Grant: Discovery Grants Program - Individual

Let p be a prime number and q be a power of p. A polynomial is a permutation polynomial of a finite field of q elements where q is a power of a prime number p, if it induces a bijective map from this finite field to itself. The study of permutation polynomials goes back to Hermite for the prime field of p elements, and to Dickson for finite field of q elements. In recent decades, the interest in permutation polynomials has significantly increased due to their applications in cryptography, coding theory and combinatorics, and to the depth of the possibilities that arise when dealing with permutation polynomials, such as bounding the number of points on certain curves and improving Weil's bound. One of the challenging problems proposed by Lidl and Muller is to find new classes of permutation polynomials. Despite the interest of numerous people on the subject for 160 years, there are only a handful of families of permutation polynomials, because there is no general strategy for finding new families of permutation polynomials. Since the existance theorems for checking whether a given polynomial is a permutation polynomial are not easy to apply then, one needs new ideas. In the proposed research, using new ideas, I plan to develop new tools to check whether a given polynomial is a permutation polynomials or not and to list all permutation polynomials over a finite field of p elements. I plan as well to find new families of permutation polynomials over a finite field of q elements where $q$ is a power of a prime number p.