Rate-1 Fully Local Somewhere Extractable Hashing from DDH

Somewhere statistically binding (SSB) hashing allows us to sample a special hashing key such that the digest statistically binds the input at m secret locations. This hash function is said to be somewhere extractable (SE) if there is an additional trapdoor that allows the extraction of the input bits at the m locations from the digest. Devadas, Goyal, Kalai, and Vaikuntanathan (FOCS 2022) introduced a variant of somewhere extractable hashing called rate-1 fully local SE hash functions. The rate-1 requirement states that the size of the digest is m+poly () (where is the security parameter). The fully local property requires that for any index i, there is a “very short” opening showing that i-th bit of the hashed input is equal to b for some b € {0,1}. The size of this opening is required to be independent of m and in particular, this means that its size is independent of the size of the digest. Devadas et al. gave such a construction from Learning with Errors (LWE). In this work, we give a construction of a rate-1 fully local somewhere extractable hash function from Decisional Diffie-Hellman (DDH) and BARGs. Under the same assumptions, we give constructions of rate-1 BARG and RAM SNARG with partial input soundness whose proof sizes are only matched by prior constructions based on LWE.