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# FactInt

## Advanced Methods for Factoring Integers

1.6.3

15 November 2019

Stefan Kohl
Email: stefan@gap-system.org
Homepage: https://stefan-kohl.github.io/

### 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 ± 1. FactInt is completely written in the GAP language and contains / requires no external binaries. It needs GAPDoc 1.6 [LN17] or higher. FactInt must be installed in the `pkg` subdirectory of the GAP distribution.

### Copyright

© 1999 - 2017 by Stefan Kohl.

FactInt is free software: you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation, either version 2 of the License, or (at your option) any later version.

FactInt is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details.

For a copy of the GNU General Public License, see the file `GPL` in the `etc` directory of the GAP distribution or see http://www.gnu.org/licenses/gpl.html.

### 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
References
Index
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