On generating elements of orders dividing p2k ±p k ∈+∈1

Abstract

In 1999 Gong and Harn proposed a new cryptosystem based on third-order characteristic sequences with period p 2k ∈+∈p k ∈+∈1 for a large prime p and fixed k. In order to find key parameters and therefore to construct a polynomial whose characteristic sequence is equal to p 2k ∈+∈p k ∈+∈1 one should generate a prime p such that the prime factorization of the number p 2k ∈+∈p k ∈+∈1 is known. In this paper we propose new, efficient methods for finding the prime p and the factorization of the aforementioned number. Our algorithms work faster in practice than those proposed before. Moreover, when used for generating of XTR key parameters, they are a significant improvement over the Lenstra-Verheul Algorithm. Our methods have been implemented in C++ using LiDIA and numerical test are presented. © 2008 Springer Berlin Heidelberg.