ecm(1)
NAME
ecm - integer factorization using ECM, P-1 or P+1
SYNOPSIS
ecm [options] B1 [B2min-B2max | B2]
DESCRIPTION
ecm is an integer factoring program using the Elliptic Curve Method (ECM), the P-1 method, or the P+1 method. The following sections describe parameters relevant to these algorithms.
STEP 1 AND STEP 2 BOUND PARAMETERS
B1
B1 is the step 1 bound. It is a mandatory parameter. It can be
given either in integer format (for example 3000000) or in
floating-point format (3000000.0 or 3e6). The largest possible B1
value is 9007199254740996 for P-1, and ULONG_MAX or
9007199254740996 (whichever is smaller) for ECM and P+1. All primes
2 <= p <= B1 are processed in step 1.
B2
B2 is the step 2 bound. It is optional: if omitted, a default value
is computed from B1, which should be close to optimal. Like B1, it
can be given either in integer or in floating-point format. The
largest possible value of B2 is approximately 9e23, but depends on
the number of blocks k if you specify the -k option. All primes B1
<= p <= B2 are processed in step 2. If B2 < B1, no step 2 is
performed.
B2min-B2max
alternatively one may use the B2min-B2max form, which means that
all primes B2min <= p <= B2max should be processed. Thus specifying
B2 only corresponds to B1-B2. The values of B2min and B2max may be
arbitrarily large, but their difference must not exceed
approximately 9e23, subject to the number of blocks k.
FACTORING METHOD
-pm1
Perform P-1 instead of the default method (ECM).
-pp1
Perform P+1 instead of the default method (ECM).
GROUP AND INITIAL POINT PARAMETERS
-x0 x
[ECM, P-1, P+1] Use x (arbitrary-precision integer or rational) as
initial point. For example, -x0 1/3 is valid. If not given, x is
generated from the sigma value for ECM, or at random for P-1 and
P+1.
-sigma s
[ECM] Use s (arbitrary-precision integer) as curve generator. If
omitted, s is generated at random.
-A a
[ECM] Use a (arbitrary-precision integer) as curve parameter. If
omitted, is it generated from the sigma value.
-go val
[ECM, P-1, P+1] Multiply the initial point by val, which can any
valid expression, possibly containing the special character N as
place holder for the current input number. Example:
ecm -pp1 -go "N^2-1" 1e6 < composite2000
STEP 2 PARAMETERS
-k k
[ECM, P-1, P+1] Perform k blocks in step 2. For a given B2 value,
increasing k decreases the memory usage of step 2, at the expense
of more cpu time.
-treefile file
Stores some tables of data in disk files to reduce the amount of
memory occupied in step 2, at the expense of disk I/O. Data will be
written to files file.1, file.2 etc. Does not work with fast stage
2 for P+1 and P-1.
-power n
[ECM, P-1] Use x^n for Brent-Suyamas extension (-power 1 disables
Brent-Suyamas extension). The default polynomial is chosen
depending on the method and B2. For P-1 and P+1, disables the fast
stage 2. For P-1, n must be even.
-dickson n
[ECM, P-1] Use degree-n Dicksons polynomial for Brent-Suyamas
extension. For P-1 and P+1, disables the fast stage 2. Like for
-power, n must be even for P-1.
-maxmem n
Use at most n megabytes of memory in stage 2.
-ntt, -no-ntt
Enable or disable the Number-Theoretic Transform code for
polynomial arithmetic in stage 2. With NTT, dF is chosen to be a
power of 2, and is limited by the number suitable primes that fit
in a machine word (which is a limitation only on 32 bit systems).
The -no-ntt variant uses more memory, but is faster than NTT with
large input numbers. By default, NTT is used for P-1, P+1 and for
ECM on numbers of size at most 30 machine words.
OUTPUT
-q
Quiet mode. Found factorizations are printed on standard output,
with factors separated by white spaces, one line per input number
(if no factor was found, the input number is simply copied).
-v
Verbose mode. More information is printed, more -v options increase
verbosity. With one -v, the kind of modular multiplication used,
initial x0 value, step 2 parameters and progress, and expected
curves and time to find factors of different sizes for ECM are
printed. With -v -v, the A value for ECM and residues at the end of
step 1 and step 2 are printed. More -v print internal data for
debugging.
-timestamp
Print a time stamp whenever a new ECM curve or P+1 or P-1 run is
processed.
MODULAR ARITHMETIC OPTIONS
Several algorithms are available for modular multiplication. The
program tries to find the best one for each input; one can force a
given method with the following options.
-mpzmod
Use GMPs mpz_mod function (sub-quadratic for large inputs, but
induces some overhead for small ones).
-modmuln
Use Montgomerys multiplication (quadratic version). Usually best
method for small input.
-redc
Use Montgomerys multiplication (sub-quadratic version).
Theoretically optimal for large input.
-nobase2
Disable special base-2 code (which is used when the input number is
a large factor of 2^n+1 or 2^n-1, see -v).
