On Black-Box Ring Extraction and Integer Factorization

The black-box extraction problem over rings has (at least) two important interpretations in cryptography: An efficient algorithm for this problem implies (i) the equivalence of computing discrete logarithms and solving the Diffie-Hellman problem and (ii) the in-existence of secure ring-homomorphic encryption schemes. In the special case of a finite field, Boneh/Lipton [1] and Maurer/ Raub [2] show that there exist algorithms solving the black-box extraction problem in subexponential time. It is unknown whether there exist more efficient algorithms. In this work we consider the black-box extraction problem over finite rings of characteristic n, where nhas at least two different prime factors. We provide a polynomial-time reduction from factoring nto the black-box extraction problem for a large class of finite commutative unitary rings. Under the factoring assumption, this implies the in-existence of certain efficient generic reductions from computing discrete logarithms to the Diffie-Hellman problem on the one side, and might be an indicator that secure ring-homomorphic encryption schemes exist on the other side.