New permutation trinomials constructed from fractional polynomials

Abstract

Permutation trinomials over finite fields consititute an active research due to their simple algebraic form, additional extraordinary properties and their wide applications in many areas of science and engineering. In the present paper, six new classes of permutation trinomials over finite fields of even characteristic are constructed from six fractional polynomials. Further, three classes of permutation trinomials over finite fields of characteristic three are raised. Distinct from most of the known permutation trinomials which are with fixed exponents, our results are some general classes of permutation trinomials with one parameter in the exponents. Finally, we propose a few conjectures.

A note on permutation polynomials over finite fields

“…In the same paper, they also proposed the following two conjectures, which can be used to obtain two new classes of permutation trinomials with the form (1). More recent progress on permutation trinomials can be found in [2,6,7,8,9,10,12,13,14,15,16,20].…”

A conjecture on permutation trinomials over finite fields of characteristic two

“…Let u = a −1 + a −2 m and v = a −1 · a −2 m , then we have 1) a 6 (b 4 +b 2 +1) Notice that if (4) has two or more solutions in µ 2 m +1 for some t in µ 2 m +1 , then F (x) must have a quadratic factor, say x 2 + ax + b. If so, by Lemma 3, the coefficients a, b must satisfy the relation (11). Then, to prove Conjecture 1, we only need to show that (11) has no solution in F 2 m if gcd(m, 5) = 1, To this end, define G(x, y) = x 5 + (y 2 + y)x 3 + x 2 + (y 3 + y 2 )x + y 5 + y 2 + 1.…”

“…If so, by Lemma 3, the coefficients a, b must satisfy the relation (11). Then, to prove Conjecture 1, we only need to show that (11) has no solution in F 2 m if gcd(m, 5) = 1, To this end, define G(x, y) = x 5 + (y 2 + y)x 3 + x 2 + (y 3 + y 2 )x + y 5 + y 2 + 1.…”

Permutation polynomials, fractional polynomials, and algebraic curves

“…In [9] the authors presented the following conjecture about permutation trinomials in characteristic 3. (1) Let q = 3 k , k even, and…”

“…The equation of C can be rewritten as 4,8,9,15,16,20, 21}. None of its roots is in µ q+1 and therefore C does not contains points of type (x, y) ∈ µ 2 q+1 ; that is, g(x) permutes µ q+1 .…”