Factoring via the Elliptic Curve Method
---------------------------------------


Included in this package is an implementation of the elliptic curve method
for factoring integers.  This implementation is intended to be called from
inside a program supplied by the user; however, two sample programs, 
ecm-test and factor, are included.


COMPILING:

To make this software, do the following:

(1) Edit Makefile to ensure that NTLPREFIX is correctly set.
(2) Type 'make'.

Note: for (1), if the NTL header files are located in /opt/local/include/NTL/
and the library is named /opt/local/lib/libntl.a, then NTLPREFIX should be 
set to /opt/local.

Note: for best results, compile NTL to use GMP as its underlying large integer
package (LIP).  The Makefile assumes you have done this.  If you aren't using
GMP, you may need to remove the '-lgmp' from the NTLLIB line.


INTERFACE:

Please read the header file ZZFactoring.h and examine the program ecm-test.cpp
to learn how to use of the interface.  The essential points are:

1. Include the header file ZZFactoring.h.

2. Declare variables for the factoriation (of type vec_pair_ZZ_long) and the
   integer to factor (of type ZZ).

3. Call factor(factorization,n) to do the factoring.


CHOICE OF INTEGER:

Any integer can be chosen but composites that are the product of 2 or more
large factors will be the most difficult (take the most time) to factor.  

If you are interested in testing for smoothness, read the header file
ZZFactoring.h to learn how to get factor() to stop after the small factors
have been found.


PERFORMANCE:

The algorithm has been optimized on an Athalon XP1700 machine with 256MB
of RAM so it should be as fast as it can be on any similar machine.  

The following table lists the time required to find factors of the specified
size (on average using the Athalon XP1700 machine described above):

   48 bit factor:    5 seconds
   64 bit factor:  1.5 minutes
   80 bit factor:   20 minutes
   96 bit factor:  3.5 hours
  112 bit factor:   30 hours

Note that the time required to find a factor depends almost entirely on the
size of the factor to be found and very little on the size of the number 
itself.  During optimization, integers as large as 500 bits were factored.



Questions, comments, problems, or other concerns can be emailed to:
  chris.studholme@utoronto.ca

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