On Succinct Non-interactive Arguments in Relativized Worlds

Succinct non-interactive arguments of knowledge (SNARKs) are cryptographic proofs with strong efficiency properties. Applications of SNARKs often involve proving computations that include the SNARK verifier, a technique called recursive composition. Unfortunately, SNARKs with desirable features such as a transparent (public-coin) setup are known only in the random oracle model (ROM). In applications this oracle must be heuristically instantiated and used in a non-black-box way. In this paper we identify a natural oracle model, the low-degree random oracle model, in which there exist transparent SNARKs for all NP computations relative to this oracle. Informally, letting O be a low-degree encoding of a random oracle, and assuming the existence of (standard-model) collision-resistant hash functions, there exist SNARKs relative to O for all languages in NPO. Such a SNARK can directly prove a computation about its own verifier. To analyze this model, we introduce a more general framework, the linear code random oracle model (LCROM). We show how to obtain SNARKs in the LCROM for computations that query the oracle, given an accumulation scheme for oracle queries. Then we construct such an accumulation scheme for the special case of a low degree random oracle.