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T-函数是由Klimov和Shamir在2002年提出的一类新的非线性函数,这种函数软硬件实现速度快、效率高,而且所生成的序列线性复杂度高、稳定性强,故有望代替线性反馈移位寄存器,成为新的序列密码设计的非线性驱动环节,多项式函数作为一类密码学中常用的T-函数,其可逆性、周期性一直是相关研究中的重要问题,Klimov利用函数的代数正规型给出了多项式函数f(x)=a_0+a_1x+…+a_dx~d mod 2~n是单圈的充分条件,同时借助于“bit-slice”方法和参数的概念给出了广义多项式函数f(x)=a_0⊕a_1x⊕…⊕…a_dx~d mod 2~n是置换的充分条件.进一步地,刘卓军等借助于徐克舰的2-adic整数的乘法公式,给出了函数f(x)=a_0⊕a_1x⊕…⊕a_dx~d mod 2~n单圈性的判定定理.本文利用1-Lipschitz函数模2-微分理论,发展使用模4-微分确定遍历变换的技术,并结合“bit-slice”方法,给出函数遍历性判定的一种新方法,进而给出了此类函数单圈性判定定理的一个新证明.
T-function is a new kind of nonlinear function proposed by Klimov and Shamir in 2002. This kind of function is fast and efficient in hardware and software implementation, and the generated sequence has high linear complexity and strong stability, so it is expected to replace Linear feedback shift register has become a new nonlinear driving link in the design of sequence cryptography. Polynomial functions are commonly used as a kind of T-function in cryptography. Its reversibility and periodicity have always been important issues in related researches. Klimov uses the function Gives the sufficient condition for the polynomial function f (x) = a_0 + a_1x + ... + a_dx ~ d mod 2 ~ n to be a single circle, given by the concept of the “bit-slice” method and parameters The generalized polynomial function f (x) = a_0⊕a_1x⊕ ... ⊕ ... a_dx ~ d mod 2 ~ n is a sufficient condition for permutation.Furthermore, Liu Zhuojun et al. Give a The decision theorem for the unicyclicity of the function f (x) = a_0⊕a_1x⊕ ... ⊕ a_dx ~ d mod 2 ~ n is developed in this paper by using the modulo 2-differential theory of the 1-Lipschitz function and developing a technique that uses a 4-differentiate deterministic traversal transform , And combined with the “bit-slice” method, a new method for determining ergodicity of functions is given. Such a determination function theorem lap a new proof.