×

zbMATH — the first resource for mathematics

A class of new permutation trinomials. (English) Zbl 1380.05002
Summary: In this paper, we characterize the coefficients of \(f(x)=x+a_1 x^{q(q-1)+1}+a_2 x^{2(q-1)+1}\) in \(\mathbb{F}_{q^2}[x]\) for even \(q\) that lead \(f(x)\) to be a permutation of \(\mathbb{F}_{q^2}\). We transform the problem into studying some low-degree equations with variable in the unit circle, which are intensively investigated with some parameterization techniques. From the numerical results, the coefficients that lead \(f(x)\) to be a permutation appear to be completely characterized in this paper. It is also demonstrated that some permutations proposed in this paper are quasi-multiplicative (QM) inequivalent to the previously known permutation trinomials.

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, 51-67, (2011) · Zbl 1281.11102
[2] Alahmadi, A.; Akhazmi, H.; Helleseth, T.; Hijazi, R.; Muthana, N. M.; Solé, P., On the lifted zetterberg code, Des. Codes Cryptogr., 80, 3, 561-576, (2016) · Zbl 1348.94090
[3] Berlekamp, E. R.; Rumsey, H.; Solomon, G., On the solution of algebraic equations over finite fields, Inf. Control, 10, 6, 553-564, (1967) · Zbl 0166.04803
[4] Carlitz, L.; Wells, C., The number of solutions of a special system of equations in a finite field, Acta Arith., 12, 77-84, (1966) · Zbl 0147.04003
[5] Dickson, L. E., Linear groups with an exposition of the Galois field theory, (1958), Dover New York
[6] Dodunekov, S. M.; Nilsson, J. E.M., Algebraic decoding of the zetterberg codes, IEEE Trans. Inf. Theory, 38, 5, 1570-1573, (1992) · Zbl 0756.94014
[7] Dickson, L. E., The analytic representation of substitutions on a power of a prime number of letters with a discussion of the linear group, Ann. Math., 11, 65-120, (1986) · JFM 28.0135.03
[8] Ding, C.; Qu, L.; Wang, Q.; Yuan, J.; Yuan, P., Permutation trinomials over finite fields with even characteristic, SIAM J. Discrete Math., 29, 1, 79-92, (2015) · Zbl 1352.11102
[9] Ding, C.; Helleseth, T., Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59, 5898-5904, (2013) · Zbl 1364.94652
[10] Ding, C.; Yuan, J., A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113, 1526-1535, (2006) · Zbl 1106.05016
[11] Fernando, N., A note on permutation binomials and trinomials over finite fields, available online: · Zbl 1394.11080
[12] Gupta, R.; Sharma, R. K., Some new classes of permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 41, 89-96, (2016) · Zbl 1372.11108
[13] Hermite, Ch., Sur LES fonctions de sept lettres, C. R. Acad. Sci. Paris, 57, 750-757, (1863)
[14] Hou, X., Permutation polynomials over finite fields - a survey of recent advances, Finite Fields Appl., 32, 82-119, (2015) · Zbl 1325.11128
[15] Hou, X., A survey of permutation binomials and trinomials over finite fields, (Kyureghyan, G.; Mullen, G. L.; Pott, A., Topics in Finite Fields, Proceedings of the 11th International Conference on Finite Fields and Their Applications, vol. 632, (2015), AMS), 177-191 · Zbl 1418.11153
[16] Hou, X., A class of permutation trinomials over finite fields, Acta Arith., 162, 51-64, (2014) · Zbl 1294.11210
[17] Hou, X., Determination of a type of permutation trinomials over finite fields II, Finite Fields Appl., 35, 16-35, (2015) · Zbl 1343.11093
[18] Laigle-Chapuy, Y., Permutation polynomial and applications to coding theory, Finite Fields Appl., 13, 58-70, (2007) · Zbl 1107.11048
[19] Li, N.; Helleseth, T., Several classes of permutation trinomials from niho exponents, Cryptogr. Commun., 9, 693-705, (2017) · Zbl 1369.11089
[20] Li, N.; Helleseth, T., New permutation trinomials from niho exponents over finite fields with even characteristic · Zbl 1402.05005
[21] Li, K.; Qu, L.; Chen, X., New classes of permutation binomials and permutation trinomials over finite fields, Finite Fields Appl., 43, 69-85, (2017) · Zbl 1351.11078
[22] Lee, J. B.; Park, Y. H., Some permutation trinomials over finite fields, Acta Math. Sci., 17, 250-254, (1997) · Zbl 0921.11062
[23] Lidl, R.; Niederreiter, H., Finite fields, Encycl. Math. Appl., (1997), Cambridge University Press
[24] Ma, J.; Zhang, T.; Feng, T.; Ge, G., Some new results on permutation polynomials over finite fields, Des. Codes Cryptogr., 83, 425-443, (2017) · Zbl 1369.11091
[25] Niederreiter, N.; Robinson, K. H., Complete mappings of finite fields, J. Aust. Math. Soc., 33, 197-212, (1982) · Zbl 0495.12018
[26] Niho, Y., Multi-valued cross-correlation functions between two maximal linear recursive sequences, (1972), Univ. Southern Calif. Los Angeles, PhD dissertation
[27] Park, Y. H.; Lee, J. B., Permutation polynomials and group permutation polynomials, Bull. Aust. Math. Soc., 63, 67-74, (2001) · Zbl 0981.11039
[28] Rivest, R. L.; Shamir, A.; Adelman, L. M., A method for obtaining digital signatures and public-key cryptosystems, Commun. ACM, 21, 120-126, (1978) · Zbl 0368.94005
[29] Schwenk, J.; Huber, K., Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34, 759-760, (1998)
[30] Rosendahl, P., Niho type cross-correlation functions and related equations, (2004), Univ. Turku Turku, Finland, PhD dissertation
[31] Wan, D.; Lidl, R., Permutation polynomials of the form \(x^r f(x^{\frac{q - 1}{d}})\) and their group structure, Monatshefte Math., 112, 149-163, (1991) · Zbl 0737.11040
[32] Wu, D.; Yuan, P.; Ding, C.; Ma, Y., Permutation trinomials over \(\mathbb{F}_{2^m}\), Finite Fields Appl., 46, 38-56, (2017) · Zbl 1431.11131
[33] Williams, K. S., Note on cubics over GF\((2^n)\) and GF\((3^n)\), J. Number Theory, 7, 4, 361-365, (1975) · Zbl 0321.12029
[34] Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 560-574, (2011) · Zbl 1258.11100
[35] Yuan, P.; Ding, C., Further results on permutation polynomials over finite fields, Finite Fields Appl., 27, 88-103, (2014) · Zbl 1297.11148
[36] Zha, Z.; Hu, L.; Fan, S., Further results on permutation trinomials over finite fields with even characteristic, Finite Fields Appl., 45, 43-52, (2017) · Zbl 1362.05006
[37] Zieve, M., On some permutation polynomials over \(\mathbb{F}_q\) of the form \(x^r h(x^{\frac{q - 1}{d}})\), Proc. Am. Math. Soc., 137, 2209-2216, (2009) · Zbl 1228.11177
[38] Zieve, M., Permutation polynomials on \(\mathbb{F}_q\) induced form Rédei function bijections on subgroups of \(\mathbb{F}_q\), available online:
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.