Key Space of Cryptosystem Based on Chebyshev Polynomials over Finite Field

A recently proposed public key cryptosystem based on Chebyshev polynomials suggests a new approach to data encryption. The sequence of Chebyshev polynomial is proved to be periodically, but its symmetry properties have not been investigated in depth. In this paper, a new representation of Chebyshev polynomial is introduced to study these properties, and their impacts on the cryptosystem. It is shown that the factual private key space of the cryptosystem is only half or quarter of the period of Chebyshev polynomial sequence. Proper parameters should be chosen to ensure the security of the cryptosystem.