Research On The Differential Spectrum Of Generalized Kasami Functions

Cryptographic functions,which mainly include Boolean functions and vectorial functions(S box),are the most important components of block ciphers.Their cryptographic properties are crucial to the security of the underlying ciphers.For example,S-boxes,as the unique nonlinear component in most of block ciphers,are usually selected to be functions with low difference uniformity,which can resist against differential attack strongly.In 2010,the definition of difference spectrum of the cryptographic functions was proposed by C.Blondeau et al.The difference distribution of a cryptographic function is reflected by the difference spectrum,which is useful for estimating its resistance to some variants of differential cryptanalysis.Compared with difference uniformity,the difference spectrum is a more refined characterization of the difference properties.In 2001,H.Dobbertin calculated the correlation distribution of the m sequences based on difference spectrum of the cryptographic functions.There are few results on the difference spectrum of cryptographic functions.Except for the ordinary cases,only a few difference spectrum of power functions have been completely calculated.Solving the difference spectrum of cryptographic function is very significant in cryptographic function theory.The objective of this paper is to determine the differential spectrum of the power function xp2k-pk+1 over Fpn,where p is an odd prime,n,k,e are integers with gcd(n,k)=e and n/e being odd.In fact,we need to calculate distribution of the solutions to the(x+1)p2k-pk+1xp2k-pk+1=? when ? traverses Fpn.That is to find the relationship between the difference equation and the number of solutions of the system of equations in the finite field.We calculated the differential spectrum of the power function xp2k-pk+1 by a exponential function and the system of equation over the finite field.The differential spectrum of the power function xp2k-pk+1 is (?).