-base2 n
Force use of special base-2 code, input number must divide 2^n+1 if
n > 0, or 2^|n|-1 if n < 0.
FILE I/O
The following options enable one to perform step 1 and step 2
separately, either on different machines, at different times, or using
different software (in particular, George Woltmans Prime95/mprime
program can produce step 1 output suitable for resuming with GMP-ECM).
It can also be useful to split step 2 into several runs, using the
B2min-B2max option.
-inp file
Take input from file file instead of from standard input.
-save file
Save result of step 1 in file. If file exists, an error is raised.
Example: to perform only step 1 with B1=1000000 on the composite
number in the file "c155" and save its result in file "foo", use
ecm -save foo 1e6 1 < c155
-savea file
Like -save, but appends to existing files.
-resume file
Resume residues from file, reads from standard input if file is
"-". Example: to perform step 2 following the above step 1
computation, use
ecm -resume foo 1e6
-chkpoint file
Periodically write the current residue in stage 1 to file. In case
of a power failure, etc., the computation can be continued with the
-resume option.
ecm -chkpnt foo -pm1 1e10 < largenumber.txt
LOOP MODE
The "loop mode" (option -c n) enables one to run several curves on each
input number. The following options control its behavior.
-c n
Perform n runs on each input number (default is one). This option
is mainly useful for P+1 (for example with n=3) or for ECM, where n
could be set to the expected number of curves to find a d-digit
factor with a given step 1 bound. This option is incompatible with
-resume, -sigma, -x0. Giving -c 0 produces an infinite loop until a
factor is found.
-one
In loop mode, stop when a factor is found; the default is to
continue until the cofactor is prime or the specified number of
runs are done.
-b
Breadth-first processing: in loop mode, run one curve for each
input number, then a second curve for each one, and so on. This is
the default mode with -inp.
-d
Depth-first processing: in loop mode, run n curves for the first
number, then n curves for the second one and so on. This is the
default mode with standard input.
-I n
In loop mode, multiply B1 by a factor depending on n after each
curve. Default is one which should be optimal on one machine, while
-I 10 could be used when trying to factor the same number
simultaneously on 10 identical machines.
SHELL COMMAND EXECUTION
These options allow for executing shell commands to supplement functionality to GMP-ECM.
MISCELLANEOUS
-stage1time n
Add n seconds to stage 1 time. This is useful to get correct
expected time with -v if part of stage 1 was done in another run.
-h, --help
Display a short description of ecm usage, parameters and command
line options.
-printconfig
Prints configuration parameters used for the compilation and exits.
INPUT SYNTAX
The input numbers can have several forms: Raw decimal numbers like 123456789. Comments can be placed in the file: everything after "//" is ignored, up to the end of line. Line continuation. If a line ends with a backslash character "\", it is considered to continue on the next line. Common arithmetic expressions can be used. Example: 3*5+2^10. Factorial: example 53!. Multi-factorial: example 15!3 means 15*12*9*6*3. Primorial: example 11# means 2*3*5*7*11. Reduced primorial: example 17#5 means 5*7*11*13*17. Functions: currently, the only available function is Phi(x,n).
EXIT STATUS
The exit status reflects the result of the last ECM curve or P-1/P+1
attempt the program performed. Individual bits signify particular
events, specifically:
Bit 0
0 if normal program termination, 1 if error occurred
Bit 1
0 if no proper factor was found, 1 otherwise
Bit 2
0 if factor is composite, 1 if factor is a probable prime
Bit 3
0 if cofactor is composite, 1 if cofactor is a probable prime
Thus, the following exit status values may occur:
0
Normal program termination, no factor found
1
Error
2
Composite factor found, cofactor is composite
6
Probable prime factor found, cofactor is composite
8
Input number found
10
Composite factor found, cofactor is a probable prime
14
Probable prime factor found, cofactor is a probable prime
BUGS
Report bugs to <[email protected]>, after checking <http://www.loria.fr/~zimmerma/records/ecmnet.html> for bug fixes or new versions.
AUTHORS
Pierrick Gaudry <gaudry at lix dot polytechnique dot fr> contributed efficient assembly code for combined mul/redc; Jim Fougeron <jfoug at cox dot net> contributed the expression parser and several command-line options; Laurent Fousse <laurent at komite dot net> contributed the middle product code, the autoconf/automake tools, and is the maintainer of the Debian package; Alexander Kruppa <(lastname)[email protected]> contributed estimates for probability of success for ECM, the new P+1 and P-1 stage 2 (with P.-L. Montgomery), new AMD64 asm mulredc code, and some other things; Dave Newman <david.(lastname)@jesus.ox.ac.uk> contributed the Kronecker-Schoenhage and NTT multiplication code; Jason S. Papadopoulos contributed a speedup of the NTT code Paul Zimmermann <zimmerma at loria dot fr> is the author of the first version of the program and chief maintainer of GMP-ECM. Note: email addresses have been obscured, the required substitutions should be obvious.
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