##
**Combinatorial properties of differentially 2-uniform substitutions.**
*(Russian.
English summary)*
Zbl 1475.94159

Summary: A combinatorial approach to the investigation and methods of construction of differentially 2-uniform substitutions of the vector space over the finite field \(F_2\) is proposed. Necessary and sufficient conditions for the family of sets associated with a differentially 2-uniform substitution to be a symmetric block design are given. It is shown that a substitution is differentially 2-uniform if and only if it is a solution of a similarity equations system connecting a family of translations with a family of unequal weights involutions. We suggest methods of construction of differentially 2-uniform substitutions by means of the Cayley table of an additive group of finite field \(F_{2^m}\).

### MSC:

94A60 | Cryptography |

60C05 | Combinatorial probability |

05B05 | Combinatorial aspects of block designs |

### Keywords:

differentially 2-uniform substitutions; family of sets associated with a substitution; \((\alpha, \beta)\)-configurations; unequal weights involutions### References:

[1] | Nyberg K., “Differentially uniform mappings for cryptography”, EUROCRYPT’93, Lect. Notes Comput. Sci., 765, 1994, 55-64 · Zbl 0951.94510 |

[2] | Sachkov V. N., “Probability distributions of number of configurations and discordances of random permutations from regular cyclic classes”, Probabilistic methods in Discrete Mathematics, VSP, Utreht, 2002, 23-40 |

[3] | Sachkov V. N., “Tsepi Markova iteratsionnykh sistem preobrazovanii”, Trudy po diskretnoi matematike, 6, Fizmatlit, M., 2002, 165-183 |

[4] | Sachkov V. N., Kombinatornye metody diskretnoi matematiki, Nauka, M., 1977 |

[5] | Tang D., Carlet C., Tang X., Differentially 4-uniform bijections by permuting the inverse functions, Cryptology ePrint Aechive, rep. 2013/639 · Zbl 1329.94079 |

[6] | Carlet C., Charpin P., Zinoviev V., “Codes, bent functions and permutations suitable for DES-like cryptosystems”, Designs, Codes and Cryptography, 15:2 (1998), 125-156 · Zbl 0938.94011 · doi:10.1023/A:1008344232130 |

[7] | Riordan Dzh., Vvedenie v kombinatornyi analiz, Izd-vo inostrannoi literatury, M., 1963 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.