一类三元幕函数的差分均匀度

The Differential Uniformity of a Class of Ternary Power Function

摘要: S盒(S-boxes)作为分组密码算法中唯一的非线性组件，其性质的好坏对密码算法的安全性至关重 要。 为衡量S盒抵抗差分密码攻击性质的好坏，Nyberg在欧洲密码年会上提出了差分均匀度的 概念。 差分均匀度越小，IS盒的差分密码性质越好。 因此，找寻具有较低差分均匀度的函数来 构造S盒成为了如今密码学研究领域的一个热点。 具有低差分的幕函数因为其特殊的代数结构和 对硬件消耗低等特点，所以往往作为设计S盒的备选函数。 本文，我们研究了F_3n 上的一类幕函数，其中n是偶数。 然后通过对奇特征有限域上二次方程的解的个数进行分析，我们确定了三元幕函数F 差分均匀度的上界。 结果表明F 是一个差分均匀度不超过9的函数。

Abstract: Substitution boxes (S-boxes) as the only nonlinear component in block ciphers algo- rithm, the quality of its properties is crucial to the security of cryptographic algorithm- s. In order to measure the properties of S-boxes to resist differential cryptography attacks, Nyberg proposed the concept of differential uniformity at the European Cryp- tography Annual Conference. The lower the differential uniformity of F is, the better the differential cryptographic properties of S-boxes have. Therefore, finding a function with low differential uniformity to construct S-boxes has become a hot topic in the field of cryptography research today. Power functions with low differential uniformi- ty are often used as alternative functions for S-boxes design because of their special algebraic structure and low hardware consumption. In this paper, we study a class of power functions over F_3n , where n is an even. Then, by considering the number of solutions on the quadratic equation over finite field with odd characteristic, the upper bound of the differential uniformity is determined. The results show that F is a function with differential uniformity no more than 9.