The problem of a computationally feasible method of finding the discrete logarithm in a (large) finite field is discussed, presenting the main algorithms in this direction. Some cryptographic schemes based on the discrete logarithm are presented. Finally, the theory of linear recurring sequences is outlined, in relation to its applications in cryptology.

This paper shows that the Discrete Logarithm Problem is in ZPP (where MCSP is the Minimum Circuit Size Problem). This result improves the previous bound that the Discrete Logarithm Problem is in BPP Allender et al. (2006). In doing so, this paper helps classify the relative difficulty of the Minimum Circuit Size Problem.

For suitable composite modulus n and suitable base a, the discrete logarithm hash function x + ax mod n is collision free and one way if factoring n is hard. Further results on the relation between the discrete logarithm problem and factoring are given. Some complexity theory issues are considered.

We define three hard problems in the theory of elliptic divisibility sequences (EDS Association, EDS Residue and EDS Discrete Log), each of which is solvable in subexponential time if and only if the elliptic curve discrete logarithm problem is solvable in sub-exponential time. We also relate the problem of EDS Association to the Tate pairing and the MOV, Frey-Rück and Shipsey EDS attacks on th...

In this paper I propose the new fully homomorphic public-key encryption scheme without bootstrapping that is based on the discrete logarithm assumption and computational Diffie–Hellman assumption of multivariate polynomials on octonion ring. The key size of this scheme and complexity for enciphering /deciphering become to be not so large to handle. keywords: fully homomorphic public-key encrypt...