A rigorous subexponential algorithm for computation of class groups

Hafner, James L.; McCurley, Kevin S.

doi:10.1090/S0894-0347-1989-1002631-0

A rigorous subexponential algorithm for computation of class groups
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by James L. Hafner and Kevin S. McCurley

J. Amer. Math. Soc. 2 (1989), 837-850

DOI: https://doi.org/10.1090/S0894-0347-1989-1002631-0

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Abstract:

Let $C( - d)$ denote the Gauss Class Group of quadratic forms of a negative discriminant $- d$ (or equivalently, the class group of the imaginary quadratic field $Q(\sqrt { - d} )$). We give a rigorous proof that there exists a Las Vegas algorithm that will compute the structure of $C( - d)$ with an expected running time of $L{(d)^{\sqrt 2 + o(1)}}$ bit operations, where $L(d) = {\text {exp}}(\sqrt {\log d\;\log \log d} )$. Thus, of course, also includes the computation of the class number $h( - d)$, the cardinality of $C( - d)$.

References

Don Coppersmith and Shmuel Winograd, Matrix multiplication via arithmetic progressions, J. Symbolic Comput. 9 (1990), no. 3, 251–280. MR 1056627, DOI 10.1016/S0747-7171(08)80013-2
P. D. Domich, R. Kannan, and L. E. Trotter Jr., Hermite normal form computation using modulo determinant arithmetic, Math. Oper. Res. 12 (1987), no. 1, 50–59. MR 882842, DOI 10.1287/moor.12.1.50

Disquisitiones arithmeticæ

Dorian Goldfeld, Gauss’s class number problem for imaginary quadratic fields, Bull. Amer. Math. Soc. (N.S.) 13 (1985), no. 1, 23–37. MR 788386, DOI 10.1090/S0273-0979-1985-15352-2
T. C. Hu and R. D. Young, Integer programming and network flows, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0263420
Loo Keng Hua, Introduction to number theory, Springer-Verlag, Berlin-New York, 1982. Translated from the Chinese by Peter Shiu. MR 665428
Ravindran Kannan and Achim Bachem, Polynomial algorithms for computing the Smith and Hermite normal forms of an integer matrix, SIAM J. Comput. 8 (1979), no. 4, 499–507. MR 573842, DOI 10.1137/0208040
Donald E. Knuth, The art of computer programming. Vol. 2, 2nd ed., Addison-Wesley Series in Computer Science and Information Processing, Addison-Wesley Publishing Co., Reading, Mass., 1981. Seminumerical algorithms. MR 633878

Algorithms in number theory

H. W. Lenstra Jr., On the calculation of regulators and class numbers of quadratic fields, Number theory days, 1980 (Exeter, 1980) London Math. Soc. Lecture Note Ser., vol. 56, Cambridge Univ. Press, Cambridge, 1982, pp. 123–150. MR 697260
Kevin S. McCurley, Cryptographic key distribution and computation in class groups, Number theory and applications (Banff, AB, 1988) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 265, Kluwer Acad. Publ., Dordrecht, 1989, pp. 459–479. MR 1123090
Władysław Narkiewicz, Classical problems in number theory, Monografie Matematyczne [Mathematical Monographs], vol. 62, Państwowe Wydawnictwo Naukowe (PWN), Warsaw, 1986. MR 961960
Carl Pomerance, Fast, rigorous factorization and discrete logarithm algorithms, Discrete algorithms and complexity (Kyoto, 1986) Perspect. Comput., vol. 15, Academic Press, Boston, MA, 1987, pp. 119–143. MR 910929
A. Schönhage and V. Strassen, Schnelle Multiplikation grosser Zahlen, Computing (Arch. Elektron. Rechnen) 7 (1971), 281–292 (German, with English summary). MR 292344, DOI 10.1007/bf02242355

Schnelle Berechnung von Kettenbruchenentwicklungen

1

H. Zantema, Class numbers and units, Computational methods in number theory, Part II, Math. Centre Tracts, vol. 155, Math. Centrum, Amsterdam, 1982, pp. 213–234. MR 702518
Alexander Schrijver, Theory of linear and integer programming, Wiley-Interscience Series in Discrete Mathematics, John Wiley & Sons, Ltd., Chichester, 1986. A Wiley-Interscience Publication. MR 874114
Martin Seysen, A probabilistic factorization algorithm with quadratic forms of negative discriminant, Math. Comp. 48 (1987), no. 178, 757–780. MR 878705, DOI 10.1090/S0025-5718-1987-0878705-X
Daniel Shanks, Class number, a theory of factorization, and genera, 1969 Number Theory Institute (Proc. Sympos. Pure Math., Vol. XX, State Univ. New York, Stony Brook, N.Y., 1969) Amer. Math. Soc., Providence, R.I., 1971, pp. 415–440. MR 0316385
Daniel Shanks, Five number-theoretic algorithms, Proceedings of the Second Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1972) Congressus Numerantium, No. VII, Utilitas Math., Winnipeg, Man., 1973, pp. 51–70. MR 0371855
Douglas H. Wiedemann, Solving sparse linear equations over finite fields, IEEE Trans. Inform. Theory 32 (1986), no. 1, 54–62. MR 831560, DOI 10.1109/TIT.1986.1057137