ecm - integer factorization using ECM, P-1 or P+1
ecm [options] B1 [B2min-B2max | B2]
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 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 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
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.
Perform P-1 instead of the default method (ECM).
Perform P+1 instead of the default method (ECM).
Perform trial division up to n, before P-1, P+1 or ECM. In loop
mode (see option -c), trial division is only performed in the first
GROUP AND INITIAL POINT PARAMETERS
[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
[ECM] Use s (arbitrary-precision integer) as curve generator. If
omitted, s is generated at random.
[ECM] Use a (arbitrary-precision integer) as curve parameter. If
omitted, is it generated from the sigma value.
[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
[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.
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.
[ECM, P-1] Use x^n for Brent-Suyama´s extension (-power 1 disables
Brent-Suyama´s 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.
[ECM, P-1] Use degree-n Dickson´s polynomial for Brent-Suyama´s
extension. For P-1 and P+1, disables the fast stage 2. Like for
-power, n must be even for P-1.
Use at most n megabytes of memory in stage 2.
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.
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).
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
Print a time stamp whenever a new ECM curve or P+1 or P-1 run is
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.
Use GMP´s mpz_mod function (sub-quadratic for large inputs, but
induces some overhead for small ones).
Use Montgomery´s multiplication (quadratic version). Usually best
method for small input.
Use Montgomery´s multiplication (sub-quadratic version).
Theoretically optimal for large input.
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).
Force use of special base-2 code, input number must divide 2^n+1 if
n > 0, or 2^|n|-1 if n < 0.
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 Woltman´s 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
Take input from file file instead of from standard input.
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
Like -save, but appends to existing files.
Resume residues from file, reads from standard input if file is
"-". Example: to perform step 2 following the above step 1
ecm -resume foo 1e6
Periodically write the current residue in stage 1 to file. In case
of a power failure, etc., the computation can be continued with the
ecm -chkpnt foo -pm1 1e10 < largenumber.txt
The “loop mode” (option -c n) enables one to run several curves on each
input number. The following options control its behavior.
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.
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.
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.
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.
In loop mode, in the second and following runs, output only
expressions that have at most n characters. Default is -ve 0.
In loop mode, increment B1 by n after each curve.
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 optins allow for executing shell commands to supplement
functionality to GMP-ECM.
Execute command cmd to test primality if factors and cofactors
instead of GMP-ECM´s own functions. The number to test is passed
via stdin. An exit code of 0 is interpreted as “probably prime”, a
non-zero exit code as “composite”.
Executes command cmd whenever a factor is found by P-1, P+1 or ECM.
The input number, factor and cofactor are passed via stdin, each on
a line. This could be used i.e. to mail new factors automatically:
ecm -faccmd ´mail -s “$HOSTNAME found a factor”
email@example.com´ 11e6 < cunningham.in
Executes command cmd before each ECM curve, P-1 or P+1 attempt on a
number is started. If the exit status of cmd is non-zero, GMP-ECM
terminates immediately, otherwise it continues normally. GMP-ECM is
stopped while cmd runs, offering a way for letting GMP-ECM sleep
for example while the system is otherwise busy.
Run the program in “nice” mode (below normal priority).
Run the program in “very nice” mode (idle priority).
Multiply the default step 2 bound B2 by the floating-point value f.
Example: -B2scale 0.5 divides the default B2 by 2.
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.
Force cofactor output in decimal (even if expressions are used).
Display a short description of ecm usage, parameters and command
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).
The exit status reflects the result of the last ECM curve or P-1/P+1
attempt the program performed. Individual bits signify particular
0 if normal program termination, 1 if error occured
0 if no proper factor was found, 1 otherwise
0 if factor is composite, 1 if factor is a probable prime
0 if cofactor is composite, 1 if cofactor is a probable prime
Thus, the following exit status values may occur:
Normal program termination, no factor found
Composite factor found, cofactor is composite
Probable prime factor found, cofactor is composite
Input number found
Composite factor found, cofactor is a probable prime
Probable prime factor found, cofactor is a probable prime
Report bugs to <firstname.lastname@example.org>, after checking
<http://www.loria.fr/~zimmerma/records/ecmnet.html> for bug fixes or
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
Alexander Kruppa <(lastname)email@example.com> 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.