A probabilistic factorization algorithm with quadratic forms of negative discriminant

@article{Seysen1987APF,
  title={A probabilistic factorization algorithm with quadratic forms of negative discriminant},
  author={M. Seysen},
  journal={Mathematics of Computation},
  year={1987},
  volume={48},
  pages={757-780}
}
  • M. Seysen
  • Published 1987
  • Mathematics
  • Mathematics of Computation
We propose a probabilistic algorithm for factorization of an integer N with run time (expVlog N log log N)V54 +?(1). Asymptotically, our algorithm will be as fast as the wellknown factorization algorithm of Morrison and Brillhart. The latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis. Our new algorithm will be analyzed under the assumption of the Extended Riemann Hypothesis and it will be of Las Vegas type. On input N, the new algorithm… Expand
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References

SHOWING 1-10 OF 62 REFERENCES
A Monte Carlo factoring algorithm with linear storage
We present an algorithm which will factor an integer n quite efficiently if the class number h(-n) is free of large prime divisors. The running time T(n) (number of compositions in the class group)Expand
Refined Analysis and Improvements on Some Factoring Algorithms
  • C. Schnorr
  • Mathematics, Computer Science
  • J. Algorithms
  • 1982
TLDR
It is shown how to speed up the factorization of n by using preprocessed lists of those numbers in [−u, u] and [n - u, n + u], 0⪡ u ⪡ n which only have small prime factors, and it is proved that this algorithm with probability sol1 2 detects a proper factor of every composite n within o( exp 6 ln n ln lnLn n ) steps. Expand
Factoring integers with elliptic curves
TLDR
This paper is devoted to the description and analysis of a new algorithm to factor positive integers that depends on the use of elliptic curves and it is conjectured that the algorithm determines a non-trivial divisor of a composite number n in expected time at most K( p)(log n)2. Expand
Asymptotically fast factorization of integers
The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer n (the required input is the integer n together with an auxiliary sequence of random numbers). ItExpand
Worst-Case Complexity Bounds for Algorithms in the Theory of Integral Quadratic Forms
  • J. Lagarias
  • Computer Science, Mathematics
  • J. Algorithms
  • 1980
TLDR
This paper establishes explicit polynomial worst-case running time bounds for algorithms to solve certain of the problems in the theory of integral quadratic forms developed by Gauss. Expand
On a problem of Oppenheim concerning “factorisatio numerorum”
Let f(n) denote the number of factorizations of the natural number n into factors larger than 1 where the order of the factors does not count. We say n is “highly factorable” if f(m)<f(n) for all m <Expand
Gaussian elimination is not optimal
t. Below we will give an algorithm which computes the coefficients of the product of two square matrices A and B of order n from the coefficients of A and B with tess than 4 . 7 n l°g7 arithmeticalExpand
Analytic methods in the analysis and design of number-theoretic algorithms
This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoreticExpand
Theorems on factorization and primality testing
TLDR
This paper is concerned with the problem of obtaining theoretical estimates for the number of arithmetical operations required to factorize a large integer n or test it for primality, and uses a multi-tape Turing machine for this purpose. Expand
On the number of positive integers . . .
  • 1966
1. Introduction Let P(x,y) denote the number of integers specified in the title. A number of estimates and asymptotic formulae for this function have been given (cf. [1] and the literature mentionedExpand
...
1
2
3
4
5
...