| In cryptosystems,block ciphers encrypt text usually by using substitution boxes(S-boxes).Functions used to the S-boxes should have low differential uniformity,high nonlinearity,and algebraic degree larger than 3.Moreover,those functions should be permutations of Substitution-Permutation Network.In present paper,we construct several kinds of permutations of low differential uniformity and with good cryptographic properties.These functions can be used to design S-boxes for block ciphers.Inspiring by the ideal of permutation induced by the Inverse Function on a subfield of the finite field F2n with cardinality 2n,we construct our permutations of low differential uniformity and with high nonlinearity and algebraic degree.Chapter 3 of this paper is the main part for constructing the permutations of low differential uniformity.Firstly,we define F(x)as (?) where U is subset of the finite field F2n,which satisfies certain conditions.All of cycle sets of H(x)=(x+ε)-1+s related to G(x)=x-1 are computed according to the definition of the cycle sets.In addition,some properties of those cycle sets are also studied.Then a sufficient condition that F(x)is a permutation of the finite field F2n is obtained.Next we find that,when F(x)satisfies certain conditions,from case to case,the differential uniformity of F(x)is 4 or 6.Thus,the permutations of low differential uniformity are obtained.In chapter 4,we discuss other cryptographic properties of F(x)obtained above.We prove that all these functions have the optimal algebraic degree and high nonlinearity.Further,we compute the nonlinearity,the expended Walsh spectrum and differential spectrum of F(x)for different parameters.From these messages we show that,among the permutations of differential uniformity 4 we constructed,there are two new families not appeared in literature;on the other hand,all of the permutations of differential uniformity 6 we constructed are CCZ-equivalent to none of the known ones.In this way,it is proved that the functions we obtained can be used to design S-boxes for block ciphers. |
