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0.8.0 | Nov 22, 2021 |

#**295** in Cryptography

**134** downloads per month

Used in rdf-proofs

**Apache-2.0**

2.5MB

**59K**
SLoC

# Composite proof system

The goal of this crate is to allow creating and combining zero knowledge proofs by executing several protocols as sub-protocols.

The idea is to represent each relation to be proved as a

, and any relations between
`Statement`

s as a `Statement`

. Both of these types contain public (known to both prover
and verifier) information and are contained in a `MetaStatement`

whose goal is to unambiguously
define what needs to be proven. Some `ProofSpec`

s are specific to either the prover or the verifier
as those protocols require prover and verifier to use different public parameters. An example is Groth16
based SNARK protocols where the prover needs to have a proving key and the verifier needs to
have a verifying key. Both the prover and verifier can know both the proving and verifying key but
they don't need to. Thus for such protocols, there are different `Statement`

s for prover and verifier,
like `Statement`

and `SaverProver`

are statements for prover and verifier respectively,
executing SAVER protocol.`SaverVerifier`

Several

s might need same public parameters like proving knowledge of several BBS+
from the same signer, or verifiable encryption of several messages for the same decryptor. Its not
very efficient to pass the same parameters to each `Statement`

especially when using this code's WASM
bindings as the same values will be serialized and deserialized every time. To avoid this, caller can
put all such public parameters as `Statement`

in an array and then reference those by their index
while creating an `SetupParams`

. This array of `Statement`

is then included in the `SetupParams`

and used by the prover and verifier during proof creation and verification respectively.`ProofSpec`

A common requirement is to prove equality of certain

s of certain `Witness`

s. This
is done by using the `Statement`

meta-statement. For each set of `EqualWitnesses`

s (from the same or different `Witness`

s)
that need to proven equal, a `Statement`

is created which is a set of witness references `EqualWitnesses`

.
Each `WitnessRef`

contains the `WitnessRef`

index and the `Statement`

index in that `Witness`

and
thus uniquely identifies any `Statement`

across `Witness`

s. The `Statement`

meta-statement is also
used to prove predicates over signed messages in zero knowledge, when doing a range-proof over a
signed message (using BBS+), the `EqualWitnesses`

will refer `EqualWitnesses`

s from `Witness`

statement and `Statement ::`PoKBBSSignatureG1

`Statement``::`BoundCheckLegoGroth16

statement. Following are some illustrations of `EqualWitnesses`

` ┌────────────────────────────┐ ┌──────────────────────────────┐ ┌────────────────────────────┐
│ PokBBSSignatureG1 │ │ PokBBSSignatureG1 │ │ PokBBSSignatureG1 │
│ Statement 1 │ │ Statement 2 │ │ Statement 3 │
├────────────────────────────┤ ├──────────────────────────────┤ ├────────────────────────────┤
│ A1, A2, A3, A4, A5 │ │ B1, B2, B3, B4 │ │ C1, C2, C3, C4, C5, C6 │
└─────────▲──────────────────┘ └─────▲────────▲───────────────┘ └─▲────────────────▲─────────┘
│ │ │ │ │
│ │ │ │ │
│ │ │ │ │
│ │ │ │ │
│ ┌-───────────────┴────────┴───┬───────────────────┼──────┬─────────┴──────────────────┐
└────────────┼(0, 2), (1, 1), (2, 0) ├───────────────────┘ │ (2, 3), (3, 4) │
├-────────────────────────────┤ ├────────────────────────────┤
│ EqualWitnesses │ │ EqualWitnesses │
│ MetaStatement 1 │ │ MetaStatement 2 │
│ A3, B2 and C1 are equal │ │ B4 and C5 are equal │
└─────────────────────────────┘ └────────────────────────────┘
`

` For proving certain messages from ``3` `BBS``+` signatures are equal`.` Here there `2` sets of equalities`,`
`1.` message `A3` from 1st signature`,` `B2` from 2nd signature and `C1` from 3rd signature
`2.` message `B4` from 2nd signature and `C5` from 3rd signature
Thus `3` statements`,` one `for` each signature`,` and `2` meta statements`,` one `for` each equality

` ┌────────────────────────────┐ ┌──────────────────────────────┐ ┌────────────────────────────┐
│ PokBBSSignatureG1 │ │ BoundCheckLegoGroth16 │ │ SAVER │
│ Statement 1 │ │ Statement 2 │ │ Statement 3 │
├────────────────────────────┤ ├──────────────────────────────┤ ├────────────────────────────┤
│ A1, A2, A3, A4, A5 │ │ B1 │ │ C1 │
└─────────▲───────▲──────────┘ └─────▲────────-───────────────┘ └───────────────▲────-───────┘
│ |─────────────────| │ │
│ | │ │
│ |──-│-────────────────────| │
│ │ | |───|
│ ┌-───────────────┴────────-───┬────────|───────────────────────────-|─────────────────┐
└────────────┼(0, 2), (1, 0) | |─────────────────│── (0, 4), (2, 1) │
├-────────────────────────────┤ ├────────────────────────────┤
│ EqualWitnesses │ │ EqualWitnesses │
│ MetaStatement 1 │ │ MetaStatement 2 │
│ A3 and B1 are equal │ │ A5 and C1 are equal │
└─────────────────────────────┘ └────────────────────────────┘
`

