A New Method of Construction of Permutation Trinomials with Coefficients 1

Abstract

Permutation polynomials over finite fields are an interesting and constantly active research subject of study for many years. They have important applications in areas of mathematics and engineering. In recent years, permutation binomials and permutation trinomials attract people's interests due to their simple algebraic forms. By reversely using Tu's method for the characterization of permutation polynomials with exponents of Niho type, we construct a class of permutation trinomials with coefficients 1 in this paper.As applications, two conjectures of [19] and a conjecture of [13] are all special cases of our result. To our knowledge, the construction method of permutation polynomials by polar decomposition in this paper is new. Moreover, we prove that in new class of permutation trinomials, there exists a permutation polynomial which is EA-inequivalent to known permutation polynomials for all m ≥ 2. Also we give the explicit compositional inverses of the new permutation trinomials for a special case.

A note on the paper arXiv:2112.14547

“…A polynomial f (X) ∈ F q [X] is called a permutation polynomial if the function c → f (c) permutes F q . The recent paper [2] purports to provide new classes of permutation polynomials, and to resolve three conjectures from the literature. Here we show that the permutation polynomials in that paper are in fact well-known, and that the conjectures had been resolved previously.…”

“…Here we show that the permutation polynomials in that paper are in fact well-known, and that the conjectures had been resolved previously. 1 We also give a new proof of the main result of [2], which is significantly simpler and more direct than all previous proofs, and which demonstrates the general method of producing permutation polynomials that was introduced in [11].…”

“…Remark. The above statement includes all permutation polynomials that can be inferred from any interpretation of [2,Thm. 3.1].…”

“…1 Shortly after [2] appeared on the arXiv, I emailed the content of this note to the third and fifth authors of that paper (I was not able to find email addresses for the other authors), and suggested that [2] should be revised in light of the content of this note. Since I did not receive a reply, and no new version of [2] has been posted, I am posting this note now, in order to help members of the permutation polynomials community avoid spending further time and effort on rediscovering known results.…”

“…the latter result does not require the d i to be positive, and does not say that its expressions for the d i 's should be interpreted as congruences mod (q 2 − 1). However, we assume that the authors of [2] intended to state their result as above. The positivity condition is needed in order to make their result be true (and indeed, negative d i 's would not yield polynomials), and after imposing positivity then the congruences becomes natural, since such congruences do not affect whether f (X) permutes F q 2 .…”