Man Linux: Main Page and Category List


       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.


           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).

       -t n
           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


       -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

       -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


       -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-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.

       -dickson n
           [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.

       -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.


           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


       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).

       -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.


       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 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.

       -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.

           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.

       -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 GMP-ECM.

       -prpcmd cmd
           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”.

       -faccmd cmd
           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”
                     ´ 11e6 <

       -idlecmd cmd
           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).

       -B2scale f
           Multiply the default step 2 bound B2 by the floating-point 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.

           Force cofactor output in decimal (even if expressions are used).

       -h, --help
           Display a short description of ecm usage, parameters and command
           line options.


       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
       events, specifically:

       Bit 0
           0 if normal program termination, 1 if error occured

       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:

           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 <>, after checking
       <> 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 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)> 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)> 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.