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**PROGRAM:**

**NAME**

ecm - integer factorization using ECM, P-1 or P+1

**SYNOPSIS**

**ecm**[

**options**]

__B1__[

__B2min__-

__B2max__|

__B2__]

**DESCRIPTION**

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__

__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__

__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.

__B2min__

**-**

__B2max__

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__.

**FACTORING** **METHOD**

**-pm1**

Perform P-1 instead of the default method (ECM).

**-pp1**

Perform P+1 instead of the default method (ECM).

**GROUP** **AND** **INITIAL** **POINT** **PARAMETERS**

**-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 P+1.

**-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

**STEP** **2** **PARAMETERS**

**-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.

**OUTPUT**

**-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 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.

**-mpzmod**

Use GMP's mpz_mod function (sub-quadratic 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 (sub-quadratic version). Theoretically optimal for

large input.

**-nobase2**

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.

**FILE** **I/O**

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

**LOOP** **MODE**

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.

**-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**

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**.

**-d**

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.

**SHELL** **COMMAND** **EXECUTION**

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”

[email protected]' 11e6 < cunningham.in

**-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.

**MISCELLANEOUS**

**-n**

Run the program in “nice” mode (below normal priority).

**-nn**

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.

**-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.

**INPUT** **SYNTAX**

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)__.

**EXIT** **STATUS**

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

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