## Cryptology ePrint Archive: Report 2019/1229

Transparent SNARKs from DARK Compilers

Benedikt Bünz and Ben Fisch and Alan Szepieniec

Abstract: We construct a new polynomial commitment scheme for univariate and multivariate polynomials over finite fields, with logarithmic size evaluation proofs and verification time, measured in the number of coefficients of the polynomial. The underlying technique is a \emph{Diophantine Argument of Knowledge} (DARK), leveraging integer representations of polynomials and groups of unknown order. Security is shown from the strong RSA and the adaptive root assumptions. Moreover, the scheme does not require a trusted setup if instantiated with class groups. We apply this new cryptographic compiler to a restricted class of algebraic linear IOPs, which we call \emph{Polynomial IOPs}, to obtain doubly-efficient public-coin interactive arguments of knowledge for any NP relation with succinct communication. With linear preprocessing, the online verifier's work is logarithmic in the circuit complexity of the relation.

There are many existing examples of Polynomial IOPs (PIOPs) dating back to the first PCP (BFLS, STOC'91). We present a generic compilation of any PIOP using our DARK polynomial commitment scheme. In particular, compiling the PIOP from \textsf{PLONK} (GWC, ePrint'19), an improvement on \textsf{Sonic} (MBKM, CCS'19), yields a public-coin interactive argument with quasi-linear preprocessing, quasi-linear (online) prover time, logarithmic communication, and logarithmic (online) verification time in the circuit size. Applying Fiat-Shamir results in a SNARK, which we call \textsf{\textbf{Supersonic}}.

\textsf{Supersonic} is also concretely efficient with 10KB proofs and under $100$ms verification time for circuits with 1 million gates (estimated for 120-bit security). Most importantly, this SNARK is \emph{transparent}: it does not require a trusted setup. We obtain zk-SNARKs by applying a hiding variant of our polynomial commitment scheme with zero-knowledge evaluations. \textsf{Supersonic} is the first complete zk-SNARK system that has both a practical prover time as well as asymptotically \emph{logarithmic} proof size and verification time.

Category / Keywords: cryptographic protocols / Zero-Knowledge, SNARK, IOP, RSA, class groups

Date: received 19 Oct 2019, last revised 18 Nov 2019

Contact author: alan at nervos org,benedikt@cs stanford edu,bfisch@cs stanford edu

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2019/1229

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