×

zbMATH — the first resource for mathematics

Complete permutation polynomials with the form \((x^{p^m} - x + \delta)^s + a x^{p^m} + b x\) over \(\mathbb{F}_{p^n}\). (English) Zbl 1412.11138
Summary: In this paper, several classes of complete permutation polynomials with the form \((x^{p^m} - x + \delta)^s + a x^{p^m} + b x\) over \(\mathbb{F}_{p^n}\) are proposed by the AGW criterion and determining the number of solutions of some equations. Our results also enrich constructions of known permutation polynomials.

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] Bartoli, D.; Giulietti, M.; Zini, G., On monomial complete permutation polynomials, Finite Fields Appl., 41, 132-158, (2016) · Zbl 1372.11107
[3] Bassalygo, L.; Zinoviev, V., Permutation and complete permutation polynomials, Finite Fields Appl., 33, 198-211, (2015) · Zbl 1368.11126
[4] 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
[5] Feng, D.; Feng, X.; Zhang, W., Loiss: a byte-oriented stream cipher, (IWCC’11 Proceedings of the Third International Conference on Coding and Cryptology, (2011), Springer), 109-125 · Zbl 1272.94029
[6] Gupta, R.; Sharma, R. K., Further results on permutation polynomials of the form \((x^{p^m} - x + \delta)^s + x\) over \(\mathbb{F}_{p^{2 m}}\), Finite Fields Appl., 50, 196-208, (2018) · Zbl 1400.11155
[7] Helleseth, T.; Zinoviev, V., New Kloosterman sums identities over \(\mathbb{F}_{2^m}\) for all \(m\), Finite Fields Appl., 9, 2, 187-193, (2003) · Zbl 1081.11077
[8] 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
[9] Li, L.; Li, C.; Li, C.; Zeng, X., New classes of complete permutation polynomials, Finite Fields Appl., 55, 177-201, (2019) · Zbl 1401.05016
[10] Li, L.; Wang, S.; Li, C.; Zeng, X., Permutation polynomials \((x^{p^m} - x + \delta)^{s_1} +(x^{p^m} - x + \delta)^{s_2} + x\) over \(\mathbb{F}_{p^n}\), Finite Fields Appl., 51, 31-61, (2018) · Zbl 1385.05006
[11] Lidl, R.; Niederreiter, H., Finite Fields, Encycl. Math. Appl., vol. 20, (1997), Cambridge University Press: Cambridge University Press Cambridge
[12] Muratović-Ribić, A.; Pasalic, E., A note on complete polynomials over finite fields and their applications in cryptography, Finite Fields Appl., 25, 306-315, (2014) · Zbl 1302.11096
[13] (Mullen, G. L.; Panario, D., Handbook of Finite Fields, (2013), CRC Press: CRC Press Boca Raton) · Zbl 1319.11001
[14] Niederreiter, H.; Robinson, K. H., Complete mappings of finite fields, J. Aust. Math. Soc. A, 33, 2, 197-212, (1982) · Zbl 0495.12018
[15] Niho, Y., Multi-Valued Cross-Correlation Functions Between Two Maximal Linear Recursive Sequences, (1972), University of Southern: University of Southern California, Los Angeles, PhD dissertation
[16] Samardjiska, S.; Gligoroski, D., Quadratic permutation polynomials, complete mappings and mutually orthogonal Latin squares, Math. Slovaca, 67, 5, 1129-1146, (2017) · Zbl 1442.11165
[17] Schnorr, C. P.; Vaudenay, S., Black box cryptanalysis of hash networks based on multipermutations, (Advances in Cryptology-Eurocrypt’94, (1995), Springer), 47-57 · Zbl 0909.94013
[18] Specification of SMS4, block cipher for WLAN products-SMS4, (in Chinese), available at
