Disjoint Linear Codes And Their Application In The Construction Of Cryptographic Functions

Disjoint linear codes play an important role in the construction of cryptographic functions with good cryptographic properties.In this thesis,we mainly study how to generate a large number of disjoint linear codes quickly and efficiently,and how to construct S-boxes by using disjoint linear codes.The following research results have been achieved:(1)A new construction of disjoint linear codes is presented in this thesis.When n is not divisible by k,2k-1 matrices Mi of k×k are obtained by means of one primitive polynomial of degree k,and 2m-1 matrices Mji of k×m are obtained by means of one primitive polynomial of degree m,where i?0,1,…,2k-2,m=n-uk,j=0,1,…,2m-2 and u=[n/k]-1.A large number of disjoint[n,k]linear codes can be generated by combining with such identity matrix of k×k,null matrix of k x k and null matrix of k×m.When n is divisible by k,2k-1 matrices of k x k are obtained by means of only one primitive polynomial of degree k,where i=0,1,…,2k-2.All the disjoint[n,k]linear codes can be generated by combining with such Mi,identity matrix of k×k and null matrix of k×k.The number of disjoint[n,k,1]linear codes generated by our work is N(n,k,1)=1+(?)2n-ik It reaches the upper bound when n is divisible by k.It is 2m-k-1 less than the upper bound when n is not divisible by k.The new construction can obtain the sets of disjoint linear codes with currently best known cardinality.The computational cost of this construction is much smaller than that of other constructions,especially when it is n?k.The number of disjoint[n,k,2]linear codes generated by our work is N(n,k,2)=?2n-ik-u.(2)Three new constructions of(n,n)S-boxes,when n is even,are presented in this thesis.The first construction needs to use disjoint linear codes.The second construction needs to use MM class functions.The third construction needs to use disjoint linear codes and M-M class functions.The(n,n)S-boxes generated by the first construction and the second construction are balanced,and the(n,n)S-box generated by the third construction is not balanced.The nonlinearities of the(n,n)S-boxes generated by the first construction and the third construction are better than that of the(n,n)S-box generated by the second con-struction.The nonlinearity of(n,n)S-boxes generated by the first constructions is the best one.