On a class of permutation trinomials in characteristic 2

Abstract

Recently, Tu, Zeng, Li, and Helleseth considered trinomials of the form \(f(X)=X+aX^{q(q-1)+ 1}+bX^{2(q-1)+ 1}\in \mathbb {F}_{q^{2}}[X]\), where q is even and \(a,b\in \mathbb {F}_{q^{2}}^{*}\). They found sufficient conditions on a, b for f to be a permutation polynomial (PP) of \(\mathbb {F}_{q^{2}}\) and they conjectured that the sufficient conditions are also necessary. The conjecture has been confirmed by Bartoli using the Hasse-Weil bound. In this paper, we give an alternative solution to the question. We also use the Hasse-Weil bound, but in a different way. Moreover, the necessity and sufficiency of the conditions are proved by the same approach.

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Correspondence to Xiang-dong Hou.

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This article is part of the Topical Collection on Special Issue on Boolean Functions and Their Applications

Appendix

Appendix

In (3.15),

$$\begin{array}{@{}rcl@{}} F_{4}&=&a_{1}+b+b^{2}+a_{1} b^{2}+b^{3}+b^{4}+a_{1} b^{4}+b^{5}+b^{6}+a_{1} b^{6}+b^{7}+b^{8}\\ &&+({a_{1}^{3}} +{a_{1}^{2}} b+{a_{1}^{3}} b^{4} +{a_{1}^{2}} b^{5})k+({a_{1}^{4}}+{a_{1}^{5}}+{a_{1}^{4}} b+{a_{1}^{4}} b^{2}+{a_{1}^{5}} b^{2}+{a_{1}^{4}} b^{3})k^{2}\\ &&+({a_{1}^{7}} +{a_{1}^{6}} b)k^{3}+{a_{1}^{8}} k^{4}, \end{array} $$
(A1)
$$\begin{array}{@{}rcl@{}} F_{3}&=& 1+{a_{1}^{2}}+{a_{1}^{3}}+{a_{1}^{4}}+b+a_{1} b+{a_{1}^{4}} b+b^{2}+a_{1} b^{2}+{a_{1}^{3}} b^{2}+b^{3}+b^{4}+{a_{1}^{2}}b^{4}\\ &&+b^{5}+a_{1} b^{5}+b^{6}+a_{1} b^{6}+b^{7}+({a_{1}^{2}} +{a_{1}^{4}} +{a_{1}^{5}} +{a_{1}^{2}} b +{a_{1}^{4}} b^{2} +{a_{1}^{2}} b^{4}+{a_{1}^{2}}b^{5})k\\ &&+({a_{1}^{4}} +{a_{1}^{4}} b +{a_{1}^{5}} b +{a_{1}^{4}} b^{2} +{a_{1}^{5}} b^{2} +{a_{1}^{4}} b^{3})k^{2}+({a_{1}^{6}} +{a_{1}^{6}} b)k^{3}, \end{array} $$
(A2)
