Permutation polynomials from piecewise functions. (English) Zbl 1331.11108

Yuan, Pingzhi; Zheng, Yanbin

Finite Fields Appl. 35, 215-230 (2015).

Summary: Permutation polynomials are an interesting subject of mathematics and have applications in other areas of mathematics and engineering. In this paper, we present some classes of explicit permutation polynomials of the forms similar to \((x^{p^k}+ax+\delta)^{\frac{p^n-1}{d}+1}-ax\), \(d = 2, 3, 4, 6\). The results here generalize the similar results obtained in [N. Li et al., Finite Fields Appl. 22, 16–23 (2013; Zbl 1285.05004), Z. Zha and L. Hu, Finite Fields Appl. 18, No. 4, 781–790 (2012; Zbl 1288.11111)] greatly.

MSC:

11T06

Polynomials over finite fields

Keywords:

permutation polynomials; finite fields

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]	Ding, C.; Yuan, J., A family of skew Hadamard difference sets, J. Comb. Theory, Ser. A, 113, 1526-1535, (2006) · Zbl 1106.05016
[3]	Helleseth, T.; Zinoviev, V., New Kloosterman sums identities over \(\mathbb{F}_{2^m}\) for all m, Finite Fields Appl., 9, 187-193, (2003) · Zbl 1081.11077
[4]	Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite Fields Appl., 13, 58-70, (2007) · Zbl 1107.11048
[5]	Li, N.; Helleseth, Tor; Tang, X., Further results on a class of permutation polynomials over finite fields, Finite Fields Appl., 22, 16-23, (2013) · Zbl 1285.05004
[6]	Lidl, R.; Niederreiter, H., Finite fields, Encycl. Math. Appl., vol. 20, (1997), Cambridge University Press Cambridge
[7]	Lidl, R.; Niederreiter, H., Introduction to finite fields and their applications, (1986), Cambridge University Press Cambridge · Zbl 0629.12016
[8]	Mullen, G. L., Permutation polynomials over finite fields, (Proc. Conf. Finite Fields and Their Applications, Lect. Notes Pure Appl. Math., vol. 141, (1993), Marcel Dekker), 131-151 · Zbl 0808.11069
[9]	Mullen, G. L.; Wang, Q., Permutation polynomials in one variable, (Mullen, G. L.; Panario, D., Handbook of Finite Fields, (2013), CRC Press), 215-229
[10]	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
[11]	Schwenk, J.; Huber, K., Public key encryption and digital signatures based on permutation polynomials, Electron. Lett., 34, 759-760, (1998)
[12]	Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 560-574, (2011) · Zbl 1258.11100
[13]	Yuan, P.; Ding, C., Further results on permutation polynomials over finite fields, Finite Fields Appl., 27, 88-103, (2014) · Zbl 1297.11148
[14]	Yuan, P.; Ding, C., Permutation polynomials of the form \(L(x) + S_{2 k}^a + S_{2 k}^b\) over \(\mathbb{F}_{q^{3 k}}\), Finite Fields Appl., 29, 106-117, (2014)
[15]	Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 781-790, (2012) · Zbl 1288.11111
[16]	Zieve, M. E., Some families of permutation polynomials over finite fields, Int. J. Number Theory, 4, 851-857, (2008) · Zbl 1204.11180
[17]	Zieve, M. E., Classes of permutation polynomials based on cyclotomy and an additive analogue, (Additive Number Theory, (2010), Springer), 355-361 · Zbl 1261.11081