Search

Constructing permutation polynomials is a hot topic in the area of finite fields, and permutation polynomials have many applications in different areas. Recently, several classes of permutation trinomials were constructed. In 2015, Hou surveyed the achievements of permutation polynomials and novel methods. But, very few were known at that time. Recently, many permutation binomials and trinomials have been constructed. Here we survey the significant contribution made to the construction of permutation trinomials over finite fields in recent years. Emphasis is placed on significant results and novel methods. The covered material is split into three aspects: the existence of permutation trinomials of the respective forms xrh(xs), λ1xa + λ2xb + λ3xc and x + xs(qm-1)+1 + xt(qm-1)+1, with Niho-type exponents s, t. [ABSTRACT FROM AUTHOR]

In this paper, we extend the concepts of Neutrosophy to Boolean function and define ClassicalBalanced, AntiBalanced and NeutroBalanced functions. We consider functions of the form f(x) = Tr(xd), where the exponent d may be Gold exponent, Kasami exponent, Welch exponent or any arbitrary positive integer. We, for different values of d, examine nature of these functions with respect to the above stated three categories. [ABSTRACT FROM AUTHOR]

A polynomial can represent every function from a finite field to itself. The functions which are also permutations of the field give rise to permutation polynomials, which have potential applications in cryptology and coding theory. Permutation polynomials over finite rings are studied with respect to the sequences they generate. The sequences obtained through some permutation polynomials are tested for randomness by carrying out known statistical tests. Random number generation plays a major role in cryptography and hence permutation polynomials may be used as random number generators. [ABSTRACT FROM AUTHOR]

Orthogonal properties of Latin squares represented by permutation polynomials are discussed. Pairs of bivariate polynomials over small rings are considered. [ABSTRACT FROM AUTHOR]

Our work is motivated by a recent paper of Rivest [6], concerning permutation polynomials over the rings Zn with n = 2w. Permutation polynomials over finite fields and the rings Zn have lots of applications, including cryptography. For the special case n = 2w, a characterization has been obtained in [6] where it is shown that such polynomials can form a Latin square (0 ≤ x, y ≤ n - 1) if and only if the four univariate polynomials P(x, 0), P(x, 1), P(0, y) and P(1, y) are permutation polynomials. Further, it is shown that pairs of such polynomials will never form Latin squares. In this paper, we consider bivariate polynomials P(x, y) over the rings Zn when n ≠ 2w. Based on preliminary numerical computations, we give complete results for linear and quadratic polynomials. Rivest's result holds in the linear case while there are plenty of counterexamples in the quadratic case. [ABSTRACT FROM AUTHOR]