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msieve - Number Field Sieve implementation by Jason Papadopoulos

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MSIEVE: A Library for Factoring Large Integers Jason Papadopoulos

Introduction

Msieve is the result of my efforts to understand and optimize how integers are factored using the most powerful modern algorithms.

This documentation corresponds to version 1.46 of the Msieve library. Do not expect to become a factoring expert just by reading it. I've included a relatively complete list of references that you can and should look up if you want to treat the code as more than a black box to solve your factoring problems.

What Msieve Does

Factoring is the study (half math, half engineering, half art form) of taking big numbers and expessing them as the product of smaller numbers. If I find out 15 = 3 * 5, I've performed an integer factorization on the number 15. As the number to be factored becomes larger, the difficulty involved in completing its factorization explodes, to the point where you can invent secret codes that depend on the difficulty of factoring and reasonably expect your encrypted data to stay safe.

There are plenty of algorithms for performing integer factorization. The Msieve library implements most of them from scratch, and relies on optional external libraries for the rest of them. Trial division and Pollard Rho is used on all inputs; if the result is less than 25 digits in size, tiny custom routines do the factoring. For larger numbers, the code switches to the GMP-ECM library and runs the P-1, P+1 and ECM algorithms, expending a user-configurable amount of effort to do so. If these do not completely factor the input number, the library switches to the heavy artillery. Unless told otherwise, Msieve runs the self-initializing quadratic sieve algorithm, and if this doesn't factor the input number then you've found a library problem. If you know what you're doing, Msieve also contains a complete implementation of the number field sieve, that has helped complete some of the largest public factorization efforts known. Information specific to the quadratic sieve implementation is contained in Readme.qs, while the number field sieve variant is described in Readme.nfs

The maximum size of numbers that can be given to the library is hardwired at compile time. Currently the code can handle numbers up to 275 digits; however, you should bear in mind that I don't expect the library to be able to complete a factorization larger than about 120 digits by itself. The larger size inputs can only really be handled by the number field sieve, and only part of the NFS code (the final part) is efficient and robust enough to deal with problems that large.

Msieve was written with several goals in mind:

```- To be as fast as possible. I claim (without proof) that for
completely factoring general inputs between 40 and 100 digits
in size, Msieve is faster than any other code implementing any
other algorithm. I realize that's a tall order, and that I'll
probably have to eat those words, but a *lot* of effort has gone
into making Msieve fast.

To be as portable as possible. The code is written in C and is
completely self contained. It has its own basic multiple precision
library (which can be used in other applications) and is written
in as machine-independent a manner as possible. I've verified that
the source code compiles and runs correctly on 32- or 64-bit Intel
x86, 32- and 64-bit PowerPC, and 64-bit Alpha platforms. It's
reported to work in 32-bit mode on the RS6000. It works in Windows,
Linux (several flavors), Mac OS X, and AIX. Pretty much the only
requirement for building the code is that your compiler have a
native 64-bit data type.

To be simple to use. The only input is the integer to be factored.
Everything else happens automatically.

To be free (as in beer). The entire code base is released into the
public domain. This is hobby stuff for me, and the more it's used
the better.
```

If you choose to use Msieve, please let me know how it goes. I welcome bug reports, suggestions, optimizations, ports to new platforms, complaints, boasts, whatever.

Getting Msieve

The latest version of Msieve can be found on my web page, www.boo.net/~jasonp. A precompiled Windows binary using the latest source (optimized for the AMD athlon processor) is also available there.

The source distribution comes with a unix makefile you can use if you want to build msieve from source. If you have Microsoft Visual Studio 2007, Brian Gladman has kindly provided a set of build files that will generate Windows binaries.

Using Msieve

Just to be confusing, there are two things that I call 'Msieve' interchangeably. The source distribution builds a self-contained static library 'libmsieve.a', that actually performs factorizations, and also builds a 'msieve' demo application that uses the library. The library has a very lightweight inter- face defined in msieve.h, and can be used in other applications. While the demo application is (slightly) multithreaded, most the library is single- threaded and all of its state is passed in. The linear algebra code used in the quadratic- and number field sieve is multithread aware, and the entire library is supposed to be multithread-safe.

The demo application has only one job: to act as a delivery vehicle for integers to be factored. Numbers can come from a text file, from redirected input, from the command line directly, or can be manually typed in one at a time. Batch input is also supported, so that you can just point the application to a text file with a collection of numbers, one per line. By default, all output goes to a logfile and a summary goes to the screen. For the complete list of options, try 'msieve -h'.

