Asymptotically fast factorization of integers

Type: Article

Publication Date: 1981-01-01

Citations: 127

DOI: https://doi.org/10.1090/s0025-5718-1981-0595059-1

Abstract

The paper describes a "probabilistic algorithm" for finding a factor of any large composite integer <italic>n</italic> (the required input is the integer <italic>n</italic> together with an auxiliary sequence of random numbers). It is proved that the expected number of operations which will be required is <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis exp left-brace beta left-parenthesis ln n ln ln n right-parenthesis Superscript 1 slash 2 Baseline right-brace right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mi>exp</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mo fence="false" stretchy="false">{</mml:mo> <mml:mi>β<!-- β --></mml:mi> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo stretchy="false">(</mml:mo> <mml:mi>ln</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:mi>ln</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>ln</mml:mi> <mml:mo>⁡<!-- ⁡ --></mml:mo> <mml:mi>n</mml:mi> <mml:msup> <mml:mo stretchy="false">)</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>2</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> <mml:mo fence="false" stretchy="false">}</mml:mo> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O(\exp \{ \beta {(\ln n\ln \ln n)^{1/2}}\} )</mml:annotation> </mml:semantics> </mml:math> </inline-formula> for some constant <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="beta greater-than 0"> <mml:semantics> <mml:mrow> <mml:mi>β<!-- β --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\beta &gt; 0</mml:annotation> </mml:semantics> </mml:math> </inline-formula>. Asymptotically, this algorithm is much faster than any previously analyzed algorithm for factoring integers; earlier algorithms have all required <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="upper O left-parenthesis n Superscript alpha Baseline right-parenthesis"> <mml:semantics> <mml:mrow> <mml:mi>O</mml:mi> <mml:mo stretchy="false">(</mml:mo> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:msup> <mml:mi>n</mml:mi> <mml:mi>α<!-- α --></mml:mi> </mml:msup> </mml:mrow> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> <mml:annotation encoding="application/x-tex">O({n^\alpha })</mml:annotation> </mml:semantics> </mml:math> </inline-formula> operations where <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="alpha greater-than 1 slash 5"> <mml:semantics> <mml:mrow> <mml:mi>α<!-- α --></mml:mi> <mml:mo>&gt;</mml:mo> <mml:mn>1</mml:mn> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mo>/</mml:mo> </mml:mrow> <mml:mn>5</mml:mn> </mml:mrow> <mml:annotation encoding="application/x-tex">\alpha &gt; 1/5</mml:annotation> </mml:semantics> </mml:math> </inline-formula>.

Locations

  • Mathematics of Computation - View - PDF

Similar Works

Action Title Year Authors
+ Approximating the number of integers without large prime factors 2005 Koji Suzuki
+ PDF Chat Approximating the number of integers free of large prime factors 1997 Simon C. Hunter
Jonathan Sorenson
+ Fast Factoring of Integers 2005 Gordon Chalmers
+ Factorization of Integers 2008 James Simshaw
+ PDF Chat Sieving the positive integers by small primes 1988 D. A. Goldston
Kevin S. McCurley
+ Deterministic Integer Factorization Algorithms 2013 N. A. Carella
+ A new method of factorization of very large numbers 2017 Jamel Ghanouchi
+ PDF Chat On 𝑘-free integers with small prime factors 1975 D. G. Hazlewood
+ Aurifeuillian factorization 2005 Andrew Granville
P. A. B. Pleasants
+ Computational methods for factoring large integers 1988 M. C. Wunderlich
+ PDF Chat Factoring large integers 1974 Robert S. Lehman
+ PDF Chat On integers free of large prime factors 1986 Adolf Hildebrand
Gérald Tenenbaum
+ The Factorization of Integers 1982 Hua Loo Keng
+ Correction to "Factorization of Integers" 1964
+ PDF Chat The number of primes is finite 1999 Miodrag Živković
+ Exponential Methods of Factoring Integers 2019 Samuel S. Wagstaff
+ A central limit theorem for random factorizations of integers 2011 Hsien‐Kuei Hwang
Svante Janson
+ Factoring integers 2013 Joachim von zur Gathen
Jürgen Gerhard
+ On the greatest prime factor of (𝑎𝑏+1)(𝑎𝑐+1) 2002 Pietro Corvaja
Umberto Zannier
+ PDF Chat Estimates of the least prime factor of a binomial coefficient 1993 P. Erdős
C. B. Lacampagne
J. L. Selfridge