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Permutation polynomials and group permutation polynomials

Published online by Cambridge University Press:  17 April 2009

Young Ho Park
Affiliation:
Department of Mathematics, Kangwon National University, Chuncheon 200–701, Korea, e-mail: yhpark@math.kangwon.ac.kr
June Bok Lee
Affiliation:
Department of Mathematics, Yonsei University, Seoul 120–749, Korea, e-mail: leejb@bubble.yonsei.ac.kr
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Abstract

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Permutation polynomials of the form xτf (x3) over a finite field give rise to group permutation polynomials. We give a group theoretic criterion and some other criteria in terms of symmetric functions and power functions.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 2001

References

[1]Brison, O. J., ‘On group-permutation polynomials’, Portugal Math. 59 (1993), 335383.Google Scholar
[2]Carlitz, L. and Wells, C., ‘The number of solutions of a special system of equations in a finite field’, Acta Arith. 12 (1966), 7784.CrossRefGoogle Scholar
[3]Dickson, L. E., Linear Groups with an exposition of the Galois field theory (Dover, New York, 1958).Google Scholar
[4]Lidl, R. and Mullen, G. L., ‘When does a polynomial over a finite field permute the elements of the field?’, Amer. Math. Monthly 100 (1988), 243246.CrossRefGoogle Scholar
[5]Lidl, R. and Mullen, G. L., ‘When does a polynomial over a finite field permute the elements of the field? II’, Amer. Math. Monthly 95 (1993), 7174.CrossRefGoogle Scholar
[6]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia Math. Appl. 20 (Addison-Wesley, Reading, MA, 1983).Google Scholar
[7]Matthews, R., ‘Permutation properties of the polynomials over a finite field’, Proc. Amer. Math. Soc. 120 (1994), 4751.Google Scholar
[8]Mullen, G. L., ‘Permutation polynomials: a matrix analog of Schur's conjecture and a survey of recent results’, Finite Fields Appl. 1 (1995), 242258.CrossRefGoogle Scholar
[9]Niederreiter, N. and Robinson, K. H., ‘Complete mappings of finite fields’, J. Austral. Math. Soc. 33 (1982), 197212.CrossRefGoogle Scholar
[10]Park, Y. H. and Lee, J. B., ‘Permutation polynomials with exponents in an arithmetic progression’, Bull. Austral. Math. Soc. 57 (1998), 243252.CrossRefGoogle Scholar
[11]Turnwald, G., ‘A new criterion for permutation polynomials’, Finite Fields Appl. 1 (1995), 6482.CrossRefGoogle Scholar
[12]Wan, D. and Lidl, R., ‘Permutation polynomials of the form x τf (x (q−1)/d) and their group structures’, Montash. Math. 112 (1991), 149163.CrossRefGoogle Scholar