Bit security of the CDH problems over finite fields

Abstract

It is a long-standing open problem to prove the existence of (deterministic) hard-core predicates for the Computational Diffie- Hellman (CDH) problem over finite fields, without resorting to the generic approaches for any one-way functions (e.g., the Goldreich-Levin hard-core predicates). Fazio et al. (FGPS, Crypto ’13) made important progress on this problem by defining a weaker Computational Diffie- Hellman problem over Fp2, i.e., Partial-CDH problem, and proving, when allowing changing field representations, the unpredictability of every single bit of one of the coordinates of the secret Diffie-Hellman value. In this paper, we show that all the individual bits of the CDH problem over Fp2 and almost all the individual bits of the CDH problem over Fpt for t > 2 are hard-core.