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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 algorithms.

B1B1is 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 possibleB1value is 9007199254740996 for P-1, and ULONG_MAX or 9007199254740996 (whichever is smaller) for ECM and P+1. All primes 2 <= p <=B1are processed in step 1.B2B2is the step 2 bound. It is optional: if omitted, a default value is computed fromB1, which should be close to optimal. LikeB1, it can be given either in integer or in floating-point format. The largest possible value ofB2is approximately 9e23, but depends on the number of blockskif you specify the-koption. All primesB1<= p <=B2are processed in step 2. IfB2<B1, no step 2 is performed.B2min-B2maxalternatively one may use theB2min-B2maxform, which means that all primesB2min<= p <=B2maxshould be processed. Thus specifyingB2only corresponds toB1-B2. The values ofB2minandB2maxmay be arbitrarily large, but their difference must not exceed approximately 9e23, subject to the number of blocksk.

-pm1Perform P-1 instead of the default method (ECM).-pp1Perform P+1 instead of the default method (ECM).-tnPerform trial division up ton, before P-1, P+1 or ECM. In loop mode (see option-c), trial division is only performed in the first run.

-x0x[ECM, P-1, P+1] Usex(arbitrary-precision integer or rational) as initial point. For example,-x01/3is valid. If not given,xis generated from the sigma value for ECM, or at random for P-1 and P+1.-sigmas[ECM] Uses(arbitrary-precision integer) as curve generator. If omitted,sis generated at random.-Aa[ECM] Usea(arbitrary-precision integer) as curve parameter. If omitted, is it generated from the sigma value.-goval[ECM, P-1, P+1] Multiply the initial point byval, 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

-kk[ECM, P-1, P+1] Performkblocks in step 2. For a givenB2value, increasingkdecreases the memory usage of step 2, at the expense of more cpu time.-treefilefileStores 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 filesfile.1,file.2 etc. Does not work with fast stage 2 for P+1 and P-1.-powern[ECM, P-1] Use x^nfor Brent-Suyama´s extension (-power1disables 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,nmust be even.-dicksonn[ECM, P-1] Use degree-nDickson´s polynomial for Brent-Suyama´s extension. For P-1 and P+1, disables the fast stage 2. Like for-power,nmust be even for P-1.-maxmemnUse at mostnmegabytes of memory in stage 2.-ntt,-no-nttEnable 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.

-qQuiet 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).-vVerbose mode. More information is printed, more-voptions 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-vprint internal data for debugging.-timestampPrint a time stamp whenever a new ECM curve or P+1 or P-1 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.-mpzmodUse GMP´s mpz_mod function (sub-quadratic for large inputs, but induces some overhead for small ones).-modmulnUse Montgomery´s multiplication (quadratic version). Usually best method for small input.-redcUse Montgomery´s multiplication (sub-quadratic version). Theoretically optimal for large input.-nobase2Disable 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).-base2nForce use of special base-2 code, input number must divide 2^n+1 ifn> 0, or 2^|n|-1 ifn< 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 theB2min-B2maxoption.-inpfileTake input from filefileinstead of from standard input.-savefileSave result of step 1 infile. Iffileexists, an error is raised. Example: to perform only step 1 withB1=1000000 on the composite number in the file "c155" and save its result in file "foo", use ecm -save foo 1e6 1 < c155-saveafileLike-save, but appends to existing files.-resumefileResume residues fromfile, reads from standard input iffileis "-". Example: to perform step 2 following the above step 1 computation, use ecm -resume foo 1e6-chkpointfilePeriodically write the current residue in stage 1 tofile. In case of a power failure, etc., the computation can be continued with the-resumeoption. ecm -chkpnt foo -pm1 1e10 < largenumber.txt

The “loop mode” (option-cn) enables one to run several curves on each input number. The following options control its behavior.-cnPerformnruns on each input number (default is one). This option is mainly useful for P+1 (for example withn=3) or for ECM, wherencould 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-c0produces an infinite loop until a factor is found.-oneIn 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.-bBreadth-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.-dDepth-first processing: in loop mode, runncurves for the first number, thenncurves for the second one and so on. This is the default mode with standard input.-venIn loop mode, in the second and following runs, output only expressions that have at mostncharacters. Default is-ve0.-inIn loop mode, incrementB1bynafter each curve.-InIn loop mode, multiplyB1by a factor depending onnafter each curve. Default is one which should be optimal on one machine, while-I10could 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.-prpcmdcmdExecute commandcmdto 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”.-faccmdcmdExecutes commandcmdwhenever 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” me@myaddress.com´ 11e6 < cunningham.in-idlecmdcmdExecutes commandcmdbefore each ECM curve, P-1 or P+1 attempt on a number is started. If the exit status ofcmdis non-zero, GMP-ECM terminates immediately, otherwise it continues normally. GMP-ECM is stopped whilecmdruns, offering a way for letting GMP-ECM sleep for example while the system is otherwise busy.

-nRun the program in “nice” mode (below normal priority).-nnRun the program in “very nice” mode (idle priority).-B2scalefMultiply the default step 2 boundB2by the floating-point valuef. Example:-B2scale0.5divides the defaultB2by 2.-stage1timenAddnseconds to stage 1 time. This is useful to get correct expected time with-vif part of stage 1 was done in another run.-cofdecForce cofactor output in decimal (even if expressions are used).-h,--helpDisplay 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: example53!. Multi-factorial: example15!3means 15*12*9*6*3. Primorial: example11#means 2*3*5*7*11. Reduced primorial: example17#5means 5*7*11*13*17. Functions: currently, the only available function isPhi(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: 0 Normal program termination, no factor found 1 Error 2 Composite factor found, cofactor is composite 6 Probable prime factor found, cofactor is composite 8 Input number found 10 Composite factor found, cofactor is a probable prime 14 Probable prime factor found, cofactor is a probable prime

Report bugs to <ecm-discuss@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 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)al@loria.fr> 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.