Author(s)

Boris Verkhovsky

Computer Science Department, New Jersey Institute of Technology, University Heights, Newark, USA.

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.

Diffie-Hellman Key Exchange; ElGamal Cryptosystem; Generator; Generator of Pseudo-Random Numbers; Information Hiding; Primitive Element; Safe Prime

B. Verkhovsky, "Deterministic Algorithm Computing All Generators: Application in Cryptographic Systems Design," International Journal of Communications, Network and System Sciences, Vol. 5 No. 11, 2012, pp. 715-719. doi: 10.4236/ijcns.2012.511074.