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. Allhave a preferred size of number they are best at factoring, and all of themhave plenty of implementations available. Msieve is no exception: it can with high probability find the complete factorization of any input number up to about 125 digits in size. The actual number of digits supported is much higher (up to 164 digits), but problems larger than 125 digits are likely to fail.
Trial division 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 more powerful methods. Prior to version 1.04, those methods were limited to the quadratic sieve. From that point on, however, an implementation of the number field sieve is also available. Information specific to the quadratic sieve implementation is contained in Readme.qs, while the number field sieve variant is described in Readme.nfs
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.