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Permutation polynomials $$(x^{p^m} - x + \delta)^{s_1} +(x^{p^m} - x + \delta)^{s_2} + x$$ over $$\mathbb{F}_{p^n}$$. (English) Zbl 1385.05006
Summary: In this paper, we propose several classes of permutation polynomials with the form $$(x^{p^m} - x + \delta)^{s_1} +(x^{p^m} - x + \delta)^{s_2} + x$$ over finite fields. The permutation behavior of the proposed polynomials is investigated by the AGW criterion and determination of the number of solutions to certain equations over finite fields.

##### MSC:
 05A05 Permutations, words, matrices 11T06 Polynomials over finite fields 11T55 Arithmetic theory of polynomial rings over finite fields
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##### 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] Cepak, N.; Charpin, P.; Pasalic, E., Permutations via linear translators, Finite Fields Appl., 45, 19-42, (2017) · Zbl 1376.12003 [3] 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 [4] 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 [5] (Mullen, G. L.; Panario, D., Handbook of Finite Fields, (2013), CRC Press Boca Raton) · Zbl 1319.11001 [6] 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 [7] 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 [8] 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 [9] 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 [10] 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 [11] Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 6, 560-574, (2011) · Zbl 1258.11100 [12] Yuan, P.; Ding, C., Further results on permutation polynomials over finite fields, Finite Fields Appl., 27, 88-103, (2014) · Zbl 1297.11148 [13] Yuan, P.; Zheng, Y., Permutation polynomials from piecewise functions, Finite Fields Appl., 35, 215-230, (2015) · Zbl 1331.11108 [14] Zheng, D.; Chen, Z., More classes of permutation polynomials of the form $$(x^{p^m} - x + \delta)^s + L(x)$$, Appl. Algebra Eng. Commun. Comput., 28, 3, 215-223, (2017) · Zbl 1366.05005 [15] 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 [16] Zeng, X.; Zhu, X.; Li, N.; Liu, X., Permutation polynomials over $$\mathbb{F}_{2^n}$$ of the form $$(x^{2^i} + x + \delta)^{s_1} +(x^{2^i} + x + \delta)^{s_2} + x$$, Finite Fields Appl., 47, 256-268, (2017) · Zbl 1369.11092 [17] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 4, 781-790, (2012) · Zbl 1288.11111 [18] 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
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