$$\begin{array}{@{}rcl@{}} F_{2}&=& b+a_{1} b+{a_{1}^{3}} b+{a_{1}^{4}} b+b^{2}+a_{1} b^{2}+{a_{1}^{3}} b^{2}+{a_{1}^{4}} b^{2}+b^{3}+{a_{1}^{2}} b^{3}+{a_{1}^{3}}b^{3}+b^{4}\\ &&+{a_{1}^{2}} b^{4}+{a_{1}^{3}} b^{4}+b^{5}+a_{1} b^{5}+{a_{1}^{2}} b^{5}+b^{6}+a_{1} b^{6}+{a_{1}^{2}} b^{6}+b^{7}+b^{8}\\ &&+({a_{1}^{3}} +{a_{1}^{4}} +{a_{1}^{3}}b +{a_{1}^{3}} b^{2} +{a_{1}^{5}} b^{2} +{a_{1}^{3}} b^{3} +{a_{1}^{4}} b^{3})k\\ &&+({a_{1}^{4}} +{a_{1}^{4}} b +{a_{1}^{4}} b^{2} +{a_{1}^{6}} b^{2}+{a_{1}^{4}} b^{3})k^{2}+{a_{1}^{7}} k^{3}+{a_{1}^{8}} k^{4}, \end{array} $$
(A3)
$$\begin{array}{@{}rcl@{}} F_{1}&=& a_{1}+{a_{1}^{2}}+{a_{1}^{3}}+{a_{1}^{4}}+b+a_{1} b+{a_{1}^{3}} b+{a_{1}^{4}} b+a_{1} b^{2}+{a_{1}^{2}} b^{2}+b^{3}\\ &&+{a_{1}^{2}} b^{3}+{a_{1}^{3}} b^{3}+a_{1} b^{4}+{a_{1}^{3}} b^{4}+b^{5}+a_{1} b^{5}+{a_{1}^{2}} b^{5}+a_{1} b^{6}+b^{7}\\ &&+(1+{a_{1}^{3}} +{a_{1}^{4}} +{a_{1}^{5}} +b +a_{1}b +{a_{1}^{3}} b +{a_{1}^{4}} b +b^{2} +a_{1} b^{2} +{a_{1}^{4}} b^{2}\\ &&+b^{3} +{a_{1}^{3}} b^{3} +{a_{1}^{4}} b^{3} +b^{4} +{a_{1}^{3}} b^{4} +b^{5} +a_{1} b^{5} +b^{6} +a_{1} b^{6} +b^{7})k\\ &&+({a_{1}^{2}} +{a_{1}^{4}} +{a_{1}^{2}} b +{a_{1}^{4}} b +{a_{1}^{4}} b^{2} +{a_{1}^{4}} b^{3} +{a_{1}^{2}} b^{4} +{a_{1}^{2}} b^{5})k^{2}\\ &&+({a_{1}^{4}} +{a_{1}^{6}} +{a_{1}^{4}} b +{a_{1}^{5}} b +{a_{1}^{4}} b^{2} +{a_{1}^{5}} b^{2} +{a_{1}^{4}} b^{3})k^{3}+({a_{1}^{6}} +{a_{1}^{6}} b)k^{4}, \end{array} $$
(A4)
$$\begin{array}{@{}rcl@{}} F_{0}&=& b+{a_{1}^{4}} b+b^{2}+{a_{1}^{4}} b^{2}+b^{5}+b^{6}+(a_{1} +{a_{1}^{2}} +{a_{1}^{3}} +{a_{1}^{4}} +b +a_{1}b +{a_{1}^{3}} b +{a_{1}^{4}} b\\ &&+{a_{1}^{2}} b^{2} +{a_{1}^{5}} b^{2} +b^{3} +{a_{1}^{2}} b^{3} +{a_{1}^{3}} b^{3} +a_{1} b^{4} +{a_{1}^{3}} b^{4} +b^{5} +a_{1}b^{5} +{a_{1}^{2}} b^{5} +b^{7})k\\ &&+(a_{1} +{a_{1}^{2}} +{a_{1}^{4}} +{a_{1}^{5}} +b +{a_{1}^{3}} b +{a_{1}^{4}} b +{a_{1}^{5}}b +b^{2} +a_{1} b^{2} +{a_{1}^{3}} b^{2} +{a_{1}^{4}} b^{2} +{a_{1}^{6}} b^{2}\\ &&+b^{3} +{a_{1}^{3}} b^{3} +{a_{1}^{4}} b^{3} +b^{4} +a_{1} b^{4} +{a_{1}^{2}} b^{4} +{a_{1}^{3}} b^{4} +b^{5} +b^{6} +a_{1} b^{6} +b^{7} +b^{8})k^{2}\\ &&+({a_{1}^{3}} +{a_{1}^{5}} +{a_{1}^{2}} b +{a_{1}^{4}} b +{a_{1}^{5}} b^{2} +{a_{1}^{4}} b^{3} +{a_{1}^{3}} b^{4} +{a_{1}^{2}} b^{5})k^{3}\\ &&+({a_{1}^{4}} +{a_{1}^{5}} +{a_{1}^{6}}+{a_{1}^{4}} b +{a_{1}^{4}} b^{2} +{a_{1}^{5}} b^{2} +{a_{1}^{4}} b^{3} )k^{4}+({a_{1}^{7}} +{a_{1}^{6}} b )k^{5}+{a_{1}^{8}} k^{6}. \end{array} $$
(A5)

