Sharp

Abstract

We provide optimized range proofs, called Sharp, in discrete logarithm and hidden order groups, based on square decomposition. In the former setting, we build on the paradigm of Couteau et al. (Eurocrypt '21) and optimize their range proof (from now on, CKLR) in several ways: (1) We introduce batching via vector commitments and an adapted ∑;-protocol. (2) We introduce a new group switching strategy to reduce communication. (3) As repetitions are necessary to instantiate CKLR in standard groups, we provide a novel batch shortness test that allows for cheaper repetitions. The analysis of our test is nontrivial and forms a core technical contribution of our work. For example, for λ = 128 bit security and B = 64 bit ranges for N = 1 (resp. N = 8) proof(s), we reduce the proof size by 34% (resp. 75%) in arbitrary groups, and by 66% (resp. 88%) in groups of order 256-bit, compared to CKLR.