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On Interactive Oracle Proofs for Boolean R1CS Statements

Publicat a:Lecture Notes In Computer Science. 13411 LNCS 230-247 - 2022-01-01 13411 LNCS(), DOI: 10.1007/978-3-031-18283-9_11

Autors: Cascudo I; Giunta E

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The framework of interactive oracle proofs (IOP) has been used with great success to construct a number of efficient transparent zk-SNARKs in recent years. However, these constructions are based on Reed-Solomon codes and can only be applied directly to statements given in the form of arithmetic circuits or R1CS over large enough fields F. This motivates the question: what is the best way to apply these IOPs to statements that are naturally written as R1CS over small fields, and more concretely, the binary field F2 ? While one can just see the system as one over an extension field F2e containing F2, this seems wasteful, as it uses e bits to encode just one “information” bit. In fact, in FC21 the work BooLigero devised a way to apply the well-known Ligero while being able to encode e bits into one element of F2e. In this paper, we introduce a new protocol for F2 -R1CS which among other things relies on a more efficient embedding which (for practical parameters) allows to encode ≥ e/ 4 bits into an element of F2e. Our protocol makes then black box use of lincheck and rowcheck protocols for the larger field. Using the lincheck and rowcheck introduced in Aurora and Ligero respectively we obtain 1.31 - 1.65 × smaller proofs for Aurora and 3.71 × for Ligero. We also estimate the reduction of prover time by a factor of 24.7 × for Aurora and between 6.9 - 32.5 × for Ligero without interactive repetitions. Our methodology uses the notion of reverse multiplication friendly embeddings introduced in the area of secure multiparty computation, combined with a new IOPP to test linear statements modulo a subspace V≤F2e which may be of independent interest.

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