多输出弹性k-旋转对称布尔函数的存在性与构造

多输出布尔函数可由多个单输出布尔函数表示,在分组密码中有着广泛的应用.多输出k-旋转对称布尔函数(k-RSBF)是多输出旋转对称布尔函数(RSBF)的扩展.本文首先研究多输出旋转对称函数和多输出k-旋转对称函数的轨道分布情况,给出了计算两类函数中长度相同轨道个数的方法.其次研究了平衡多输出k-旋转对称布尔函数的存在性,给出了在选择合适的k的前提下,n=pr、n=2pr和n=2r时,平衡(n,m)k-RSBF的构造方法.之后研究弹性多输出k-旋转对称布尔函数的存在性,分别给出了r≥3,n=2r,2≤m≤2r-r,k=2时1阶弹性(n,m)k-RSBF的构造方法,以及p为奇素数,r≥2,n=pr,2≤m≤p-1,k=p时1阶弹性(n,m)k-RSBF的构造方法.最后我们还对两种方法得到的1阶弹性多输出k-旋转对称布尔函数进行仿真测试.

Existences and Constructions of Resilient Vectorial k-Rotation Symmetric Boolean Functions

Vectorial Boolean functions are widely used in block ciphers,they can be represented by multiple single output Boolean functions.Vectorial k-rotation symmetric Boolean functions(k-RSBF)are the extension of vectorial rotation symmetric Boolean functions(RSBF).This paper presents the orbital distributions of vectorial rotation symmetric Boolean functions and vectorial k-rotation symmetric Boolean functions.It also gives a method to compute the number of orbits of the same size in two functions.It is proved that there exists balanced(n,m)k-RSBF for suitable k,where n=pr,n=2 pr and n=2r,and some instances are constructed.The existence of resilient(n,m)k-RSBF is also proved,and some specific functions are constructed,including 1-resilient(2r,m) 2-RSBF,where2≤m≤2r-r,r≥3 and 1-resilient(pr,m) p-RSBF,where 2≤m≤p-1,r≥2,and p is an odd prime.Finally,two examples of the 1-resilient(n,m) k-RSBFs are obtained using the two methods.