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by jwmerrill 3837 days ago
I don't work for the NSA—I just have a laptop, and a copy of Mathematica. The factorization took about 10 seconds.

Mathematica is really a pretty remarkable piece of software. I doubt it's world class for factorization, but it gives you easy access to a lot of pretty darn good algorithms for a wide range of problems.

There's 1 sentence on the implementation of FactorInteger in the docs, FWIW:

http://reference.wolfram.com/language/tutorial/SomeNotesOnIn...
2 comments
archgoon 3836 days ago
pari-gp is freely available for linux, and it's source is available if you're interested in their implementation.

http://pari.math.u-bordeaux.fr/

The relevant file is

  src/basemath/ifactor1.c
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waynecochran 3837 days ago
Okay ... wow ... how big do the numbers need to get to bog down Mathematica?
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Pyxl101 3837 days ago
The numbers that we're talking about are not big. The number you specified:

641071800653367850802176606120792275422168080497001121

OK, so take a 64 bit integer. That's the typical size of an integer in a modern programming language. A signed integer can represent 2^63-1 which is 9223372036854775807.

Imagine a "big integer" which is a composite of multiple 64-bit integers. With a big integer made up of two regular integers, you can represent the value 170141183460469231731687303715884105727. With three, you get to 3138550867693340381917894711603833208051177722232017256447 if I'm doing the math right. That's already in the ballpark of the number you mentioned - just three regular integers together.

Granted, I'm not talking about how difficult the factoring is, but it's worth noting that the size of the numbers are pretty small to a computer. Modern computers are really fast and can brute force a large number of computations. Mathematica probably has a fairly optimized general purpose factoring algorithm as well.

A number like this might be more difficult to factor:

  16158503035655503650357438344334975980222051334857
  74201606517271376232756943394544659860070576145673
  18443589804609490097470597795752454605475440761932
  24141560315438683650498045875098875194826053398028
  81919203378413839610932130987808091904716923808523
  52908229260181525214437879457705329043037761995619
  65192760957166694834171210342487393282284747428088
  01766316102903890282966551309635423015707512929643
  20885583629718018592309286787991755761508229522018
  48806616643615613562842355410104862578550863465661
  73483927129032834896752299863417649931910776258319
  47186677718010677166148023226592393024760740967779
  26805529798115327
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