Security of Public-Key Cryptosystems Based on Chebyshev Polynomials Over Z/pkZ

A public-key cryptosystem using Chebyshev polynomials defined on a finite set has recently been developed, which is a kind of chaos-based cryptography. The security of this cryptosystem relies on the difficulty of finding the degree of Chebyshev polynomials from given parameters. In this brief, we propose polynomial time algorithms to identify the degree of Chebyshev polynomials modulo a prime power. We demonstrate that the cryptosystem based on Chebyshev polynomials modulo a prime power is not secure. This result also means that there are no commutative polynomials for constructing public-key cryptosystems modulo a prime power.