Nova: Recursive Zero-Knowledge Arguments from Folding Schemes

Abstract

We introduce a new approach to realize incrementally verifiable computation (IVC), in which the prover recursively proves the correct execution of incremental computations of the form $y=F^{(\ell )}(x)$, where F is a (potentially non-deterministic) computation, x is the input, y is the output, and $\ell > 0$. Unlike prior approaches to realize IVC, our approach avoids succinct non-interactive arguments of knowledge (SNARKs) entirely and arguments of knowledge in general. Instead, we introduce and employ folding schemes, a weaker, simpler, and more efficiently-realizable primitive, which reduces the task of checking two instances in some relation to the task of checking a single instance. We construct a folding scheme for a characterization of NP and show that it implies an IVC scheme with improved efficiency characteristics: (1) the “recursion overhead” (i.e., the number of steps that the prover proves in addition to proving the execution of F) is a constant and it is dominated by two group scalar multiplications expressed as a circuit (this is the smallest recursion overhead in the literature), and (2) the prover’s work at each step is dominated by two multiexponentiations of size O(|F|), providing the fastest prover in the literature. The size of a proof is O(|F|) group elements, but we show that using a variant of an existing zkSNARK, the prover can prove the knowledge of a valid proof succinctly and in zero-knowledge with $O(\log {|F|})$ group elements. Finally, our approach neither requires a trusted setup nor FFTs, so it can be instantiated efficiently with any cycles of elliptic curves where DLOG is hard.