On the Generalization of Butterfly Structure

Yongqiang Li,Shizhu Tian,Yuyin Yu,Mingsheng Wang

doi:10.46586/tosc.v2018.i1.160-179

Abstract

Butterfly structure was proposed in CRYPTO 2016 [PUB16], and it cangenerate permutations over F22n from power permutations over F2n for odd n. Afterthat, a generalized butterfly structure was proposed in IEEE IT [CDP17], which cangenerate permutations over F22n from any permutation over F2n . There is also anothergeneralization which was given in [FFW17]. Up to now, three constructions based onbutterfly structure and Gold type permutations are proposed. In the present paper,we give a construction which contains the three previous constructions as special casesand also generates new permutations with good cryptographic properties. Moreover,we give a characterization of the number of solutions of a special system of linearequations in a more general way, which is useful to investigate the cryptographicproperties of quadratic functions obtained with butterfly construction based on Goldexponents.

Highlights

S(ubstitution)-boxes play an important role in symmetric ciphers since they serve as the confusion part and in most cases are the only nonlinear components of round functions
Constructing permutations with low differential uniformity and high nonlinearity is of particular interest in the study of cryptographic functions
The lower bound for functions on F2n is 2, and the functions that achieve this bound are called almost perfect nonlinear (APN) functions

Summary

Introduction

S(ubstitution)-boxes play an important role in symmetric ciphers since they serve as the confusion part and in most cases are the only nonlinear components of round functions. The core part of the proof of previous constructions relies on the determination of the number of roots of a system of linear equations of the following type a1x2i + a2x + b1y2i + b2y = 0, a3x2i + a4x + b3y2i + b4y = 0, where aj, bj ∈ F2n , 1 ≤ j ≤ 4 are some particular elements derived from the differences of special quadratic functions This system was studied case by case in previous constructions [LW14, PUB16, CDP17, FFW17]. 3, several results concerning the number of solutions of a special system of linear equations are given, which are very helpful to characterize the properties of quadratic functions obtained with butterfly construction based on Gold exponents.

Preliminaries

On the number of solutions to a special system of linear equations

Experimental results

Conclusion