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Cyclotomy and permutation polynomials of large indices. (English) Zbl 1331.11107
Summary: We use cyclotomy to construct new classes of permutation polynomials over finite fields. This allows us to generate permutation polynomials in an algorithmic way and also to unify several previous constructions. Many permutation polynomials constructed in this way have large indices.

##### MSC:
 11T06 Polynomials over finite fields 11T22 Cyclotomy
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##### References:
 [1] Akbary, A.; Alaric, S.; Wang, Q., On some classes of permutation polynomials, Int. J. Number Theory, 4, 1, 121-133, (2008) · Zbl 1218.11108 [2] Akbary, A.; Ghioca, D.; Wang, Q., On permutation polynomials of prescribed shape, Finite Fields Appl., 15, 195-206, (2009) · Zbl 1220.11145 [3] Akbary, A.; Ghioca, D.; Wang, Q., On constructing permutations of finite fields, Finite Fields Appl., 17, 1, 51-67, (2011) · Zbl 1281.11102 [4] Akbary, A.; Wang, Q., On some permutation polynomials, Int. J. Math. Math. Sci., 16, 2631-2640, (2005) · Zbl 1092.11046 [5] Akbary, A.; Wang, Q., A generalized Lucas sequence and permutation binomials, Proc. Amer. Math. Soc., 134, 1, 15-22, (2006) · Zbl 1137.11355 [6] Akbary, A.; Wang, Q., On polynomials of the form $$x^r f(x^{(q - 1) / l})$$, Int. J. Math. Math. Sci., 2007, (2007), Article ID 23408, 7 pp · Zbl 1135.11341 [7] Fernando, N.; Hou, X., A piecewise construction of permutation polynomial over finite fields, Finite Fields Appl., 18, 1184-1194, (2012) · Zbl 1254.05008 [8] Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite Fields Appl., 13, 58-70, (2007) · Zbl 1107.11048 [9] Hou, X., Two classes of permutation polynomials over finite fields, J. Combin. Theory Ser. A, 118, 2, 448-454, (2011) · Zbl 1230.11146 [10] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permute the elements of the field?, Amer. Math. Monthly, 95, 243-246, (1988) · Zbl 0653.12010 [11] Lidl, R.; Mullen, G. L., When does a polynomial over a finite field permute the elements of the field? II, Amer. Math. Monthly, 100, 71-74, (1993) · Zbl 0777.11054 [12] Lidl, R.; Niederreiter, H., Finite fields, Encyclopedia Math. Appl., (1997), Cambridge University Press [13] Mullen, G. L., Permutation polynomials over finite fields, (Finite Fields, Coding Theory, and Advances in Communications and Computing, (1993), Marcel Dekker New York), 131-151 · Zbl 0808.11069 [14] Mullen, G. L.; Wang, Q., Permutation polynomials of one variable, (Handbook of Finite Fields, (2013), Chapman and Hall/CRC), Section 8.1 [15] Niederreiter, H.; Winterhof, A., Cyclotomic $$\mathcal{R}$$-orthomorphisms of finite fields, Discrete Math., 295, 161-171, (2005) · Zbl 1078.11068 [16] Wan, D.; Lidl, R., Permutation polynomials of the form $$x^r f(x^{(q - 1) / d})$$ and their group structure, Monatsh. Math., 112, 149-163, (1991) · Zbl 0737.11040 [17] 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, Lecture Notes in Comput. Sci., vol. 4893, (2007), Springer-Verlag), 119-128 · Zbl 1154.11342 [18] Wang, Q., On generalized Lucas sequences, (Combinatorics and Graphs: The Twentieth Anniversary Conference of IPM, May 15-21, 2009, Contemp. Math., vol. 531, (2010), Amer. Math. Soc.), 127-141 · Zbl 1246.11039 [19] Yuan, P.; Ding, C., Permutation polynomials over finite fields from a powerful lemma, Finite Fields Appl., 17, 6, 560-574, (2011) · Zbl 1258.11100 [20] Zha, Z.; Hu, L., Two classes of permutation polynomials over finite fields, Finite Fields Appl., 18, 4, 781-790, (2012) · Zbl 1288.11111 [21] Zieve, M., Some families of permutation polynomials over finite fields, Int. J. Number Theory, 4, 851-857, (2008) · Zbl 1204.11180 [22] Zieve, M., On some permutation polynomials over $$\mathbb{F}_q$$ of the form $$x^r h(x^{(q - 1) / d})$$, Proc. Amer. Math. Soc., 137, 7, 2209-2216, (2009) · Zbl 1228.11177 [23] Zieve, M., Classes of permutation polynomials based on cyclotomy and an additive analogue, (Additive Number Theory, (2010), Springer New York), 355-361 · Zbl 1261.11081
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