Orthomorphic Permutation Polynomials of Degree 2d+1 over Finite Field F2n

Orthomorphic permutations have important application in the design of cryptosystems. Based on the one-to-one corresponding relationship between orthomorphic permutations and orthomorphic permutation polynomials, this paper studies orthomorphic permutation polynomials of degree 2^d+1 over finite field F₂n. By the congruence theory and the distributive law of degrees for multiplying polynomials, the necessary condition that the polynomials of degree 2^d+1 are orthomorphism is obtained exactly. Meanwhile, this paper proves that there does not exist orthomorphic permutation polynomials of degree 5 over finite field F₂n. The above results not only provide a method to judge orthomorphic permutation polynomials, but also make a contribution to the enumeration of the kind of polynomials.