×

zbMATH — the first resource for mathematics

Some classes of monomial complete permutation polynomials over finite fields of characteristic two. (English) Zbl 1314.11073
A polynomial \(f(x)\in \mathbb F_q[x]\) is a complete permutation polynomial if both \(f(x)\) and \(f(x)+x\) are permutations. These arise in the study of orthogonal Latin squares.
The authors consider the case \(q=2^n\) and \(f(x)=ax^d\). They construct four new classes of complete permutation monomials. One class is an extension of a result of Z. Tu, X. Zeng and L. Hu [Finite Fields Appl. 25, 182–193 (2014; Zbl 1284.05012)]. The other three have the form: \[ n=rk\qquad d=1+(2^{rk}-1)/(2^k-1)\qquad (k,r)=1, \] for \(r=4,6\) and \(10\). The coefficient \(a\) is chosen so that an associated polynomial is a Dickson polynomial.

MSC:
11T06 Polynomials over finite fields
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Akbary, A.; Ghioca, D.; Wang, Q., On constructing permutations of finite fields, Finite Fields Appl., 17, 1, 51-67, (2011) · Zbl 1281.11102
[2] Charpin, P.; Kyureghyan, G. M., Cubic monomial bent functions: a subclass of \(\mathcal{M}\), SIAM J. Discrete Math., 22, 2, 650-665, (2008) · Zbl 1171.11062
[3] Coulter, R.; Henderson, M.; Matthews, R., A note on constructing permutation polynomials, Finite Fields Appl., 15, 5, 553-557, (2009) · Zbl 1215.11112
[4] Ding, C.; Xiang, Q.; Yuan, J.; Yuan, P., Explicit classes of permutation polynomials of \(\mathbb{F}_{3^{3 m}}\), Sci. China Ser. A, 53, 4, 630-647, (2009)
[5] Dobbertin, H., One-to-one highly nonlinear power functions on \(G F(2^n)\), Appl. Algebra Eng. Commun. Comput., 9, 2, 139-152, (1998) · Zbl 0924.94026
[6] Dobbertin, H.; Felke, P.; Helleseth, T.; Rosendahl, P., Niho type cross-correlation functions via dickson polynomials and Kloosterman sums, IEEE Trans. Inf. Theory, 52, 2, 613-627, (2006) · Zbl 1178.94220
[7] Hou, X., A new approach to permutation polynomials over finite fields, Finite Fields Appl., 18, 3, 492-521, (2012) · Zbl 1273.11169
[8] Hou, X.; Ly, T., Necessary conditions for reversed dickson polynomials to be permutational, Finite Fields Appl., 16, 6, 436-448, (2010) · Zbl 1209.11103
[9] Kyureghyan, G.; Suder, V., On inversion in \(\mathbb{Z}_{2^n - 1}\), Finite Fields Appl., 25, 234-254, (2014) · Zbl 1355.11108
[10] Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite Fields Appl., 13, 1, 58-70, (2007) · Zbl 1107.11048
[11] Li, N.; Helleseth, T.; Tang, X., Further results on a class of permutation polynomials over finite fields, Finite Fields Appl., 22, 16-23, (2013) · Zbl 1285.05004
[12] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permute the elements of the field?, Am. Math. Mon., 95, 3, 243-246, (1988) · Zbl 0653.12010
[13] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permute the elements of the field? II, Am. Math. Mon., 100, 1, 71-74, (1993) · Zbl 0777.11054
[14] Lidl, R.; Niederreiter, H., Finite fields, Encycl. Math. Appl., (1997), Cambridge University Press
[15] Marcos, J. E., Specific permutation polynomials over finite fields, Finite Fields Appl., 17, 2, 105-112, (2011) · Zbl 1261.11080
[16] Mullen, G. L.; Niederreiter, H., Dickson polynomials over finite fields and complete mappings, Can. Math. Bull., 30, 1, 19-27, (1987) · Zbl 0576.12020
[17] Niederreiter, H.; Robinson, K. H., Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 2, 197-212, (1982) · Zbl 0495.12018
[18] Niho, Y., Multivalued cross-correlation functions between two maximal linear recursive sequences, (1972), Univ. Southern Calif. Los Angeles, PhD dissertation
[19] Qu, L.; Tan, Y.; Tan, C. H.; Li, C., Constructing differentially 4-uniform permutations over \(\mathbb{F}_{2^{2 k}}\) via the switching method, IEEE Trans. Inf. Theory, 59, 7, 4675-4686, (2013) · Zbl 1364.94565
[20] Shallue, C. J.; Wanless, I. M., Permutation polynomials and orthomorphism polynomials of degree six, Finite Fields Appl., 20, 84-92, (2013) · Zbl 1276.11197
[21] Tu, Z.; Zeng, X.; Hu, L., Several classes of complete permutation polynomials, Finite Fields Appl., 25, 182-193, (2014) · Zbl 1284.05012
[22] Tu, Z.; Zeng, X.; Hu, L.; Li, C., A class of binomial permutation polynomials, (2013)
[23] Wan, D., On a problem of Niederreiter and Robinson about finite fields, J. Aust. Math. Soc. A, 41, 3, 336-338, (1986) · Zbl 0607.12009
[24] Yuan, J.; Ding, C., Four classes of permutation polynomials of \(\mathbb{F}_{2^m}\), Finite Fields Appl., 13, 4, 869-876, (2007) · Zbl 1167.11045
[25] Yuan, J.; Ding, C.; Wang, H.; Pieprezyk, J., Permutation polynomials of the form \((x^p + x + \delta)^s + L(x)\), Finite Fields Appl., 14, 2, 482-492, (2008)
[26] Yuan, Y.; Tong, Y.; Zhang, H., Complete mapping polynomials over finite field \(\mathbb{F}_{16}\), (Arithmetic of Finite Fields, Lect. Notes Comput. Sci., vol. 4547, (2007), Springer Berlin), 147-158 · Zbl 1213.11193
[27] Zieve, M. E., On some permutation polynomials over \(F_q\) of the form \(x^r h(x^{(q - 1) / d})\), Proc. Am. Math. Soc., 137, 2209-2216, (2009) · Zbl 1228.11177
[28] Zieve, M. E., A class of permutation trinomials related to Rédei functions, (2013)
[29] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 4, 781-790, (2012) · Zbl 1288.11111
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.