An Improved Cryptanalysis Algorithm for Chebyshev Map-Based Discrete Logarithm Problem

Chebyshev map is a chaotic map frequently used in design of cryptography schemes and cryptosystems based on the hardness of the Chebyshev map-based discrete logarithm (CMDL) problem. The properties of Chebyshev map have great impact on the security of these cryptosystems. It has been known that the polynomial sequences generated by Chebyshev map defined on finite fields exhibit strong periodical features which may be utilized for cryptanalysis. This paper presents the periodical properties of Chebyshev polynomial sequences. Based on the properties, an improved cryptanalysis algorithm is proposed for the CMDL problem. It turns out that a chebyshev map-based cryptosystem using Chebyshev prime number as its modulus will have better security, where the Chebyshev prime number is defined as the prime number p satisfying that ( p + 1 ) / 2 or ( p - 1 ) / 2 is also a prime number. In support of cryptanalysis, fast algorithms to calculate the value of a Chebyshev polynomial and find the minimal period of a Chebyshev polynomial sequence are proposed, too. An example is given to show the process of cryptanalysis. Computational results have shown that only a small fraction of prime numbers are valid Chebyshev prime numbers.