

ecm [options] B1 [B2minB2max  B2]  
 
ecm is an integer factoring program using the Elliptic Curve Method (ECM), the P1 method, or the P+1 method. The following sections describe parameters relevant to these algorithms.
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 floatingpoint format (3000000.0 or 3e6). The largest possible B1 value is 9007199254740996 for P1, 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 floatingpoint 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.B2minB2max
alternatively one may use the B2minB2max form, which means that all primes B2min <= p <= B2max should be processed. Thus specifying B2 only corresponds to B1B2. 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.
pm1
Perform P1 instead of the default method (ECM).pp1
Perform P+1 instead of the default method (ECM).t n
Perform trial division up to n, before P1, P+1 or ECM. In loop mode (see option c), trial division is only performed in the first run.
x0 x
[ECM, P1, P+1] Use x (arbitraryprecision 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 P1 and P+1.sigma s
[ECM] Use s (arbitraryprecision integer) as curve generator. If omitted, s is generated at random.A a
[ECM] Use a (arbitraryprecision integer) as curve parameter. If omitted, is it generated from the sigma value.go val
[ECM, P1, 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^21" 1e6 < composite2000
k k
[ECM, P1, 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 P1.power n
[ECM, P1] Use x^n for BrentSuyama's extension (power 1 disables BrentSuyama's extension). The default polynomial is chosen depending on the method and B2. For P1 and P+1, disables the fast stage 2. For P1, n must be even.dickson n
[ECM, P1] Use degreen Dickson's polynomial for BrentSuyama's extension. For P1 and P+1, disables the fast stage 2. Like for power, n must be even for P1.maxmem n
Use at most n megabytes of memory in stage 2.ntt, nontt
Enable or disable the NumberTheoretic 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 nontt variant uses more memory, but is faster than NTT with large input numbers. By default, NTT is used for P1, P+1 and for ECM on numbers of size at most 30 machine words.
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 P1 run is processed.
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 GMP's mpz_mod function (subquadratic for large inputs, but induces some overhead for small ones).modmuln
Use Montgomery's multiplication (quadratic version). Usually best method for small input.redc
Use Montgomery's multiplication (subquadratic version). Theoretically optimal for large input.nobase2
Disable special base2 code (which is used when the input number is a large factor of 2^n+1 or 2^n1, see v).base2 n
Force use of special base2 code, input number must divide 2^n+1 if n > 0, or 2^n1 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 GMPECM). It can also be useful to split step 2 into several runs, using the B2minB2max 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
The loop mode
(option c n) enables one to run several curves on each input number. The following options control its behavior. \( rqc 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 ddigit 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
Breadthfirst 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
Depthfirst 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.ve n
In loop mode, in the second and following runs, output only expressions that have at most n characters. Default is ve 0.i n
In loop mode, increment B1 by n after each curve.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.
These optins allow for executing shell commands to supplement functionality to GMPECM.
prpcmd cmd
Execute command cmd to test primality if factors and cofactors instead of GMPECM's own functions. The number to test is passed via stdin. An exit code of 0 is interpreted asprobably prime \( lq, a nonzero exit code as \( rqcomposite \( lq. \( rqfaccmd cmd
Executes command cmd whenever a factor is found by P1, 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 \( lqme@myaddress.com' 11e6 < cunningham.in \( rq
idlecmd cmd
Executes command cmd before each ECM curve, P1 or P+1 attempt on a number is started. If the exit status of cmd is nonzero, GMPECM terminates immediately, otherwise it continues normally. GMPECM is stopped while cmd runs, offering a way for letting GMPECM sleep for example while the system is otherwise busy.
n
Run the program innice \( lqmode (below normal priority). \( rqnn
Run the program invery nice \( lqmode (idle priority). \( rqB2scale f
Multiply the default step 2 bound B2 by the floatingpoint value f. Example: B2scale 0.5 divides the default B2 by 2.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.cofdec
Force cofactor output in decimal (even if expressions are used).h, help
Display a short description of ecm usage, parameters and command line options.printconfig
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
// \( lqis ignored, up to the end of line. \( rqLine continuation. If a line ends with a backslash character
\ \( lq, it is considered to continue on the next line. \( rqCommon arithmetic expressions can be used. Example: 3*5+2^10.
Factorial: example 53!.
Multifactorial: 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 P1/P+1 attempt the program performed. Individual bits signify particular events, specifically:
Bit 0
0 if normal program termination, 1 if error occuredBit 1
0 if no proper factor was found, 1 otherwiseBit 2
0 if factor is composite, 1 if factor is a probable primeBit 3
0 if cofactor is composite, 1 if cofactor is a probable primeThus, the following exit status values may occur:
0
Normal program termination, no factor found1
Error2
Composite factor found, cofactor is composite6
Probable prime factor found, cofactor is composite8
Input number found10
Composite factor found, cofactor is a probable prime14
Probable prime factor found, cofactor is a probable prime
Report bugs to <ecmdiscuss@lists.gforge.inria.fr>, after checking <http://www.loria.fr/~zimmerma/records/ecmnet.html> for bug fixes or new versions.
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 commandline 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)al@loria.fr> contributed estimates for probability of success for ECM, the new P+1 and P1 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 KroneckerSchoenhage 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 GMPECM.
Note: email addresses have been obscured, the required substitutions should be obvious.
April 22, 2003  ECM (1)  03/17/2010 
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