Imprimitive permutations groups generated by the round functions of key-alternating block ciphers and truncated differential cryptanalysis

TL;DR

This paper investigates the structure of permutation groups generated by key-alternating block cipher round functions, revealing their block systems are linear subspace translates and establishing conditions for primitivity, with implications for cryptanalysis.

Contribution

It proves all block systems are linear subspace translates and provides a primitivity condition applicable to AES, linking group theory to cryptanalysis.

Findings

Block systems are translates of linear subspaces.

A condition guarantees the group is primitive, including AES.

Connection established between group structure and truncated differential cryptanalysis.

Abstract

We answer a question of Paterson, showing that all block systems for the group generated by the round functions of a key-alternating block cipher are the translates of a linear subspace. Following up remarks of Paterson and Shamir, we exhibit a connection to truncated differential cryptanalysis. We also give a condition that guarantees that the group generated by the round functions of a key-alternating block cipher is primitive. This applies in particular to AES.