Version 2.2"
RFC 8017
DOI 10.17487/RFC8017
November 2016
[RFC8174]
Leiba, B.
"Ambiguity of Uppercase vs Lowercase in RFC 2119 Key Words"
BCP 14
RFC 8174
DOI 10.17487/RFC8174
May 2017
11.2.
Informative References
[AGJOP21]
Addanki, S.
Garbe, K.
Jaffe, E.
Ostrovsky, R.
, and
A. Polychroniadou
"Prio+: Privacy Preserving Aggregate Statistics via Boolean Shares"
2021
[BBCGGI19]
Boneh, D.
Boyle, E.
Corrigan-Gibbs, H.
Gilboa, N.
, and
Y. Ishai
"Zero-Knowledge Proofs on Secret-Shared Data via Fully Linear PCPs"
CRYPTO 2019
2019
[BBCGGI21]
Boneh, D.
Boyle, E.
Corrigan-Gibbs, H.
Gilboa, N.
, and
Y. Ishai
"Lightweight Techniques for Private Heavy Hitters"
IEEE S&P 2021
2021
[CGB17]
Corrigan-Gibbs, H.
and
D. Boneh
"Prio: Private, Robust, and Scalable Computation of Aggregate Statistics"
NSDI 2017
2017
[DAP]
Geoghegan, T.
Patton, C.
Rescorla, E.
, and
C. A. Wood
"Distributed Aggregation Protocol for Privacy Preserving Measurement"
Work in Progress
Internet-Draft, draft-ietf-ppm-dap-01
11 July 2022
[Dou02]
Douceur, J.
"The Sybil Attack"
IPTPS 2002
2002
[Dwo06]
Dwork, C.
"Differential Privacy"
ICALP 2006
2006
[ENPA]
"Exposure Notification Privacy-preserving Analytics (ENPA) White Paper"
2021
[EPK14]
Erlingsson, Ú.
Pihur, V.
, and
A. Korolova
"RAPPOR: Randomized Aggregatable Privacy-Preserving Ordinal Response"
CCS 2014
2014
[GI14]
Gilboa, N.
and
Y. Ishai
"Distributed Point Functions and Their Applications"
EUROCRYPT 2014
2014
[OriginTelemetry]
"Origin Telemetry"
2020
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
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.
Prio3Aes128Count
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
Prio3Aes128Sum
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
Prio3Aes128Histogram
buckets: [1, 10, 100]
verify_key: "01010101010101010101010101010101"
upload_0:
measurement: 50
nonce: "01010101010101010101010101010101"
public_share: >-
input_share_0: >-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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]
Poplar1Aes128
Sharding
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
Preparation, Aggregation, and Unsharding
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
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1bda898326421778e72467244acb17f472a4e61040801ea8170611e914421029a53aa0
1b494ff834cf2e01239409a37b99623c332eebf34778eee2356d44b95a1681d1f394f7
fe66cad88d59fcb458283e84d455
agg_share_1: >-
3adfa00910309869e33eff177f0c3a72a31d7ca7a350804a71f63db45643b64d45a5b7
8099d5484dbf4be493c53cba3c8ab4bc344e5a27ea7177a62e66b2ea8d3f800be22b35
33b8f4c3bd759e0568b293064e2eb39246bfe0f1e8b0031aa2892d87798b701f637950
2a748188e83b81168cf60101259b7cf6bfcb72681c1b5e05b7fa4315f2fdc2005f2c34
e425767cd9bde88718db98dbb534e80b8d5b19dd3f7fe157e8f9ee16ebbdefd65ac55f
e4b6b007cb30d1fedc6bf65c84669dc3bb51140cb887111dca92bb46a5e97e2e0c6b08
0199352772a6034ba7d7c17b2b98
agg_result: [0, 0, 0, 0, 0, 1, 0]
Authors' Addresses
Richard L. Barnes
Cisco
Email:
rlb@ipv.sx
Christopher Patton
Cloudflare
Email:
chrispatton+ietf@gmail.com
Phillipp Schoppmann
Google
Email:
schoppmann@google.com
Datatracker
draft-irtf-cfrg-vdaf-03
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Authors
Richard Barnes
Christopher Patton
Phillipp Schoppmann
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draft-patton-cfrg-vdaf
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