Abstract
In this paper we consider the problem of computation of a basis for a finite abelian group G with N elements. We present a deterministic algorithm such that given a generating set for G and the prime factorization of N, it computes a basis of G.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beynon, W.M., Iliopoulos, C.S.: Computing a basis for a finite abelian p-group. Information Processing Letters 20, 161–163 (1985)
Borges-Quintana, M., Borges-Trenard, M.A., MartÃnez-Moro, E.: On the use of gröbner bases for computing the structure of finite abelian groups. In: Ganzha, V.G., Mayr, E.W., Vorozhtsov, E. (eds.) CASC 2005. LNCS, vol. 3718, pp. 52–64. Springer, Heidelberg (2005)
Buchmann, J., Schmidt, A.: Computing the structure of a finite abelian group. Mathematics of Computation 74, 2017–2026 (2005)
Chen, L.: Algorithms and their complexity analysis for some problems in finite group. Journal of Sandong Normal University 2, 27–33 (1984) (in Chinese)
Chen, L., Fu, B.: Linear Sublinear Time Algorithms for the Basis of Abelian groups. Theoretical Computer Science (2010), doi:10.1016/j.tcs.2010.06.011
Cheung, K.H., Mosca, M.: Decomposing finite abelian groups. Journal of Quantum Information and Computation 1(3), 26–32 (2001)
Dinur I., Grigorescu E., Kopparty S., Sudan M.:, Decodability of Group Homomorphisms beyond the Johnson Bound, ECCC Report, No 20 (2008) and in 40th STOC 275-284 (2008)
Gathen, J., Gerhard, J.: Modern Computer Algebra. Cambridge University Press, Cambridge (1999)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)
Iliopoulos, C.S.: Analysis of algorithms on problems in general abelian groups. Information Processing Letters 20, 215–220 (1985)
Iliopoulos, C.S.: Worst-case complexity bounds on algorithms for computing the canonical structure of finite abelian groups and the Hermite and Smith normal forms of an integer matrix. SIAM Journal of Computing 18(4), 658–669 (1989)
Kitaev, A.Y.: Quantum computations: algorithms and error correction. Russian Math. Surveys 52(6), 1191–1249 (1997)
Koblitz, N., Menezes, A.J.: A survey of public-key cryptosystems. SIAM Review 46(4), 599–634 (2004)
Kohel, A.R., Shparlinski, I.E.: Exponential sums and group generators for elliptic curves over finite fields. In: Bosma, W. (ed.) ANTS 2000. LNCS, vol. 1838, pp. 395–404. Springer, Heidelberg (2000)
Ledermann, W.: Introduction to group theory. Longman Group Limited, London (1973)
Lomont C.: The hidden subgroup problem - review and open problems (2004), http://arxiv.org/abs/quantph/0411037
Shor, P.W.: Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM Journal of Computing 26(5), 1484–1509 (1997)
Teske, E.: The Pohlig-Hellman Method Generalized for Group Structure Computation. Journal of Symbolic Computation 27, 521–534 (1999)
Teske, E.: A space efficient algorithm for group structure computation. Mathematics of Computation 67(224), 1637–1663 (1998)
Washington, L.C.: Elliptic Curves. Chapman and Hall, Boca Raton (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Karagiorgos, G., Poulakis, D. (2011). An Algorithm for Computing a Basis of a Finite Abelian Group. In: Winkler, F. (eds) Algebraic Informatics. CAI 2011. Lecture Notes in Computer Science, vol 6742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-21493-6_11
Download citation
DOI: https://doi.org/10.1007/978-3-642-21493-6_11
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-21492-9
Online ISBN: 978-3-642-21493-6
eBook Packages: Computer ScienceComputer Science (R0)