Research On The Construction And Properties Of Multi-output Cryptographic Functions

The security of stream cipher depends on the security strength of the key stream,and the Boolean function plays an important role in stream ciphers as a nonlinear component of the key stream generator.In order to design Boolean functions that can resist various cryptographic attacks and satisfy different requirements of various cryptographic algorithms,it is usually necessary to construct Boolean functions with multiple cryptographic criteria such as optimal or strictly almost optimal nonlinearity,resiliency,algebraic degree,etc.However,there is a restrictive relationship among various cryptographic criteria,so how to achieve the optimal trade-off between them is an important research topic.This paper focuses on the nonlinearity and resiliency of Boolean functions and vectorial Boolean functions,and the main results that have been achieved so far as follows:(1)Two methods are given to construct balanced Boolean functions and resilient Boolean functions with very high nonlinearity respectively.By giving several nonlinear small variables functions with special algebraic normal forms and analyzing the distribution of their Walsh spectrum,it is found that these nonlinear functions can be used as small functions in the improved Maiorana-McFarland(M-M)cascade construction method.Based on the above findings,two construction methods have been proposed using these small functions with special structures,and a class of balanced Boolean functions with nonlinearity 2n-1-2n/2+2n/2-2 and a class of resilient Boolean functions with nonlinearity 2n-1-2n/2+2n/2-2k Nh are obtained respectively.Where the nonlinearity of the banlanced function can reach the maximum known so far and the nonlinearity of the resilient Boolean function is the strictly almost optimal,e.g.the(44,1,2,243-221-214-212-211-210)function.(2)Two methods of constructing resilient multi-output Boolean functions with high nonlinearity are given by using disjoint linear codes and disjoint spectral functions in combination with General Maiorana-McFarland(GMM)construction method respectively.And two classes of resilient multi-output Boolean functions with high nonlinearity and high algebraic degree are obtained.In addition,the results are compared and analyzed with the highest nonlinearity of known functions,showing that the nonlinearity of our functions>2n-1-2n/2,satisfying the requirement of strictly almost optimal nonlinearity,and the nonlinearity of some functions is the highest known result at present,e.g.the(34,6,1,233-216-213-211)function,etc.(3)By combining disjoint linear codes with vectorial matrices with special properties,a method is proposed to construct resilient multi-output Boolean functions with a high nonlinearity 2n-1-2n/2-1-2k-1,which is a strictly almost optimal function.And the nonlinearity of some resilient multi-output Boolean functions constructed by our method is the maximum value known in the same condition,e.g.the(32,4,1,231-215-210)function,etc.