zbMATH — the first resource for mathematics

New methods for generating permutation polynomials over finite fields. (English) Zbl 1245.11114
Summary: We present two methods for generating linearized permutation polynomials over an extension of a finite field $$\mathbb F_q$$. These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying $$q \equiv 3\pmod 4$$, these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of C. Ding et al. [Sci. China, Ser. A 52, No. 4, 639–647 (2009; Zbl 1215.11113)] (the citation given in the paper contains errors).

MSC:
 11T06 Polynomials over finite fields 12E20 Finite fields (field-theoretic aspects)
Full Text:
References:
 [1] Akbary, A.; Alaric, S.; Wang, Q., On some classes of permutation polynomials, Int. J. number theory, 4, 121-133, (2008) · Zbl 1218.11108 [2] Blokhuis, A.; Coulter, R.; Henderson, M., Permutations amongst the Dembowski-Ostrom polynomials, (), 37-42 · Zbl 1009.11064 [3] Charpin, P.; Kyureghyan, G.M., On a class of permutation polynomials over $$\mathbb{F}_{2^n}$$, (), 368-376 · Zbl 1180.11038 [4] Coulter, R.; Henderson, M.; Matthews, R., A note on constructing permutation polynomials, Finite fields appl., 15, 553-557, (2009) · Zbl 1215.11112 [5] Coulter, R.; Henderson, M., Commutative presemifields and semifields, Adv. math., 217, 282-304, (2008) · Zbl 1194.12007 [6] Ding, C.; Xiang, Q.; Yuan, J.; Yuan, P., Explicit classes of permutation polynomials of $$\mathbb{F}_{3^{3 m}}$$, Sci. China ser. A, 53, 4, 630-647, (2009) [7] Hollmann, Henk D.L.; Xiang, Q., A class of permutation polynomials of $$\mathbb{F}_{2^m}$$ related to dickson polynomials, Finite fields appl., 11, 111-122, (2005) · Zbl 1073.11074 [8] Lidl, R.; Mullen, G.L.; Turnward, G., Dickson polynomials, Pitman monogr. surv. pure appl. math., vol. 65, (1993), Addison-Wesley · Zbl 0823.11070 [9] 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 [10] 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 [11] Lidl, R.; Niederreiter, H., Finite fields, Encyclopedia math. appl., vol. 20, (1983), Addison-Wesley Reading [12] Wan, D., Permutation polynomials over finite fields, Acta math. sinica, 10, 30-35, (1994) · Zbl 0817.11056 [13] 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 [14] Yuan, J.; Ding, C.; Wang, H.; Pieprezyk, J., Permutation polynomials of the form $$(x^p - x + \delta)^s + L(x)$$, Finite fields appl., 14, 482-492, (2008) [15] Yuan, P., More explicit classes of permutation polynomials of $$\mathbb{F}_{3^{3 m}}$$, Finite fields appl., 16, 88-93, (2010) [16] 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 engrg. comm. comput., 21, 145-150, (2010) · Zbl 1215.11116
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.