Rational Sumchecks

Rational proofs, introduced by Azar and Micali (STOC 2012) are a variant of interactive proofs in which the prover is neither honest nor malicious, but rather rational. The advantage of rational proofs over their classical counterparts is that they allow for extremely low communication and verification time. In recent work, Guo et al. (ITCS 2014) demonstrated their relevance to delegation of computation by showing that, if the rational prover is additionally restricted to being computationally bounded, then every language in NC1 admits a single-round delegation scheme that can be verified in sublinear time. We extend the Guo et al. result by constructing a single-round delegation scheme with sublinear verification for all languages in P. Our main contribution is the introduction of rational sumcheck protocols, which are a relaxation of classical sumchecks, a crucial building block for interactive proofs. Unlike their classical counterparts, rational sumchecks retain their (rational) soundness properties, even if the polynomial being verified is of high degree (in particular, they do not rely on the Schwartz-Zippel lemma). This enables us to bypass the main efficiency bottleneck in classical delegation schemes, which is a result of sumcheck protocols being inapplicable to the verification of the computation’s input level. As an additional contribution we study the possibility of using rational proofs as efficient blocks within classical interactive proofs. Specifically, we show a composition theorem for substituting oracle calls in an interactive proof by a rational protocol. ? Part of this work done while authors were visiting IDC Herzliya, supported by the European Research Council under the European Union’s Seventh Framework Programme (FP 2007-2013), ERC Grant Agreement n. 307952. ?? Work partially supported by RGC GRF grants CUHK410112 and CUHK410113. ??? Supported by the I-CORE Program of the Planning and Budgeting Committee and The Israel Science Foundation (grant No. 4/11). † Supported by ISF grant no. 1255/12 and by the ERC under the EU’s Seventh Framework Programme (FP/2007-2013) ERC Grant Agreement n. 307952. Work in part done while the author was visiting the Simons Institute for the Theory of Computing, supported by the Simons Foundation and by the DIMACS/Simons Collaboration in Cryptography through NSF grant #CNS-1523467. ‡ Work supported by the Check Point Institute for Information Security and by ISF grant no. 1255/12.