Abstract
Perfect nonlinear (PN) functions have been an interesting subject of study for a long time and have applications in coding theory, cryptography, combinatorial designs, and so on. In this paper, the planarity of the trinomialsxpk+1+ux2+vx2pkover GF(p2k) are presented. This class of PN functions are all EA-equivalent tox2.
Highlights
Let p be a prime and GF(pn) a finite field with pn elements
We propose a new family of Perfect nonlinear (PN) trinomials over GF(p2k) which are composed of inequivalent monomials x2 and xpk+1
If one of bl and bl+k equals to 0, L1(x) = blxpl + bl+kxpl+k is a monomial permutation. If both of bl and bl+k equal to 0, we get u = Vpk, which leads to a contradiction
Summary
Let p be a prime and GF(pn) a finite field with pn elements. Let f be a mapping from GF(pn) to itself. We know that APN functions are optimal over GF(2n) This concept is of interest in cryptography since differential and linear cryptanalysis exploit the uniform property of the functions which are used in many block ciphers, such as DES. In recent papers [3, 4], PN functions were used to describe new finite commutative semifields of odd order. Bierbrauer [18] introduced a general projection method to construct commutative semifields and generalized the known PN functions. Mathematical Problems in Engineering theoretic approach to prove the planarity of a function in [20] They presented new commutative semifields with two parameters and get new PN functions [21]. In their paper [22], Kyureghyan and Ozbudak constructed some new PN functions by the products of two linearized polynomials.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
