Paper 2026/2491

Shift-Invariant Permutations with Polynomial Composition: Construction and Algebraic Structure

Abstract

Shift-invariant permutations of \(\mathbb F_2^n\) are particularly attractive in symmetric cryptography because of their regularity and implementation efficiency. Constructing such permutations with an explicit algebraic structure remains a challenging problem. In this paper, we study a family of shift-invariant transformations of \(\mathbb F_2^n\) constructed from a recursively defined sequence of mappings \(\{\gamma_j\}_{j\ge 0}\) associated with landscapes. First, we prove that, under an explicit dimension condition relating \(n\) to the support of the landscape, the sequence \(\{\gamma_j\}_{j\ge 0}\) has the polynomial composition property if and only if the corresponding landscape is quasi-conserved on \(\mathbb F_2^n\) with a nonzero special index. Then, we determine the eventual zero or periodic behavior of \(\{\gamma_j\}_{j\ge0}\), including the exact index at which the sequence either first vanishes or becomes periodic, together with the least eventual period, and establish linear independence until the first linear relation occurs. When the polynomial composition property holds, we establish an explicit isomorphism between the group of permutation elements in the monoid generated by these mappings and the unit group of a quotient ring \(\mathbb F_2[z]/\langle p(z)\rangle\), where \(p(z)\) is a monomial or a binomial determined by the landscape. This framework reduces the tasks of testing bijectivity, computing iterates, and determining compositional inverses and orders of permutation elements to polynomial computations. Finally, we apply the results to mappings represented by binomials and trinomials in the quotient ring, obtaining explicit criteria and formulas, and tabulate more than one hundred representative constructions of shift-invariant permutations, recovering and unifying several previously studied families.