` For proving certain messages from a ``BBS``+` signature satisfy `2` predicates`,`
`1`) message `A3` satisfies bounds specified `in` statement `2`
`2`) message `A5` has been verifiably encrypted `as` per statement `3.`
Thus `3` statements`,` one `for` a signature`,` and one each `for` a predicate`.` `2` meta statements`,` one each
`for` proving equality of the message of the signature and the witness of the predicate

After creating the

, the prover uses a `ProofSpec`

per `Witness`

and creates a
corresponding `Statement`

. All `StatementProof`

s are grouped together in a `StatementProof`

.
The verifier also creates its `Proof`

and uses it to verify the given proof. Currently it is
assumed that there is one `ProofSpec`

per `StatementProof`

and one `Statement`

per `Witness`

and `Statement`

s appear in the same order in `StatementProof`

as `Proof`

s do in `Statement`

.`ProofSpec`

, `Statement`

and `Witness`

are enums whose variants will be entities from different
protocols. Each of these protocols are variants of the enum `StatementProof`

. `SubProtocol`

s can internally
call other `SubProtocol`

s, eg `SubProtocol`

invokes several `SaverProtocol`

s`SchnorrProtocol`

Currently supports

- proof of knowledge of a BBS or BBS+ signature and signed messages
- proof of knowledge of multiple BBS or BBS+ signature and equality of certain messages
- proof of knowledge of accumulator membership and non-membership
- proof of knowledge of Pedersen commitment opening.
- proof of knowledge of BBS or BBS+ signature(s) and that certain message(s) satisfy given bounds (range proof)
- verifiable encryption of messages in a BBS or BBS+ signature
- proof of knowledge of BBS or BBS+ signature(s) and that certain message(s) satisfy given R1CS. The R1CS is generated from Circom and the proof system used is LegoGroth16. LegoGroth16 is similar to Groth16 but in addition to the zero knowledge proof, it provides a Pedersen commitment to the witness (signed messages in our case). This commitment allows us to prove that the witness in the proof protocol are the same as the signed messages using the Schnorr proof of knowledge protocol.

See following tests for examples:

- test

proves knowledge of 3 BBS+ signatures and also that certain messages are equal among them without revealing them.`pok_of_3_bbs_plus_sig_and_message_equality` - test

proves knowledge of a BBS+ signature and also that certain messages are present and absent in the 2 accumulators respectively.`pok_of_bbs_plus_sig_and_accumulator` - test

proves knowledge of a BBS+ signature and opening of a Pedersen commitment.`pok_of_knowledge_in_pedersen_commitment_and_bbs_plus_sig` - test

shows how to request a blind BBS+ signature by proving opening of a Pedersen commitment.`requesting_partially_blind_bbs_plus_sig` - test

shows how a verifier can link separate proofs from a prover (with prover's permission) and assign a unique identifier to the prover without learning any message from the BBS+ signature. Also this identifier cannot be linked across different verifiers (intentional by the prover).`verifier_local_linkability` - test

shows proving knowledge of a BBS+ signature and that a specific message satisfies some upper and lower bounds i.e. min <= signed message <= max. This is a range proof.`pok_of_bbs_plus_sig_and_bounded_message` - test

shows how to verifiably encrypt a message signed with BBS+ such that the verifier cannot decrypt it but still ensure that it is encrypted correctly for the specified decryptor.`pok_of_bbs_plus_sig_and_verifiable_encryption` - test

shows proving knowledge of several BBS+ signatures using`pok_of_bbs_plus_sig_with_reusing_setup_params`

s. Here the same signers are used in multiple signatures thus their public params can be put as a variant of enum`SetupParams`

. Similarly test`SetupParams`

shows use of`pok_of_knowledge_in_pedersen_commitment_and_equality_with_commitment_key_reuse`

when the same commitment key is reused in several commitments and test`SetupParams`

shows use of`pok_of_bbs_plus_sig_and_verifiable_encryption_of_many_messages`

when several messages are used in verifiable encryption for the same decryptor.`SetupParams` - For R1CS/Circom, see various tests like using less than, not-equals comparison operators on messages signed with BBS+, proving that the preimage of an MiMC hash is the message signed with BBS+, sum of certain signed messages (from same or different signatures) is bounded by a given value, etc here. The Circom compiler output and circuits are here. The circuits were compiled and tested for BLS12-381 curve.

*Note*: This design is largely inspired from my work at Hyperledger Ursa.

*Note*: The design is tentative and will likely change as more protocols are integrated.

#### Dependencies

~11–22MB

~299K SLoC