International Association for Cryptologic Research

International Association
for Cryptologic Research


Anomalies and Vector Space Search: Tools for S-Box Analysis

Xavier Bonnetain
Léo Perrin
Shizhu Tian
DOI: 10.1007/978-3-030-34578-5_8
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Abstract: S-boxes are functions with an input so small that the simplest way to specify them is their lookup table (LUT). How can we quantify the distance between the behavior of a given S-box and that of an S-box picked uniformly at random?To answer this question, we introduce various “anomalies”. These real numbers are such that a property with an anomaly equal to a should be found roughly once in a set of $$2^{a}$$ random S-boxes. First, we present statistical anomalies based on the distribution of the coefficients in the difference distribution table, linear approximation table, and for the first time, the boomerang connectivity table.We then count the number of S-boxes that have block-cipher like structures to estimate the anomaly associated to those. In order to recover these structures, we show that the most general tool for decomposing S-boxes is an algorithm efficiently listing all the vector spaces of a given dimension contained in a given set, and we present such an algorithm.Combining these approaches, we conclude that all permutations that are actually picked uniformly at random always have essentially the same cryptographic properties and the same lack of structure.
  title={Anomalies and Vector Space Search: Tools for S-Box Analysis},
  booktitle={Advances in Cryptology – ASIACRYPT 2019},
  series={Advances in Cryptology – ASIACRYPT 2019},
  author={Xavier Bonnetain and Léo Perrin and Shizhu Tian},