A New Algorithm for Computing Branch Number of Non-Singular Matrices over Finite Fields

TL;DR

This paper introduces an improved algorithm for calculating the branch number of non-singular matrices over finite fields, significantly reducing computational complexity for cryptanalysis applications.

Contribution

The paper presents a novel algorithm that enhances the efficiency of computing branch numbers, outperforming classical methods with a square root complexity improvement.

Findings

The new algorithm has a computational complexity that is the square root of the classical method.

It demonstrates improved efficiency in calculating branch numbers for cryptanalysis.

Comparative analysis confirms the algorithm's superior performance.

Abstract

The notion of branch numbers of a linear transformation is crucial for both linear and differential cryptanalysis. The number of non-zero elements in a state difference or linear mask directly correlates with the active S-Boxes. The differential or linear branch number indicates the minimum number of active S-Boxes in two consecutive rounds of an SPN cipher, specifically for differential or linear cryptanalysis, respectively. This paper presents a new algorithm for computing the branch number of non-singular matrices over finite fields. The algorithm is based on the existing classical method but demonstrates improved computational complexity compared to its predecessor. We conduct a comparative study of the proposed algorithm and the classical approach, providing an analytical estimation of the algorithm's complexity. Our analysis reveals that the computational complexity of our algorithm is the square root of that of the classical approach.