NP-Complete Sets for Computing Discrete Logarithms and Integer Factorization

Abstract

We explore an NP-complete set such that the problem of breaking a cryptographic scheme reduces to the complete set, where the reduction can be given in a straightforward form like the reduction from the graph isomorphism to the subgraph isomorphism. We construct such NP-complete sets Π_DL and Π_IF for the discrete logarithm problem modulo a prime and the integer factoring problem, respectively. We also show that the decision version of Diffie-Hellman problem reduces directly to Π_DL with respect to the polynomial-time many-one reducibility. These are the first complete sets that have direct reductions from significant cryptographic primitives.