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Master Thesis Year : 2020

On the QIC of quadratic APN functions

Abstract

Vectorial Boolean functions are useful in symmetric cryptography for designing block ciphers among other primitives. One of the main attacks on these ciphers is the differential attack. Differential attacks exploit the highest values in the differential distribution table (DDT). A function is called Almost Perfect Nonlinear (APN) if all entries in the DDT belong to {0, 2}, which is optimal against the differential attack. The search for APN permutations, as well as their classification, has been an open problem for more than 25 years. All these $n$-variable permutations are known for $n$ ≤ 5, but the question remains unsolved for large values of $n$. It has been conjectured for a long time that, when $n$ is even, APN bijective functions do not exist. However, in 2009, Dillon and his coauthors have found an APN permutation of 6 variables. Our aim on this thesis is to find such functions for larger n. Dillon et al.’s approach was finding an APN permutation from a quadratic APN function which are CCZ-equivalent. Two vectorial Boolean functions are “CCZ-equivalent” if there exists an affine permutation that maps the graph of one function to the other. It preserves the differential properties of a function and thus the APN property. Our approach to find APN permutations will be the same. We propose an idea of representing a quadratic vectorial Boolean function using a cubic structure called quadratic indicator cube (QIC). Also, we describe the criteria related to this cube that is necessary and sufficient for a function to be APN. Then we present some algorithms based on backtracking to change the elements of the cube in such a way that if we start from an APN function it remains APN. We implement our modification algorithm in SageMath and then, in order to get better performances, we also implement it in C++. We also use multithreading in order to get better performances. For $n$ = 6 our algorithm outputs 13 EA-equivalence class of functions, which covers all possible classes for $n$ = 6.
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Dates and versions

hal-03135737 , version 1 (09-02-2021)

Identifiers

  • HAL Id : hal-03135737 , version 1

Cite

Shibam Ghosh. On the QIC of quadratic APN functions. Discrete Mathematics [cs.DM]. 2020. ⟨hal-03135737⟩
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