Starting with v1.08, the inputs to msieve can be integer arithmetic expressions using any of the following operators:

• - plus, minus (lowest priority) % integer remainder
• / multiply, integer divide ^ power ( ) grouping (highest priority)

Hence for example:

```    (10^53 - 1) / 9
```

gives the value:

```    11111111111111111111111111111111111111111111111111111
```

The integers used in an expression can be of any length but all intermediate results and the final result are restricted to 275 or less decimal digits.

While factoring an integer, the library can produce a very large amount of intermediate information. This information is stored in one or more auxiliary savefiles, and the savefiles can be used to restart an interrupted factorization. Note that factoring one integer and then another integer will overwrite the savefiles from the first integer.

The amount of memory that's needed will depend on the size of the number to be factored and the algorithm used. If running the quadratic sieve or the number field sieve, the memory requirements increase towards the end of a factorization, when all of the intermediate results are needed at the same time. For a 100-digit quadratic sieve factorization, most of the time Msieve needs 55-65MB of memory, with the last stage of the factorization needing 100-130MB. The final part of the number field sieve can use up incredible amounts of memory; for example, completing the factorization of a 512-bit number like an RSA key needs 2-3GB of memory.

Q. I want to factor much bigger numbers. Can't Msieve solve problems larger than you say? Q. I'm trying to break my ex-girlfriend's RSA key with Msieve, and it's not working. What's wrong? A. The quadratic sieve really is not appropriate for factoring numbers over ~110 digits in size, and the number field sieve implementation isn't even close to done. On a fast modern CPU, a 110-digit factor- ization takes nearly 120 hours for Msieve, and the time increases steeply beyond that. If you have really big factorization needs, there are essentially only two packages that you can use: GGNFS and the NFS implementation by Chris Card. Both are hosted on SourceForge (see www.sf.net/projects/ggnfs and www.sf.net/projects/factor-by-gnfs). For the largest size problems, you have to use the number field sieve; in fact, you have to use GGNFS for the first part of the factorization and then msieve for the last part.

Q. Can you make Msieve network aware? Can you make it a client-server thingy? Can I use the internet to factor numbers? A. The demo application for the Msieve library is just that, a demo. I don't know anything about network programming and I'm not qualified to build a client-server application that's safe in the face of internet threats. If you have these kinds of smarts, you can use Msieve in your own code and I'll help as much as I can. The demo is good enough for people with a few machines on a small private LAN, and this is ~100% of the user community right now.

Q. How can I modify Msieve to work on a cluster? A. Distributed sieving is so easy that you don't need high-performance parallel programming techniques or message passing libraries to do it. If you're lucky enough to have a cluster then the batch scheduling system that comes with the cluster is more than enough to implement cluster-aware sieving. Of course if you have access to that much firepower you owe it to yourself to use an NFS package of some sort.

Q. Can you modify Msieve to run on multi-core processors? A. As described above, the really intensive part of the QS and NFS algorithms is the sieving, and it's a waste of effort to multithread that. You won't save any time compared to just running two copies of Msieve. The final stage can benefit from multithreading, and the intensive parts of that are already multithread-aware. This can be improved, but multithreading more parts of the library is a low priority for me.

Q. Why put Msieve into the public domain and not make it GPL? Wouldn't GPL-ed code protect your work and encourage contributions? A. Msieve is a learning exercise, not a collaborative effort per se. I don't expect other developers to help, though several have and it's appreciated. As for protection, there's only one way to generate income from this code: use it to win factoring contests. While the number field sieve can win factoring contests, you personally do not have the resources to do so. Even if you did, this code just can't manage factorizations that big. There's no reason to put licensing restrictions on this code.

Q. Your multiple precision library sucks. Why didn't you use GMP? A. I know it sucks. Using GMP would have left MSVC users in the cold, and even building GMP is a major exercise that requires essentially a complete and up-to-date unix environment. The latest GMP did not even build on several (old, buggy or experimental) platforms that Msieve ran happily on. The nice thing about sieve-based factoring is that for big factorizations the actual multiple precision math takes about 1% of the total runtime. Since bignum performance isn't an issue but portability is, I decided against GMP. Latter-day versions of Msieve use a much- improved multiple-precision library that probably manages most of the speed gains possible if GMP was used.

Credits

Especially as the code became more useful, credit is due to several people who pushed it very hard.

Tom Womack, Greg Childers, Bruce Dodson, Hallstein Hansen, Paul Leyland and Richard Wackerbarth have continually thrown enormous size problems at the NFS postprocessing code of Msieve, and have been very patient as I've frantically tried to keep up with them. If you try to use NFS and it just works, even when other programs fail, you primarily have these guys to thank.

