×

zbMATH — the first resource for mathematics

Further results on permutation polynomials over finite fields. (English) Zbl 1297.11148
Let \(\mathbb F_{q^n}\) be a finite field. The main result of the paper is: Suppose \(g(x)\in\mathbb F_{q^n}[x]\) satisfies \(g(x)^q=g(x)\) for all \(x\in \mathbb F_{q^n}\), and let \(L(x)\) be a linearized polynomial. For every \(\delta\in \mathbb F_{q^n}\), \[ f(x)=g(x^q-x+\delta)+L(x) \] permutes \(\mathbb F_{q^n}\) if and only if \(L(x)\) does. The authors use this to unify some previous constructions of permutation polynomials and to construct new ones.
Typos have been corrected in the corrigenda [ibid. 30, 153–154 (2014; Zbl 1297.11150)].

MSC:
11T06 Polynomials over finite fields
05B10 Combinatorial aspects of difference sets (number-theoretic, group-theoretic, etc.)
PDF BibTeX XML Cite
Full Text: DOI arXiv
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] Carlet, C.; Ding, C.; Yuan, J., Linear codes from highly nonlinear functions and their secret sharing schemes, IEEE Trans. Inf. Theory, 51, 6, 2089-2102, (2005) · Zbl 1192.94114
[3] Cao, X.; Hu, L., New methods for generating permutation polynomials over finite fields, Finite Fields Appl., 17, 493-503, (2011) · Zbl 1245.11114
[4] Charpin, P.; Kyureghyan, G., When does \(F(X) + \operatorname{Tr}(H(X))\) permute \(\mathbb{F}_{p^n}\)?, Finite Fields Appl., 15, 5, 615-632, (2009) · Zbl 1229.11153
[5] Charpin, P.; Kyureghyan, G., On a class of permutation polynomials over \(\mathbb{F}_{2^n}\), (SETA 2008, Lect. Notes Comput. Sci., vol. 5203, (2008), Springer-Verlag), 368-376 · Zbl 1180.11038
[6] Ding, C.; Helleseth, T., Optimal ternary cyclic codes from monomials, IEEE Trans. Inf. Theory, 59, 9, 5898-5904, (2013) · Zbl 1364.94652
[7] Ding, C.; Yin, J., Signal sets from functions with optimum nonlinearity, IEEE Trans. Commun., 55, 5, 936-940, (2007)
[8] Ding, C.; Yuan, J., A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113, 1526-1535, (2006) · Zbl 1106.05016
[9] Ding, C.; Xiang, Q.; Yuan, J.; Yuan, P., Explicit classes of permutation polynomials over GF(\(3^{3 m}\)), Sci. China, Ser. A, 53, 639-647, (2009) · Zbl 1215.11113
[10] Kyureghyan, G., Constructing permutations of finite fields via linear translators, J. Comb. Theory, Ser. A, 118, 1052-1061, (2010) · Zbl 1241.11136
[11] Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite Fields Appl., 13, 58-70, (2007) · Zbl 1107.11048
[12] 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
[13] Hou, X., Two classes of permutation polynomials over finite fields, J. Comb. Theory, Ser. A, 118, 2, 448-454, (2011) · Zbl 1230.11146
[14] Hou, X.; Mullen, G. L.; Sellers, J. A.; Yucas, J. L., Reversed dickson polynomials over finite fields, Finite Fields Appl., 15, 6, 748-773, (2009) · Zbl 1228.11174
[15] Lidl, R.; Müller, W. B., Permutation polynomials in RSA-cryptosystems, (Advances in Cryptology, (1984), Plenum New York), 293-301
[16] Lidl, R.; Niederreiter, H., Finite fields, Encycl. Math. Appl., vol. 20, (1997), Cambridge University Press Cambridge
[17] Lidl, R.; Niederreiter, H., Introduction to finite fields and their applications, (1986), Cambridge University Press Cambridge · Zbl 0629.12016
[18] Marcos, J. E., Specific permutation polynomials over finite fields, Finite Fields Appl., 17, 105-112, (2011) · Zbl 1261.11080
[19] Mullen, G. L., Permutation polynomials over finite fields, (Proceedings of the Conference on Finite Fields and Their Applications, Lect. Notes Pure Appl. Math., vol. 141, (1993), Marcel Dekker), 131-151 · Zbl 0808.11069
[20] Mullen, G. L.; Wang, Q., Permutation polynomials in one variable, (Mullen, G. L.; Panario, D., Handbook of Finite Fields, (2013), CRC Press), 215-229
[21] 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
[22] Schwenk, J.; Huber, K., Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34, 759-760, (1998)
[23] Wang, Q., Cyclotomic mapping permutation polynomials over finite fields, (Sequences, Subsequences, and Consequences, International Workshop, SSC 2007, Los Angeles, CA, USA, May 31-June 2, 2007, Lect. Notes Comput. Sci., vol. 4893, (2007)), 119-128 · Zbl 1154.11342
[24] Yuan, J.; Carlet, C.; Ding, C., The weight distribution of a class of linear codes from perfect nonlinear functions, IEEE Trans. Inf. Theory, 52, 2, 712-717, (2006) · Zbl 1192.94128
[25] Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 560-574, (2011) · Zbl 1258.11100
[26] Zeng, X.; Zhu, X.; Hu, L., Two new permutation polynomials with the form \((x^{2^k} + x + \delta)^s + x\) over \(\mathbb{F}_{2^n}\), Appl. Algebra Eng. Commun. Comput., 21, 145-150, (2010) · Zbl 1215.11116
[27] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 781-790, (2012) · Zbl 1288.11111
[28] Zieve, M. E., Some families of permutation polynomials over finite fields, Int. J. Number Theory, 4, 851-857, (2008) · Zbl 1204.11180
[29] Zieve, M. E., Classes of permutation polynomials based on cyclotomy and an additive analogue, (Additive Number Theory, (2010), Springer), 355-361 · Zbl 1261.11081
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.