## Abstract

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 A_{1},A_{2} over GF(2)^{n} such that G = A_{2}°F°A_{1}Algorithms 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 n^{3}2^{2n}nd the second has time complexity of about n^{3}2^{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.

Original language | English |
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Title of host publication | Advances in Cryptology - EUROCRYPT 2018 - 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, 2018 Proceedings |

Editors | Jesper Buus Nielsen, Vincent Rijmen |

Publisher | Springer Verlag |

Pages | 413-442 |

Number of pages | 30 |

ISBN (Print) | 9783319783802 |

DOIs | |

State | Published - 1 Jan 2018 |

Event | 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018 - Tel Aviv, Israel Duration: 29 Apr 2018 → 3 May 2018 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 10820 LNCS |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 37th Annual International Conference on the Theory and Applications of Cryptographic Techniques, EUROCRYPT 2018 |
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Country/Territory | Israel |

City | Tel Aviv |

Period | 29/04/18 → 3/05/18 |

## Keywords

- Affine equivalence problem
- Block cipher
- Cryptanalysis
- Even-Mansour cipher
- Rank table

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)