Security of public-key cryptosystems based on commutative polynomials over ℤ²_2^k

抄録

The commutative property has been an essential characteristic in the development of public-key cryptosystems. There are essentially only two kinds of commutative polynomials: monomials and Chebyshev polynomials. By leveraging the commutative property of them, efficiently implementable public-key cryptosystems over the residue class ring ℤ_2^k have been introduced; unfortunately, however, they can be broken. Although commutative polynomials with two variables could be potential candidates for a public-key cryptosystem over the ring, the characteristics of these polynomials should be rigorously investigated. In this study, we analyzed several properties of commutative polynomials with two variables over ℤ²_2^k. More precisely, the degree period and the condition for permutation polynomials are discussed theoretically and verified experimentally. Based on the derived properties, a security analysis of a key exchange protocol using the commutative polynomials is also discussed.