More classes of permutation pentanomials over finite fields with characteristic two

Let q = 2(m). In this paper, we investigate permutation pentanomials over F-q(2) of the form f(x) = x(t) + x(1)(r)(q-1)+t + x(2)(r)(q-1)+t + x(3)(r)(q-1)+t + x(4)(r)(q-1)+t with gcd(x(4)(r) + x(3)(r) + x(2)(r) + x(1)(r) + 1, xt + x(1)(t-r) + x(2)(t-r)+ x(</span>)(t-r) + x(4)(t-r) ) = 1. We transform the problem concerning permutation property of f(x) into demonstrating that the corresponding fractional polynomial permutes the unit circle U of F-q(2) with order q + 1 via a well-known lemma, and then into showing that there are no certain solution in F-q for some highdegree equations over F-q associated with the fractional polynomial. According to numerical data, we have found all such permutations with 4 <= t < 100, 1 <= r(i )< t, i is an element of [1, 4]. Several permutation polynomials are also investigated from the fractional polynomials permuting the unit circle U found in this paper. (c) 2024 Elsevier Inc. All rights are reserved, including those for text and data mining, AI training, and similar