Three Theorems on odd degree Chebyshev polynomials and more generalized permutation polynomials over a ring of module $2^w$

TL;DR

This paper thoroughly analyzes the periodic properties of odd degree Chebyshev polynomials over rings modulo 2^w, revealing their implications for cryptographic security and generalizing to other permutation polynomials.

Contribution

It completely characterizes the orbital and degree periods of these polynomials, and demonstrates the insecurity of related key-exchange protocols.

Findings

Both periods are fully characterized.

Key-exchange protocols using these polynomials are insecure.

Generalized permutation polynomials also lead to insecure protocols.

Abstract

Odd degree Chebyshev polynomials over a ring of modulo $2^w$ have two kinds of period. One is an "orbital period". Odd degree Chebyshev polynomials are bijection over the ring. Therefore, when an odd degree Chebyshev polynomial iterate affecting a factor of the ring, we can observe an orbit over the ring. The "orbital period" is a period of the orbit. The other is a "degree period". It is observed when changing the degree of Chebyshev polynomials with a fixed argument of polynomials. Both kinds of period have not been completely studied. In this paper, we clarify completely both of them. The knowledge about them enables us to efficiently solve degree decision problem of Chebyshev polynomial over the ring, and so a key-exchange protocol with Chebyshev polynomial over the ring is not secure. In addition, we generalize the discussion and show that a key-exchange protocol with more generalized permutation polynomials which belong to a certain class is not secure.