On the primitivity of PRESENT and other lightweight ciphers

TL;DR

This paper establishes conditions under which the round functions of certain lightweight translation-based ciphers, including PRESENT, generate the alternating group, highlighting their strong algebraic properties and primitivity.

Contribution

It introduces two sufficient conditions for the primitivity of round functions and proves that under these, the group is the alternating group for specific lightweight ciphers.

Findings

Round functions of PRESENT generate the alternating group.

Conditions for primitivity are satisfied by some lightweight ciphers.

The group generated is highly symmetric, being the alternating group.

Abstract

We provide two sufficient conditions to guarantee that the round functions of a translation based cipher generate a primitive group. Furthermore, under the same hypotheses, and assuming that a round of the cipher is strongly proper and consists of m-bit S-Boxes, with m = 3; 4 or 5, we prove that such a group is the alternating group. As an immediate consequence, we deduce that the round functions of some lightweight translation based ciphers, such as the PRESENT cipher, generate the alternating group.