The Constructions Of Permutations And Bent Functions

Bent functions have been applied to cryptography,spread spectrum,coding theory,and combinatorial design.Permutations play an important role in the design of crypto-graphic transformations such as block ciphers,hash functions and stream ciphers.By using the Kronecker product this paper presents a general recursive construction method of per-mutations over finite field.As applications of our method,several infinite classes of permu-tations are obtained.By means of the permutations obtained and Maiorana-Mcfarland(M-M for short)functions we construct several infinite families of Bent functions.Chapter 1 introduces the development and the current research status of the study,as well as some basic concepts and main lemmas.Chapter 2 first found some examples of permutation functions on the finite field of F2,second we use the general recursive method of finite permutation function to construct infinite class of permutation function,and constructed the permutation function of a series of new infinite classes by Kronecker product.By using the computer,we found some new permutation functions of 3 dimensional to 9 dimensional on the finite field of F2,it laid the foundation for construction of n(n? 4)infinite dimensional Bent function on the finite field of F2.Finally,several new permutation functions on F3 and F4 are found by computer.Chapter 3 a new method of recursively constructing Bent functions is proposed by using permutation function and M-M function,and several infinite classes of Bent functions are obtained.Chapter 4 concludes the main content of this paper and puts forward some suggestions and some unsolved problems.