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# FactInt

## Advanced Methods for Factoring Integers

Version 1.5.2

September 26, 2007

Stefan Kohl
e-mail: kohl@mathematik.uni-stuttgart.de
WWW: http://www.cip.mathematik.uni-stuttgart.de/~kohlsn/
Address:
Institut für Geometrie und Topologie
Pfaffenwaldring 57
Universität Stuttgart
70550 Stuttgart
Germany

### Abstract

This package for GAP 4 provides a general-purpose integer factorization routine, which makes use of a combination of factoring methods. In particular it contains implementations of the following algorithms:

- Pollard's p-1

- Williams' p+1

- Elliptic Curves Method (ECM)

- Continued Fraction Algorithm (CFRAC)

- Multiple Polynomial Quadratic Sieve (MPQS)

It also contains code by Frank Lübeck for making use of Richard P. Brent's tables of factors of integers of the form b^k pm 1. FactInt is completely written in the GAP language and contains / requires no external binaries. It needs GAPDoc 1.0 [LN07] or higher. FactInt must be installed in the `pkg` subdirectory of the GAP distribution.

### Copyright

© 1999 - 2007 by Stefan Kohl. This package is distributed under the GNU General Public License.

### Acknowledgements

I would like to thank Bettina Eick and Steve Linton for their support and many interesting discussions.

### Contents

1. Preface
2. The General Factorization Routine
2.1 The method for `Factors`
2.1-1 Factors
2.1-2 FactInt
2.2 Getting information about the factoring process
2.2-1 InfoFactInt
3. The Routines for Specific Factorization Methods
3.1 Trial division
3.1-1 FactorsTD
3.2 Pollard's p-1
3.2-1 FactorsPminus1
3.3 Williams' p+1
3.3-1 FactorsPplus1
3.4 The Elliptic Curves Method (ECM)
3.4-1 FactorsECM
3.5 The Continued Fraction Algorithm (CFRAC)
3.5-1 FactorsCFRAC
3.6 The Multiple Polynomial Quadratic Sieve (MPQS)
3.6-1 FactorsMPQS
4. How much Time does a Factorization take?
4.1 Timings for the general factorization routine
4.2 Timings for the ECM
4.3 Timings for the MPQS
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