On differential uniformity of maps that may hide an algebraic trapdoor

TL;DR

This paper explores the differential properties of certain permutations in the affine group over binary vector spaces, focusing on a new group operation that creates an alternative vector space structure.

Contribution

It introduces a novel group operation on vector spaces and analyzes its impact on the differential uniformity of affine permutations, revealing potential algebraic trapdoors.

Findings

Identifies differential properties of permutations under the new group operation

Shows how the alternative structure can conceal algebraic trapdoors

Provides insights into the security implications of such permutations

Abstract

We investigate some differential properties for permutations in the affine group, of a vector space V over the binary field, with respect to a new group operation $\circ$, inducing an alternative vector space structure on $V$ .