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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
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 performed.
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
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 debugging.
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
Use Montgomery's multiplication (sub-quadratic version). Theoretically optimal for
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 B2min-B2max
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 computation, use
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 -resume option.
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
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 line options.
Prints configuration parameters used for the compilation and exits.
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 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 events, specifically:
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
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