The expectation of success using a Monte Carlo factoring method--some statistics on quadratic class numbers
Author: Duncan A. Buell
Journal: Math. Comp. 43 (1984), 313-327
MSC: Primary 11Y05; Secondary 11A51, 11R29, 11Y40
MathSciNet review: 744940
Abstract: A method has been proposed for factoring an integer N by using the structure of the class groups of quadratic fields of radicand for various small multipliers k. We discuss the method and an implementation of the method, and various theoretical questions which have an impact on the practical use of the method in factoring. Some of the theoretical questions relate to the nature of class numbers and class groups; we present extensive statistical results on the class numbers and class groups of imaginary quadratic fields.
- [BRIL] John Brillhart, D. H. Lehmer, J. L. Selfridge, Bryant Tuckerman, and S. S. Wagstaff Jr., Factorizations of 𝑏ⁿ±1, Contemporary Mathematics, vol. 22, American Mathematical Society, Providence, R.I., 1983. 𝑏=2,3,5,6,7,10,11,12 up to high powers. MR 715603
- [BUEL76] Duncan A. Buell, Class groups of quadratic fields, Math. Comp. 30 (1976), no. 135, 610–623. MR 0404205, https://doi.org/10.1090/S0025-5718-1976-0404205-X
- [KNUT] Donald E. Knuth, The art of computer programming, 2nd ed., Addison-Wesley Publishing Co., Reading, Mass.-London-Amsterdam, 1975. Volume 1: Fundamental algorithms; Addison-Wesley Series in Computer Science and Information Processing. MR 0378456
- [MOOD] Alexander M. Mood & Franklin A. Graybill, Introduction to the Theory of Statistics, 2nd ed., McGraw-Hill, New York, 1963, pp. 126-129.
- [POMEa] Carl Pomerance, private communication.
- [POMEb] C. Pomerance, Analysis and comparison of some integer factoring algorithms, Computational methods in number theory, Part I, Math. Centre Tracts, vol. 154, Math. Centrum, Amsterdam, 1982, pp. 89–139. MR 700260
- [POMEc] Carl Pomerance and Samuel S. Wagstaff Jr., Implementation of the continued fraction integer factoring algorithm, Congr. Numer. 37 (1983), 99–118. MR 703581
- [SCHN] C.-P. Schnorr, A Monte Carlo factoring algorithm with finite storage, Seminar on Number Theory, 1981/1982, Univ. Bordeaux I, Talence, 1982, pp. Exp. No. 40, 16. Based on joint work with H. W. Lenstra, Jr. MR 695351
- [SHAN] 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
- [ZIMM72] Horst G. Zimmer, Computational problems, methods, and results in algebraic number theory, Lecture Notes in Mathematics, Vol. 262, Springer-Verlag, Berlin-New York, 1972. MR 0323751
- John Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman & S. S. Wagstaff, Jr., Factorizations of , b = 2, 3, 5, 6, 7, 10, 11, 12, up to High Powers, Contemporary Math., Vol. 22, Amer. Math. Soc., Providence, R. I., 1983. MR 715603 (84k:10005)
- Duncan A. Buell, "Class groups of quadratic fields," Math. Comp., v. 30, 1976, pp. 610-623. MR 0404205 (53:8008)
- Donald Knuth, The Art of Computer Programming, 2nd ed., Vol. 2, Addison-Wesley, Reading, Mass., 1981, p. 323. MR 0378456 (51:14624)
- Alexander M. Mood & Franklin A. Graybill, Introduction to the Theory of Statistics, 2nd ed., McGraw-Hill, New York, 1963, pp. 126-129.
- Carl Pomerance, private communication.
- Carl Pomerance, "Analysis and comparison of some integer factoring methods," Computational Methods in Number Theory (H. W. Lenstra, Jr., R. Tijdeman, eds.), Math. Centrum, Amsterdam, 1982. MR 700260 (84i:10005)
- Carl Pomerance & Samuel S. Wagstaff, Jr., "Implementation of the continued fraction integer factoring algorithm." (To appear.) MR 703581 (85c:11124)
- C. P. Schnorr & H. W. Lenstra, Jr., "A Monte Carlo factoring algorithm with finite storage." (To appear.) (Extended abstract appears in Lecture Notes in Comput. Sci., Vol. 145, Springer-Verlag, New York, pp. 19-34.) MR 695351 (84d:10003)
- Daniel Shanks, Class Number, A Theory of Factorization, and Genera, Proc. Sympos. Pure Math., Vol. 20, Amer. Math. Soc., Providence, R. I., 1971, pp. 415-440. MR 0316385 (47:4932)
- H. G. Zimmer, Computational Problems, Methods, and Results in Algebraic Number Theory, Lecture Notes in Math., Vol. 262, Springer-Verlag, New York, 1972. MR 0323751 (48:2107)