[19] Stǎnicǎ, P.; Gangopadhyay, S.; Chaturvedi, A.; Gangopadhyay, A. K.; Maitra, S., Investigations on bent and negabent functions via the negaHadamard transform, IEEE Trans. Inf. Theory, 58, 4064-4072, (2012) · Zbl 1365.94684
[20] Tu, Z.; Zeng, X.; Hu, L., Several classes of complete permutation polynomials, Finite Fields Appl., 25, 182-193, (2014) · Zbl 1284.05012
[21] Tu, Z.; Zeng, X.; Li, C.; Helleseth, T., A classe of new permutation trinomials, Finite Fields Appl., 50, 178-195, (2018) · Zbl 1380.05002
[22] Tu, Z.; Zeng, X.; Jiang, Y., Two classes of permutation polynomials having the form \((x^{2^m} + x + \delta)^s + x\), Finite Fields Appl., 31, 12-24, (2015) · Zbl 1320.11120
[23] Tu, Z.; Zeng, X.; Li, C.; Helleseth, T., Permutation polynomials of the form \((x^{p^m} - x + \delta)^s + L(x)\) over the finite field \(\mathbb{F}_{p^{2 m}}\) of odd characteristic, Finite Fields Appl., 34, 20-35, (2015) · Zbl 1315.05008
[24] Tuxanidy, A.; Wang, Q., Compositional inverses and complete mappings over finite fields, Discrete Appl. Math., 217, 318-329, (2017) · Zbl 1372.11111
[25] Wang, L.; Wu, B.; Liu, Z., Further results on permutation polynomials of the form \((x^{p^m} - x + \delta)^s + L(x)\) over \(\mathbb{F}_{p^{2 m}}\), Finite Fields Appl., 44, 92-112, (2017) · Zbl 1352.05009
[26] Wang, L.; Wu, B., General constructions of permutation polynomials of the form \((x^{2^m} + x + \delta)^{i(2^m - 1) + 1} + x\) over \(\mathbb{F}_{2^{2 m}}\), Finite Fields Appl., 52, 137-155, (2018) · Zbl 1388.05009
[27] Wu, B.; Lin, D., On constructing complete permutation polynomials over finite fields of even characteristic, Discrete Appl. Math., 184, 213-222, (2015) · Zbl 1311.05009
[28] Wu, G.; Li, N.; Helleseth, T.; Zhang, Y., Some classes of monomial complete permutation polynomials over finite fields of characteristic two, Finite Fields Appl., 28, 148-165, (2014) · Zbl 1314.11073
[29] Xu, G.; Cao, X., Complete permutation polynomials over finite fields of odd characteristic, Finite Fields Appl., 31, 228-240, (2015) · Zbl 1320.11121
[30] Xu, X.; Li, C.; Zeng, X.; Helleseth, T., Constructions of complete permutation polynomials, Des. Codes Cryptogr., 86, 12, 2869-2892, (2018) · Zbl 1398.05012
[31] 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
[32] Yuan, J.; Ding, C.; Wang, H.; Pieprzyk, J., Permutation polynomials of the form \((x^p - x + \delta)^s + L(x)\), Finite Fields Appl., 14, 2, 482-493, (2008) · Zbl 1211.11136
[33] Yuan, P.; Zheng, Y., Permutation polynomials from piecewise functions, Finite Fields Appl., 35, 215-230, (2015) · Zbl 1331.11108
[34] 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, 2, 145-150, (2010) · Zbl 1215.11116
[35] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 4, 781-790, (2012) · Zbl 1288.11111
[36] Zha, Z.; Hu, L., Some classes of permutation polynomials of the form \((x^{p^m} - x + \delta)^s + x\) over \(\mathbb{F}_{p^{2 m}}\), Finite Fields Appl., 40, 150-162, (2016) · Zbl 1336.05005
[37] Zha, Z.; Hu, L.; Cao, X., Constructing permutations and complete permutations over finite fields via subfield-valued polynomials, Finite Fields Appl., 31, 162-177, (2015) · Zbl 1320.11123
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.