Jeff Gilchrist has done a lot of testing, feedback on 64-bit windows, and general documentation writing

Tom Cage (RIP) found lots of bugs and ground through hundreds of factorizations with early versions of Msieve.

Jay Berg did a lot of experimenting with very big factorizations in earlier Msieve versions

The regulars in the Factoring forum of www.mersenneforum.org (especially Jes, Luigi, Sam, Dennis, Sander, Mark R., Peter S., Jeff G.) have also done tons of factoring work.

Alex Kruppa and Bob Silverman all contributed useful theoretical stuff. Bob's NFS siever code has been extremely helpful in getting me to understand how things work.

I thank my lucky stars that Chris Card figured out how NFS filtering works before I had to.

Bob Silverman, Dario Alpern and especially Bill Hart helped out with the NFS square root.

'forzles' helped improve the multithreading in the linear algebra.

Falk Hueffner and Francois Glineur found several nasty bugs in earlier versions.

Brian Gladman contributed an expression evaluator, the Visual Studio build system, a lot of help with the numerical integration used in the NFS polynomial selector, and various portability fixes.

J6M did the AIX port.

Larry Soule did a lot of work incorporating GMP into the code, which I regrettably don't expect to use

I know I left out people, but that's my fault and not theirs.

The book "Prime Numbers: A Computational Perspective", by Richard Crandall and Carl Pomerance, is an excellent introduction to the quadratic sieve and many other topics in computational number theory.

Scott Contini's thesis, "Factoring Large Integers with the Self-Initializing Quadratic Sieve", is an implementer's dream; it fills in all the details of the sieving process that Crandall and Pomerance gloss over.

Wambach and Wettig's 1995 paper "Block Sieving Algorithms" gives an introduction to making sieving cache-friendly. Msieve uses very different (more efficient) algorithms, but you should try to understand these first.

Lenstra and Manasse's 1994 paper "Factoring with Two Large Primes" describes in great detail the cycle-finding algorithm that is the heart of the combining stage of Msieve. More background information on spanning trees and cycle- finding can be found in Manuel Huber's 2003 paper "Implementation of Algorithms for Sparse Cycle Bases of Graphs". This was the paper that connected the dots for me (pun intended).

There are three widely available descriptions of SQUFOF. An introductory one is Hans Riesel's section titled "Shanks' Factoring Method SQUFOF" in his book "Prime Numbers and Computer Methods for Factorization". The much more advanced one is "Square Form Factorization", a PhD dissertation by Jason Gower (which is the reference I used when implementing the algorithm). Henri Cohen's book (mentioned below) also has an extended treatment of SQUFOF. Daniel Shanks was a professor at the University of Maryland while I was a student there, and his work got me interested in number theory and computer programming. I dearly wish I met him before he died in 1996.

Brandt Kurowski's 1998 paper "The Multiple Polynomial Quadratic Sieve: A Platform-Independent Distributed Application" is the only reference I could find that describes the Knuth-Schroeppel multiplier algorithm.

Davis and Holdrige's 1983 paper "Factorization Using the Quadratic Sieve Algorithm" gives a surprising theoretical treatment of how QS works. Reading it felt like finding some kind of forgotten evolutionary offshoot, strangely different from the conventional way of implementing QS.

Peter Montgomery's paper "A Block Lanczos Algorithm for Finding Dependencies over GF(2)" revolutionized the factoring business. The paper by itself isn't enough to implement his algorithm; you really need someone else's implementation to fill in a few critical gaps.

Michael Peterson's recent thesis "Parallel Block Lanczos for Solving Large Binary Systems" gives an interesting reformulation of the block Lanczos algorithm, and gives lots of performance tricks for making it run fast.

Kaltofen's paper 'Blocked Iterative Sparse Linear System Solvers for Finite Fields' is a good condensing of Montgomery's original block Lanczos paper

Gupta's IBM technical report 'Recent Advances in Direct Methods for Solving Unsymmetric Sparse Systems of Linear Equations' doesn't have anything in an NFS context, but there's gotta be some useful material in it for factoring people

Number Field Sieve References

Matthew Briggs' 'An Introduction to the Number Field Sieve' is a very good introduction; it's heavier than C&P in places and lighter in others

Michael Case's 'A Beginner's Guide to the General Number Field Sieve' has more detail all around and starts to deal with advanced stuff

Per Leslie Jensen's thesis 'Integer Factorization' has a lot of introductory detail on NFS that other references lack

Peter Stevenhagen's "The Number Field Sieve" is a whirlwind introduction the algorithm

Steven Byrnes' "The Number Field Sieve" is a good simplified introduction as well.

