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资源说明：msieve - Number Field Sieve implementation by Jason Papadopoulos
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.
Frequently Asked Questions
--------------------------
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.
Quadratic Sieve References
--------------------------
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
Decio Luiz Gazzoni Filho adds:
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).
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