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On decomposition of sub-linearised-polynomials

Part of: Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  09 April 2009

Robert S. Coulter ,
George Havas  and
Marie Henderson
Robert S. Coulter
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716–2553, USA e-mail: [email protected]@math.udel.edu
George Havas
Affiliation:
Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland, Qld 4072, Australia e-mail: [email protected]
Marie Henderson
Affiliation:
Department of Mathematical Sciences, University of Delaware, Newark, DE 19716–2553, USA e-mail: [email protected]@math.udel.edu
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Abstract

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We give a detailed exposition of the theory of decompositions of linearised polynomials, using a well-known connection with skew-polynomial rings with zero derivative. It is known that there is a one-to-one correspondence between decompositions of linearised polynomials and sub-linearised polynomials. This correspondence leads to a formula for the number of indecomposable sub-linearised polynomials of given degree over a finite field. We also show how to extend existing factorisation algorithms over skew-polynomial rings to decompose sub-linearised polynomials without asymptotic cost.

MSC classification

Secondary: 11T55: Arithmetic theory of polynomial rings over finite fields 16H05: Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 76 , Issue 3 , June 2004 , pp. 317 - 328
Copyright
Copyright © Australian Mathematical Society 2004

References

[1]Bosma, W., Cannon, J. and Playoust, C., ‘The Magma algebra system I: The user language’, J. Symbolic Comput. 24 (1997), 235–265.CrossRefGoogle Scholar
[2]Cohn, P. M., Skew field constructions, London Math. Soc. Lecture Note Ser. 27 (Cambridge University Press, Cambridge, 1977).Google Scholar
[3]Coulter, R. S., Havas, G. and Henderson, M., ‘Functional decomposition of a class of wild polynomials’, J. Combin. Math. Combin. Comput. 28 (1998), 87–94.Google Scholar
[4]Coulter, R. S., Havas, G. and Henderson, M., ‘Giesbrecht's algorithm, the HFE cryptosystem and Ore's p s-polynomials’, in: Computer Mathematics: Proceedings of the Fifth Asian Symposium (ASCM 2001) (eds. Shirayanagi, K. and Yokoyama, K.), Lecture Notes Ser. Comput. 9 (World Scientific, River Edge, NJ, 2001) pp. 36–45.CrossRefGoogle Scholar
[5]Dorey, F. and Whaples, G., ‘Prime and composite polynomials’, J. Algebra 28 (1974), 88–101.CrossRefGoogle Scholar
[6]Engstrom, H. T., ‘Polynomial substitutions’, Amer. J. Math. 63 (1941), 249–255.CrossRefGoogle Scholar
[7]Fried, M. D. and MacRae, R. E., ‘On the invariance of chains of fields’, Illinois J. Math. 13 (1969), 165–171.CrossRefGoogle Scholar
[8]von zur Gathen, J., ‘Functional decomposition of polynomials: the tame case’, J. Symbolic Comput. 9 (1990), 281–299.CrossRefGoogle Scholar
[9]von zur Gathen, J., ‘Functional decomposition of polynomials: the wild case’, J. Symbolic Comput. 10 (1990), 437–452.CrossRefGoogle Scholar
[10]Giesbrecht, M., ‘Factoring in skew-polynomial rings over finite fields’, J. Symbolic Comput. 26 (1998), 463–486.CrossRefGoogle Scholar
[11]Henderson, M. and Matthews, R., ‘Composition behaviour of sub-linearised polynomials over a finite field’, in: Finite fields: theory, applications and algorithms (eds. Mullin, R. C. and Mullen, G. L.), Contemporary Mathematics 225 (American Mathematical Society, Providence, 1999) pp. 67–75.CrossRefGoogle Scholar
[12]Jacobson, N., Finite-dimensional division algebras over fields (1996).CrossRefGoogle Scholar
[13]Levi, H., ‘Composite polynomials with coefficients in an arbitary field of characteristic zero’, Amer. J. Math. 64 (1942), 389–400.CrossRefGoogle Scholar
[14]Lidl, R. and Niederreiter, H., Finite fields, Encyclopedia of Mathematics and its Applications 20 (Addison-Wesley, Reading, 1983).Google Scholar
[15]Odoni, R. W. K., ‘On additive polynomials over a finite field’, Proc. Edinburgh Math. Soc. 42 (1999), 1–16.CrossRefGoogle Scholar
[16]Ore, O., ‘On a special class of polynomials’, Trans. Amer. Math. Soc. 35 (1933), 559–584;CrossRefGoogle Scholar
Errata, Trans. Amer. Math. Soc. 36 (1934), 275.Google Scholar
[17]Ore, O., ‘Theory of non-commutative polynomials’, Ann. of Math. 34 (1933), 480–508.CrossRefGoogle Scholar
[18]Reiner, I., Maximal orders, London Math. Soc. Monographs 5 (1975).Google Scholar
[19]Ritt, J. F., ‘Prime and composite polynomials’, Trans. Amer. Math. Soc. 23 (1922), 51–66.CrossRefGoogle Scholar
[20]Schinzel, A., Polynomials with special regard to reducibility, Encyclopedia of Mathematics and its Applications 77 (Cambridge University Press, Cambridge, 2000).CrossRefGoogle Scholar