Permutation binomials over finite fields

Abstract

Abstract. We prove that if x m + ax n permutes the prime field F p , whereprove that if q ≥ 4 and m > n > 0 are fixed and satisfy gcd(m − n, q − 1) > 2q(log log q)/ log q, then there exist permutation binomials over F q of the form x m + ax n if and only if gcd(m, n, q − 1) = 1.

Classification of Permutation Trinomials and Quadrinomials Over Prime Fields

“…The paper [9] shows that all permutation binomials belong to the same class x r f (x q−1 d ) with d < 2 log p (under the assumption that the above conjecture is true). It turns out that most permutation trinomials and quadrinomials belong to a similar class, but, in addition, there is one new class for trinomials, and two new classes for quadrinomials.…”

“…For a fixed value of d, the class of such polynomials is closed under composition, and the paper [6] contains results on the size of the corresponding group. The paper [9] contains the theorem stating that for permutation binomials over prime fields F p , the value d is less than √ p, and the conjecture that d < 2 log p. If this conjecture is true, then the following operations on permutation binomials can be implemented in polynomial time:…”

Determination of a type of permutation binomials over finite fields

“…Carlitz and Wells' proof of the existence result relies on a bound on the Weil sum of a multiplicative character of F q [16], [7,Theorem 5.39]. Using the Hasse-Weil bound on the number of degree one places of a function field over F q [10, Theorem V.2.3], Masuda and Zieve [8] were able to make Carlitz-Wells' existence result (with k = 1) more precise. They proved that if q ≥ 4 and q−1 e > 2q(log log q)/ log q, then there exists a ∈ F * q such that x c (x q−1 e + a) is a PP of F q .…”

On some permutation polynomials over $\mathbb {F}_q$ of the form $x^r h(x^{(q-1)/d})$

“…However, for binomials the situation becomes much more mysterious. Despite the attention of numerous authors since the 1850's (cf., e.g., [5,25,20,6,7,27,31,29,32,21,34,3,24]), the known results seem far from telling the full story of permutation binomials. This brings us to the present paper.…”