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).
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 P+1.
[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
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 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.
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 <
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
Save result of step 1 in file
. If file
exists, an error is raised. Example: to perform only step 1 with
=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 computation, use
Periodically write the current residue in stage 1 to
. In case of a power failure, etc., the computation can be
continued with the -resume
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
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, 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.
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
Display a short description of ecm usage, parameters and
command line options.
Prints configuration parameters used for the compilation
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:
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
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
0 if no proper factor was found, 1 otherwise
0 if factor is composite, 1 if factor is a probable
0 if cofactor is composite, 1 if cofactor is a probable
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
Probable prime factor found, cofactor is a probable
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)firstname.lastname@example.org> 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
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.