In (3.27) and (3.28),

$$\begin{array}{@{}rcl@{}} h_{1}&=& 1+{a_{1}^{4}}+{a_{1}^{2}} b+{a_{1}^{4}} b^{2}+{a_{1}^{2}} b^{5}+b^{8}+({a_{1}^{2}} +{a_{1}^{6}} +{a_{1}^{2}} b^{2} +{a_{1}^{2}}b^{4} +{a_{1}^{2}} b^{6} )k\\ &&+({a_{1}^{4}} +{a_{1}^{6}} b +{a_{1}^{4}} b^{4} )k^{2}+({a_{1}^{6}} +{a_{1}^{6}} b^{2} )k^{3}, \end{array} $$
(A6)
$$\begin{array}{@{}rcl@{}} h_{2}&=& {a_{1}^{2}}+{a_{1}^{6}}+b+{a_{1}^{4}} b+{a_{1}^{2}} b^{2}+{a_{1}^{6}} b^{2}+b^{3}+{a_{1}^{4}} b^{3}+{a_{1}^{2}} b^{4}+{a_{1}^{4}}b^{5}+{a_{1}^{2}} b^{6}+{a_{1}^{4}} b^{7}+b^{9}\\ &&+b^{11}+({a_{1}^{4}} +{a_{1}^{8}} +{a_{1}^{4}} b^{4} )k+({a_{1}^{2}} +{a_{1}^{6}} +{a_{1}^{4}} b +{a_{1}^{8}} b +{a_{1}^{6}} b^{4}+{a_{1}^{4}} b^{5} +{a_{1}^{2}} b^{8} )k^{2}\\ &&+({a_{1}^{4}} +{a_{1}^{8}} +{a_{1}^{4}} b^{2} +{a_{1}^{8}} b^{2} +{a_{1}^{4}} b^{4} +{a_{1}^{4}} b^{6} )k^{3}+({a_{1}^{6}} +{a_{1}^{8}} b^{3} +{a_{1}^{6}} b^{4} )k^{4}\\ &&+({a_{1}^{8}} +{a_{1}^{8}} b^{2} )k^{5}. \end{array} $$
(A7)

