איתי דינור

In this paper we study the affine equivalence problem, where given two functions F,G: {0, 1}n → {0, 1}n, the goal is to determine whether there exist invertible affine transformations A₁,A₂ over GF(2)ⁿ such that G = A₂°F°A₁Algorithms for this problem have several wellknown applications in the design and analysis of Sboxes, cryptanalysis of white-box ciphers and breaking a generalized Even-Mansour scheme. We describe a new algorithm for the affine equivalence problem and focus on the variant where F,G are permutations over n-bit words, as it has the widest applicability. The complexity of our algorithm is about n32n bit operations with very high probability whenever F (or G) is a random permutation. This improves upon the best known algorithms for this problem (published by Biryukov et al. at EUROCRYPT 2003), where the first algorithm has time complexity of n³2²ⁿnd the second has time complexity of about n³2^3n/2 and roughly the same memory complexity. Our algorithm is based on a new structure (called a rank table) which is used to analyze particular algebraic properties of a function that remain invariant under invertible affine transformations. Besides its standard application in our new algorithm, the rank table is of independent interest and we discuss several of its additional potential applications.