Recovering S-boxes from the Differential Distribution Table and Affine Equivalence Classes of S-boxes with Respect to Modular Addition
DOI:
https://doi.org/10.20535/tacs.2664-29132025.2.328960Abstract
This paper considers the problem of S-box recovery from its differential distribution table (DDT) with respect to modular addition. We describe the structure of DDT for affine S-boxes and affine transformations of S-boxes. We found some unexpected internal symmetry in DDT w.r.t. modular addition, which holds for other algebraic operations, but not for bitwise addition (XOR). We describe two classes of affine transformations (affine shifts) which preserve the structure of DDT. For a recovery of S-box from its DDT we propose a backtracking-based algorithm, which is moderately effective for medium-size S-boxes. We apply our algorithm for three-bit S-boxes and describe the structure of their DDT equivalence classes; among other things, it was shown that affine shifts do not cover all DDT equivalence class members.
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