Related-Key Differential Analysis of the AES

Abstract

The Advanced Encryption Standard (AES) is considered to be the most important and widely deployed symmetric primitive. While the cipher was designed to be immune against differential and other classical attacks, this immunity does not hold in the related-key setting, and various related-key attacks have appeared over time. This work presents tools and algorithms to search for related-key distinguishers and attacks of differential nature against the AES. First, we propose two entirely different approaches to find optimal truncated differential characteristics and bounds on the minimum number of active S-boxes for all variants of the AES. In the first approach, we propose a simple MILP model that handles better linear inconsistencies with respect to the AES system of equations and that compares particularly well to previous tool-based approaches to solve this problem. The main advantage of this tool is that it can easily be used as the core algorithm to search for any attack on AES exploiting related-key differentials. Then, we design a fast and low-memory algorithm based on dynamic programming that has a very simple to understand complexity analysis and does not depend on any generic solver. This second algorithm provides us useful insight on the related-key differential search problem for AES and shows that the search space is not as big as one would expect. Finally, we build on the top of our MILP model a fully automated tool to search for the best differential MITM attacks against the AES. We apply our tool on AES-256 and find an attack on 13 rounds with only two related keys. This attack can be seen as the best known cryptanalysis against this variant if only 2 related keys are permitted.