In (3.30),

$$\begin{array}{@{}rcl@{}} d_{1}&=& {a_{1}^{2}}+{a_{1}^{6}}+a_{1}^{10}+a_{1}^{14}+b+{a_{1}^{4}} b+{a_{1}^{8}} b+a_{1}^{12} b+{a_{1}^{4}} b^{3}+a_{1}^{12} b^{3}\\ &&+{a_{1}^{2}} b^{4}+a_{1}^{10} b^{4}+b^{5}+{a_{1}^{8}} b^{5} +{a_{1}^{6}} b^{6}+{a_{1}^{8}} b^{7}+{a_{1}^{6}} b^{8}+{a_{1}^{8}} b^{11}\\ &&+{a_{1}^{6}} b^{14}+{a_{1}^{2}} b^{16}+b^{17}+{a_{1}^{4}}b^{17} +{a_{1}^{4}} b^{19}+{a_{1}^{2}} b^{20}+b^{21}\\ &&+({a_{1}^{2}} b +{a_{1}^{6}} b +a_{1}^{10} b +a_{1}^{14} b +{a_{1}^{2}} b^{3} +a_{1}^{10} b^{3} +a_{1}^{10}b^{7} +a_{1}^{10} b^{9}\\ &&+{a_{1}^{2}} b^{17} +{a_{1}^{6}} b^{17} +{a_{1}^{2}} b^{19} )k\\ &&+({a_{1}^{2}} +{a_{1}^{6}} +a_{1}^{10} +a_{1}^{14}+{a_{1}^{2}} b^{2} +{a_{1}^{6}} b^{2} +a_{1}^{10} b^{2} +a_{1}^{14} b^{2} +{a_{1}^{8}} b^{3}\\ &&+a_{1}^{12} b^{3} +{a_{1}^{6}} b^{4} +a_{1}^{10}b^{4}+a_{1}^{12} b^{5} +{a_{1}^{6}} b^{6} +{a_{1}^{6}} b^{8} +{a_{1}^{6}} b^{10} \\ && +a_{1}^{10} b^{10} +{a_{1}^{8}} b^{11} +{a_{1}^{6}} b^{12} +{a_{1}^{6}}b^{14} +{a_{1}^{2}} b^{16} +{a_{1}^{2}} b^{18} )k^{2}\\ &&+(a_{1}^{10} b^{3} +a_{1}^{14} b^{3} +a_{1}^{10} b^{5} +a_{1}^{10} b^{7} +a_{1}^{10} b^{9} )k^{3}\\ &&+({a_{1}^{6}}+a_{1}^{14} +{a_{1}^{6}} b^{2} +a_{1}^{14} b^{2} +{a_{1}^{6}} b^{4} +{a_{1}^{6}} b^{6} +{a_{1}^{6}} b^{8}\\ &&+{a_{1}^{6}} b^{10} +{a_{1}^{6}} b^{12} +{a_{1}^{6}} b^{14} )k^{4}, \end{array} $$
(A8)
$$\begin{array}{@{}rcl@{}} d_{2}&=& 1+{a_{1}^{4}}+{a_{1}^{8}}+a_{1}^{12}+{a_{1}^{2}} b+a_{1}^{10} b+b^{2}+{a_{1}^{8}} b^{2}+{a_{1}^{6}} b^{3}+a_{1}^{10}b^{3}\\ &&+{a_{1}^{8}} b^{6}+{a_{1}^{8}} b^{8}+{a_{1}^{6}} b^{11}+b^{16}+{a_{1}^{4}} b^{16}+{a_{1}^{2}} b^{17}+b^{18}\\ &&+({a_{1}^{4}} b +a_{1}^{12} b +{a_{1}^{4}} b^{3} +{a_{1}^{8}} b^{3} +{a_{1}^{4}} b^{5} +{a_{1}^{8}} b^{5} +{a_{1}^{4}} b^{7}\\ &&+{a_{1}^{8}} b^{7} +{a_{1}^{4}} b^{9} +{a_{1}^{8}} b^{9} +{a_{1}^{4}} b^{11} +{a_{1}^{4}} b^{13} +{a_{1}^{4}} b^{15} )k\\ &&+({a_{1}^{4}} +a_{1}^{12} +{a_{1}^{4}} b^{2} +a_{1}^{12} b^{2} +{a_{1}^{4}} b^{4} +{a_{1}^{4}} b^{6} +{a_{1}^{4}} b^{8}\\ &&+{a_{1}^{4}} b^{10} +{a_{1}^{4}} b^{12} +{a_{1}^{4}} b^{14} )k^{2}. \end{array} $$
(A9)

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Hou, X. On a class of permutation trinomials in characteristic 2. Cryptogr. Commun. 11, 1199–1210 (2019). https://doi.org/10.1007/s12095-018-0342-1

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Keywords

  • Finite field
  • Hasse-Weil bound
  • Permutation polynomial

Mathematics Subject Classification (2010)

  • 11T06
  • 11T55
  • 14H05