On permutation polynomials of prescribed shape. (English) Zbl 1220.11145

Akbary, Amir; Ghioca, Dragos; Wang, Qiang

Finite Fields Appl. 15, No. 2, 195-206 (2009).

Summary: We count permutation polynomials of \(\mathbb F_q\) which are sums of \(m+1 (\geq 2\)) monomials of prescribed degrees. This allows us to prove certain results about existence of permutation polynomials of prescribed shape.

MSC:

11T06

Polynomials over finite fields

Keywords:

permutation polynomials; finite fields

References:

[1]	Carlitz, L.; Wells, C., The number of solutions of a special system of equations in a finite field, Acta arith. XII, 77-84, (1966) · Zbl 0147.04003
[2]	Chu, W.; Golomb, S.W., Circular tuscan-k arrays from permutation binomials, J. combin. theory ser. A, 97, 1, 195-202, (2002) · Zbl 1009.05032
[3]	Das, P., The number of permutation polynomials of a given degree over a finite field, Finite fields appl., 8, 478-490, (2002) · Zbl 1029.11066
[4]	Fried, M.D.; Guralnick, R.; Saxl, J., Schur covers and Carlitz’s conjecture, Israel J. math., 82, 1-3, 157-225, (1993) · Zbl 0855.11063
[5]	Konyagin, S.; Pappalardi, F., Enumerating permutation polynomials over finite fields by degree, Finite fields appl., 8, 4, 548-553, (2002) · Zbl 1029.11067
[6]	Konyagin, S.; Pappalardi, F., Enumerating permutation polynomials over finite fields by degree. II, Finite fields appl., 12, 1, 26-37, (2006) · Zbl 1163.11350
[7]	Laigle-Chapuy, Y., Permutation polynomials and applications to coding theory, Finite fields appl., 13, 1, 58-70, (2007) · Zbl 1107.11048
[8]	Lidl, R.; Müller, W.B., Permutation polynomials in RSA-cryptosystems, (), 293-301
[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.; Niederreiter, H., Finite fields, Encyclopedia math. appl., (1997), Cambridge Univ. Press
[11]	A. Masuda, M.E. Zieve, Permutation binomials over finite fields, Trans. Amer. Math. Soc., in press · Zbl 1239.11139
[12]	Sun, J.; Takeshita, O.Y., Interleavers for turbo codes using permutation polynomials over integer rings, IEEE trans. inform. theory, 51, 1, 101-119, (2005) · Zbl 1280.94121
[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]	Weil, A., On some exponential sums, Proc. natl. acad. sci. USA, 27, 345-347, (1941) · Zbl 0061.06406