Resonance properties of Chebyshev chaotic sequences

A public-key cryptosystem based on Chebyshev maps has been recently proposed, where one can reap benefits of the semi-group property of Chebyshev polynomials. However, we have to confront the following two disadvantages of the semi-group property in using such a chaotic cryptosystem: (1) the semi-group property gives a lot of public keys with a fixed private key ´s´ from a public key which provides a suitable situation for known-plaintext attacks as observed in attacks to block ciphers; (2) this property implies the resonance property to be defined here, which enables us to take a statistical approach to known-plaintext attacks based on prime factorization of a large integer ´s´. This notifies us that the cryptographic resistance to such a known-plaintext attack is to be evaluated.