Permutation rotation-symmetric S-boxes, liftings and affine equivalence

TL;DR

This paper studies permutation rotation-symmetric vectorial Boolean functions that are liftings from smaller Boolean functions, generalizing the S-box used in Keccak, and explores their affine equivalence and construction methods.

Contribution

It introduces new constructions for permutation rotation-symmetric S-boxes and analyzes their affine equivalence with associated Boolean functions.

Findings

General constructions for rotation-symmetric S-boxes

Affine equivalence relationships between S-boxes and Boolean functions

Extension of Keccak's S-box framework

Abstract

In this paper, we investigate permutation rotation-symmetric (shift-invariant) vectorial Boolean functions on $n$ bits that are liftings from Boolean functions on $k$ bits, for $k\leq n$. These functions generalize the well-known map used in the current Keccak hash function, which is generated via the Boolean function on $3$ variables, $x_1+(x_2+1)x_3$. We provide some general constructions, and also study the affine equivalence between rotation-symmetric S-boxes and describe the corresponding relationship between the Boolean function they are associated with.