The Construction And Cryptographic Properties Of Permutation Polynomials From Niho Exponents

In recent years,because of their widely applications in coding,cryptography,combinatorial designs and other areas of mathematics and engineering,permutation polynomials have attracted a lot of interest.Let p be a odd prime,m be a positive integer,n=2m,a positive integer d is called Niho exponent overF_pn if it satisfies d?p^j?mod p^m-1?,0?j<m.In this paper,we investigate permutation polynomials of the form as follows:f?x?=x^?p-1??q+1+x^pq-x^q+?p-1?,x?F_{p n},where m is a positive integer,q=p^m and n=2m.We prove that when p?{3,5},the polynomial is a permutation polynomial ofF_pn if and only if m is even.In addition,when p?{3,5},l be a positive integer and gcd?2l+p,q-1?=1,we can also prove that the polynomial with a more general form:g?x?=x^{?q+1??l+?p-1?q+1}+x^?q+1??l+pq-x^{?q+1??l+q+p-1}is also a permutation polynomial ofF_pnn if and only if m is even.We also show that when p=5,the permutation polynomial we construct is not multiplicative equivalent to the known ones.In this paper,we also study the cryptography properties of the following permutation trinomials over F_2nf?x?=x+x^2m+x^{2m-1??2m-1?+1},where m is a positive integer satisfying m?0 mod 3 and n=2m.By the power sum method,we have determined the Wlash spectrum of f?x?and then obtaind its nonlinearity.The results show that when m>2,the Walsh spectrum of f?x?is 5-value and the nonlinearity is2^n-1-3?2^n2-1.In addition,we derive a lower bound of its differential uniform?_F,which is that?_F?2.Our results show that f?x?has a ideal nonlinearity,and it can be resistant against the linear cryptography attack.