Periodic Properties of Commutative Polynomials Defined by Fourth Order Recurrence Relations with Two Variables Over Z 2 k
Daisaburo Yoshioka · 2023
In constructing public-key cryptosystems, commutative property is an essential characteristic. Taking advantage of the commutative property of Chebyshev polynomials, a publickey cryptosystem over the residue class ring ${\mathbb{Z}_{{2^k}}}$ has been introduced, which can be implemented very efficiently. Unfortunately, however, the cryptosystem is broken using knowledge of the periodic properties of Chebyshev polynomials. Although commutative polynomials with two variables can be candidates for a public-key cryptosystem instead of Chebyshev polynomials, characteristics of the polynomials should be discussed carefully. In this study, we analyzed some properties of commutative polynomials with two variables over ${\mathbb{Z}_{{2^k}}}$. More precisely, the degree period and the condition for permutation polynomial are given theoretically and verified experimentally. Based on the derived properties, a security evaluation of a key-exchange protocol using the polynomials is also discussed.