Permutation Polynomials of Degree 8 Over Finite Fields of Odd Characteristic

Abstract

We give an algorithmic generalisation of Dickson’s method of classifying permutation polynomials (PPs) of a given degree $d$ over finite fields. Dickson’s idea is to formulate from Hermite’s criterion several polynomial equations satisfied by the coefficients of an arbitrary PP of degree $d$. Previous classifications of PPs of degree at most 6 were essentially deduced from manual analysis of these polynomial equations, but this approach is no longer viable for $d>6$. Our idea is to calculate some radicals of ideals generated by the polynomials, implemented by a computer algebra system. Our algorithms running in SageMath 8.6 on a personal computer work very fast to determine all PPs of degree 8 over an arbitrary finite field of odd order $q>8$. Such PPs exist if and only if $q\in \{11,13,19,23,27,29,31\}$ and are explicitly listed in normalised form.

Equivalence Relations for Computing Permutation Polynomials

“…In looking at Tables 5, and Tables 7 through 28, one should keep in mind that the specific nPPs listed in the tables are for the stated primitive polynomial and for our naming convention for elements of GF (q). Our results for degrees 7 and 8 agree with those listed in [12], [13], and [11], except for differences caused by naming conventions.…”

“…For degrees d ≤ 5, all PPs have been described, for example in [8]. More recent work [11,12,13,18,24] gives all PPs of degree d ≤ 8; however, we list our results in Table 5 for completeness. Table 5 does not have columns for d ≥ 11, because the computations become too time consuming (at least for large q).…”

“…For example, N 8 (27) = 6, 899, 256, and there are 364 nPPs, but only 6 equivalence classes as shown in Table 12. Compare our list of classes of degree 8 PPs over GF (27) with the list given in [13], which has 26 nPPs. The difference is that we often combined three of his nPPs into one class by the use of our R F,G relation.…”

“…It is customary in the literature to refer to the normalization operations applied to a PP P (x) by aP (x + b) + c, where a, b, and c are elements of the finite field, with a being nonzero. References to extended normalization applied to a PP P (x), denoted by aP (sx + b) + c, where a, b, c, and s are elements of the finite field, with both a and s are nonzero, appear, for example, in [11,12,13] and have been called linear transformations.…”

“…We say that the integer interval [r, s] has a [t, u] gap, if for all d ∈ [r, s], the expansion of (x + b) d , does not include any nonzero x e monomials, where e ∈ [t, u]. For example, the integer interval [8,13] has a [6,7] x + b 13 . That is, in each of the exhibited expansions there are no x 6 or x 7 terms.…”