We provide new constructions for zero-knowledge commit-and-prove SNARKs (CP-SNARKs) with a universal updatable SRS.
Informally, a commit-and-prove argument system is one that can efficiently prove relations over committed inputs. They have many applications, including allowing for efficient composition of proof systems with different strength points.

We first show a general technique to compile Algebraic Holographic Proofs (AHP) with special “decomposition” properties into an efficient CP-SNARK with universal and updatable SRS. We require that the polynomials in an AHP can be easily decomposed into components that refer to the committed part of the witness and the rest of the witness respectively.

We then show that some of the most efficient AHP constructions—Marlin, PLONK, and Sonic—satisfy our compilation requirements. To obtain succinct instantiations of our protocols we rely on recent advancements in compressed $Sigma$-protocol theory (Attema and Cramer, Crypto ’20). Our constructions retain the succinct proof size of the underlying AHP and only impose an additional proof size that grows logarithmically with the size of the committed component of the witness.

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