Efficient Lucas Public-Key Cryptosystems

Abstract

The Lucas sequence (or equivalently, Dickson polynomial ) has important applications in cryptology. In the first part of this paper, we will investigate distribution properties of Lucas sequences over GF(p) where p is a prime, and over Zn where n is a product of two primes. We will show that all elements of the Lucas sequence can be classified into two disjoint subsets according to their Legendre symbols. In the second part, we will show how to utilize those distinct properties of Lucas sequences to design Lucas public-key cryptographic algorithms. The performance of these cryptosystems is more efficient than cryptosystems based on the exponential function.