Reducing overdefined systems of polynomial equations derived from small scale variants of the AES via data mining methods

• Published in: Journal of Computer Virology and Hacking Techniques Vol. 20; no. 4; pp. 885 - 900
• Main Authors: Berušková, Jana, Jureček, Martin, Jurečková, Olha
• Format: Journal Article
• Language: English
• Published: Paris Springer Paris 01.11.2024; Springer Nature B.V
• Subjects: Algebra; Algorithms; Clustering; Computation; Computer Science; Data mining; Original Paper; Polynomials; Variables; Gröbner bases; Locality sensitive hashing; Partitioning around medoids; Data mining; Algebraic cryptoanalysis; AES
• ISSN: 2263-8733; 2263-8733
• DOI: 10.1007/s11416-024-00540-2

This paper deals with reducing the secret key computation time of small scale variants of the AES cipher using algebraic cryptanalysis, which is accelerated by data mining methods. This work is based on the known plaintext attack and aims to speed up the calculation of the secret key by processing the polynomial equations extracted from plaintext-ciphertext pairs. Specifically, we propose to transform the overdefined system of polynomial equations over GF(2) into a new system so that the computation of the Gröbner basis using the F4 algorithm takes less time than in the case of the original system. The main idea is to group similar polynomials into clusters, and for each cluster, sum the two most similar polynomials, resulting in simpler polynomials. We compare different data mining techniques for finding similar polynomials, such as clustering or locality-sensitive hashing (LSH). Experimental results show that using the LSH technique, we get a system of equations for which we can calculate the Gröbner basis the fastest compared to the other methods that we consider in this work. Experimental results also show that the time to calculate the Gröbner basis for the transformed system of equations is significantly reduced compared to the case when the Gröbner basis was calculated from the original non-transformed system. This paper demonstrates that reducing an overdefined system of equations reduces the computation time for finding a secret key.