Lenstra, Lenstra, Manasse and Pollard's paper 'The Number Field Sieve' is nice for historical interest

'Factoring Estimates for a 1024-bit RSA Modulus' should be required reading for anybody who thinks it would be a fun and easy project to break a commercial RSA key.

Brian Murphy's thesis, 'Polynomial Selection for the Number Field Sieve Algorithm', is simply awesome. It goes into excruciating detail on a very undocumented subject.

Thorsten Kleinjung's 'On Polynomial Selection for the General Number Field Sieve' explains in detail a number of improvements to NFS polynomial selection developed since Murphy's thesis.

Jason Gower's 'Rotations and Translations of Number Field Sieve Polynomials' describes some very promising improvements to the polynomial generation process. As far as I know, nobody has actually implemented them.

D.J. Bernstein has two papers in press and several slides on some improvements to the polynomial selection process, that I'm just dying to implement.

Aoki and Ueda's 'Sieving Using Bucket Sort' described the kind of memory optimizations that a modern siever must have in order to be fast

Dodson and Lenstra's 'NFS with Four Large Primes: An Explosive Experiment' is the first realization that maybe people should be using two large primes per side in NFS after all

Franke and Kleinjung's 'Continued Fractions and Lattice Sieving' is the only modern reference available on techniques used in a high- performance lattice siever.

Bob Silverman's 'Optimal Parametrization of SNFS' has lots of detail on parameter selection and implementation details for building a line siever

Cavallar's 'Strategies in Filtering in the Number Field Sieve' is really the only documentation on NFS postprocessing

Denny and Muller's extended abstract 'On the Reduction of Composed Relations from the Number Field Sieve' is an early attempt at NFS filtering that's been completely forgotten by now, but their techniques can work on top of ordinary NFS filtering

Montgomery's 'Square Roots of Products of Algebraic Numbers' describes the standard algorithm for the NFS square root phase

Nguyen's 'A Montgomery-Like Square Root for the Number Field Sieve' is also standard stuff for this subject; I haven't read this or the previous paper in great detail, but that's because the convetional NFS square root algorithm is still a complete mystery to me

David Yun's 'Algebraic Algorithms Using P-adic Constructions' provided a lot of useful theoretical insight into the math underlying the simplex brute-force NFS square root algorithm that msieve uses

The collection of papers

```The Development of the Number Field
Sieve' (Springer Lecture Notes In Mathematics 1554) should be
absolutely required reading -- unfortunately it's very hard to get
ahold of. It's always marked```
special order' at Amazon.com, and I figured I shouldn't even try to order as they'd get back to me in a couple of weeks saying the book wasn't available. I was very lucky to find a copy available one day, which I promptly ordered. Again, I cannot recommend this book enough; I had read lots of literature on NFS but the first time I `got' it was after reading the papers here. Modern expositions of NFS only show the algorithm as its currently implemented, and at times certain things are far from obvious. Now this book, being a historical account of NFS, shows how it progressed starting from John Pollard's initial work on SNFS, and things that looked out of place start to make sense. It's particularly enlightening to understand the initial formulation of SNFS, without the use of character columns. [NOTE: this has been reprinted and is available from bn.com, at least -JP]

As usual, a very algebraic and deep exposition can be found in Henri Cohen's book

```A Course In Computational Algebraic Number Theory'.
Certainly not for the faint of heart though. It's quite dated as
well, e.g. the SNFS section is based on the```
old' (without character columns) SNFS, but explores a lot of the underlying algebra.

In order to comprehend NFS, lots of background on algebra and algebraic number theory is necessary. I found a nice little introductory book on algebraic number theory, `The Theory of Algebraic Numbers' by Harry Pollard and Harold Diamond. It's an old book, not contaminated by the excess of abstraction found on modern books. It helped me a lot to get a grasp on the algebraic concepts. Cohen's book is hard on the novice but surprisingly useful as one advances on the subject, and the algorithmic touches certainly help.

As for papers: `Solving Sparse Linear Equations Over Finite Fields' by Douglas Wiedemann presents an alternate method for the matrix step. Block Lanczos is probably better, but perhaps Wiedemann's method has some use, e.g. to develop an embarassingly parallel algorithm for linear algebra (which, in my opinion, is the current holy grail of NFS research).