feat: Pedersen vector commitment scheme multi VK verification

[ivokub]

Description

This PR changes the Pedersen commitment verification and verification key to support batch verifying multiple commitment POKs with different keys using a folded POK.

This is a breaking change to allow the folding to work.

Previously, keys are defined as:

G <- sample random G2 element
KK_i <- sample random commitment basis vector in G1
sigma_i <- sample random scalar. must be discarded after setup (toxic waste)
pk_i, vk_i = (KK_i, sigma_i KK_i), (G, -1/sigma_i G)

and commitment and its POK for vector aa as

cmt, pok = <aa, KK_i>, <aa, sigma_i KK_i>

the verification of individual POK was

pairingcheck((cmt, pok), (G, -1/sigma G)) == 1

and batch verification

pairingcheck((pok_1, .., r^{n-1} pok_n, \sum_i r^i cmt_i), (-1/sigma_1 G, ..., -1/sigma_n G, G)

In this way however I didn't find a way to fold the POKs which may be useful to reduce the proof size in case of many commitments. However, if we change the definitions to

pk_i, vk_i = (KK_i, sigma_i KK_i), (G, -sigma_i G)
pairingcheck((pok, cmt), (G, -sigma G)) == 1
pairingcheck((cmt, .., r^{n-1} cmt, \sum_i r^i pok_i), (-sigma_1 G, ..., -sigma_n G, G)

then the prover can already send \sum_i r^i pok_i on its own.

Type of change

Please delete options that are not relevant.

• New feature (non-breaking change which adds functionality)
• Breaking change (fix or feature that would cause existing functionality to not work as expected)

How has this been tested?

• TestFoldProofs
• TestSemiFoldProofs
• ExampleBatchVerifyMultiVk

Checklist:

• I have performed a self-review of my code
• I have commented my code, particularly in hard-to-understand areas
• I have made corresponding changes to the documentation
• I have added tests that prove my fix is effective or that my feature works
• I did not modify files generated from templates
• golangci-lint does not output errors locally
• New and existing unit tests pass locally with my changes
• Any dependent changes have been merged and published in downstream modules

[ivokub]

@Tabaie - addressed comments.