TITLE:
Deterministic Algorithm Computing All Generators: Application in Cryptographic Systems Design
AUTHORS:
Boris Verkhovsky
KEYWORDS:
Diffie-Hellman Key Exchange; ElGamal Cryptosystem; Generator; Generator of Pseudo-Random Numbers; Information Hiding; Primitive Element; Safe Prime
JOURNAL NAME:
International Journal of Communications, Network and System Sciences,
Vol.5 No.11,
November
26,
2012
ABSTRACT: Primitive elements play important roles in the Diffie-Hellman protocol for establishment of secret communication keys, in the design of the ElGamal cryptographic system and as generators of pseudo-random numbers. In general, a deterministic algorithm that searches for primitive elements is currently unknown. In information-hiding schemes, where a primitive element is the key factor, there is the freedom in selection of a modulus. This paper provides a fast deterministic algorithm, which computes every primitive element in modular arithmetic with special moduli. The algorithm requires at most O(log2p) digital operations for computation of a generator. In addition, the accelerated-descend algorithm that computes small generators is described in this paper. Several numeric examples and tables illustrate the algorithms and their properties.