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<?xml version="1.0" encoding="ISO-8859-1"?>
<!DOCTYPE refentry PUBLIC "-//OASIS//DTD DocBook XML V4.1.2//EN"
                   "http://www.oasis-open.org/docbook/xml/4.1.2/docbookx.dtd">
<!-- lifted from troff+man by doclifter -->
<refentry>
<!--                                       Hey, EMACS: \-*\- nroff \-*\- -->
<!--  First parameter, NAME, should be all caps -->
<!--  Second parameter, SECTION, should be 1\-8, maybe w/ subsection -->
<!--  other parameters are allowed: see man(7), man(1) -->
<refmeta>
<refentrytitle>ECM</refentrytitle>
<manvolnum>1</manvolnum>
<refmiscinfo class='source'>April 22, 2003</refmiscinfo>
</refmeta>
<refnamediv id='name'>
<refname>ecm</refname>
<refpurpose>integer factorization using ECM, P-1 or P+1</refpurpose>
</refnamediv>
<!-- body begins here -->
<refsynopsisdiv id='synopsis'>
<cmdsynopsis>
  <command>ecm</command>    
    <arg choice='opt'><option>options</option></arg>
    <arg choice='plain'><replaceable>B1</replaceable></arg>
    <group choice='opt'><arg choice='plain'><replaceable>B2min</replaceable>-<replaceable>B2max</replaceable></arg><arg choice='plain'><replaceable>B2</replaceable></arg></group>
    <sbr/>
</cmdsynopsis>
</refsynopsisdiv>


<refsect1 id='description'><title>DESCRIPTION</title>
<para>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.</para>

</refsect1>

<refsect1 id='bounds'><title>STEP 1 AND STEP 2 BOUND PARAMETERS</title>
<variablelist remap='TP'>
  <varlistentry>
  <term><emphasis remap='B'><replaceable>B1</replaceable></emphasis></term>
  <listitem>
<para><replaceable>B1</replaceable> 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 <replaceable>B1</replaceable> value is
9007199254740996 for P-1, and ULONG_MAX or 9007199254740996 (whichever is
smaller) for ECM and P+1.  All primes 2 &lt;=
p &lt;= <replaceable>B1</replaceable> are processed in step 1.</para>

  </listitem>
  </varlistentry>
  <varlistentry>
  <term><emphasis remap='B'><replaceable>B2</replaceable></emphasis></term>
  <listitem>
<para><replaceable>B2</replaceable> is the step 2 bound. It is optional: if 
omitted, a default value is computed from <replaceable>B1</replaceable>, which 
should be close to optimal. Like <replaceable>B1</replaceable>, it can be given 
either in integer or in floating-point format. The largest possible value of 
<replaceable>B2</replaceable> is approximately 9e23, but depends on the 
number of blocks <replaceable>k</replaceable> if you specify the 
<option>-k</option> option. All primes 
<replaceable>B1</replaceable> &lt;= p &lt;= <replaceable>B2</replaceable> 
are processed in step 2. If <replaceable>B2</replaceable> &lt;
<replaceable>B1</replaceable>, no step 2 is performed.</para>

  </listitem>
  </varlistentry>
  <varlistentry>
  <term><emphasis remap='B'><replaceable>B2min</replaceable>-<replaceable>B2max</replaceable></emphasis></term>
  <listitem>
<para>alternatively one may use the 
<replaceable>B2min</replaceable>-<replaceable>B2max</replaceable> 
form, which means that all primes 
<replaceable>B2min</replaceable> &lt;= p &lt;= <replaceable>B2max</replaceable> 
should be processed. Thus specifying <replaceable>B2</replaceable> only corresponds to 
<replaceable>B1</replaceable>-<replaceable>B2</replaceable>. The values of 
<replaceable>B2min</replaceable> and <replaceable>B2max</replaceable> may be 
arbitrarily large, but their difference must not exceed approximately 9e23, 
subject to the number of blocks <replaceable>k</replaceable>.</para>
  </listitem>
  </varlistentry>
</variablelist>
</refsect1>

<refsect1 id='factoring_method'><title>FACTORING METHOD</title>
<variablelist remap='TP'>
  <varlistentry>
  <term><option>-pm1</option></term>
  <listitem>
<para>Perform P-1 instead of the default method (ECM).</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-pp1</option></term>
  <listitem>
<para>Perform P+1 instead of the default method (ECM).</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='group_and_initial_point_parameters'><title>GROUP AND INITIAL POINT PARAMETERS</title>
<variablelist remap='TP'>
  <varlistentry>
  <term><option>-x0 <replaceable>x</replaceable></option></term>
  <listitem>
<para>[ECM, P-1, P+1] Use <replaceable>x</replaceable> 
(arbitrary-precision integer or rational)
as initial point. For example, <option>-x0 1/3</option> is
valid. If not given, <replaceable>x</replaceable> is generated from the sigma
value for ECM, or at random for P-1 and P+1.</para>

  </listitem>
  </varlistentry>
  <varlistentry>
  <term><option>-sigma <replaceable>s</replaceable></option></term>
  <listitem>
<para>[ECM] Use <replaceable>s</replaceable> (arbitrary-precision integer) as
curve generator. If omitted, <replaceable>s</replaceable> is generated at
random.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-A <replaceable>a</replaceable></option></term>
  <listitem>
<para>[ECM] Use <replaceable>a</replaceable> (arbitrary-precision integer) as
curve parameter. If omitted, is it generated from the sigma value.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-go <replaceable>val</replaceable></option></term>
  <listitem>
<para>[ECM, P-1, P+1] Multiply the initial point by
<replaceable>val</replaceable>, which can any valid expression,
possibly containing the special character N as place holder for the current
input number. Example:
<programlisting>ecm -pp1 -go "N^2-1" 1e6 &lt; composite2000</programlisting>
</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='step_2_parameters'><title>STEP 2 PARAMETERS</title>
<variablelist remap='TP'>
  <varlistentry>
  <term><option>-k <replaceable>k</replaceable></option></term>
  <listitem>
<para>[ECM, P-1, P+1] Perform <replaceable>k</replaceable> blocks in step 2.
For a given <replaceable>B2</replaceable> value, increasing 
<replaceable>k</replaceable> decreases the memory usage of step 2, at the 
expense of more cpu time.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-treefile <replaceable>file</replaceable></option></term>
  <listitem>
<para>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 <replaceable>file</replaceable>.1, <replaceable>file</replaceable>.2 etc.
Does not work with fast stage 2 for P+1 and P-1.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-power <replaceable>n</replaceable></option></term>
  <listitem>
<para>[ECM, P-1]
Use x^<replaceable>n</replaceable> for Brent-Suyama's extension
(<option>-power 1</option> 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, <replaceable>n</replaceable> must be even.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-dickson <replaceable>n</replaceable></option></term>
  <listitem>
<para>[ECM, P-1]
Use degree-<replaceable>n</replaceable> Dickson's polynomial for Brent-Suyama's extension.
For P-1 and P+1, disables the fast stage 2.
Like for <option>-power</option>, <replaceable>n</replaceable> must be 
even for P-1.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-maxmem <replaceable>n</replaceable></option></term>
  <listitem>
<para>Use at most <replaceable>n</replaceable> megabytes of memory in 
      stage 2.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-ntt</option></term>
  <term><option>-no-ntt</option></term>
  <listitem>
<para>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.
</para>
  </listitem>
  </varlistentry>
</variablelist>
</refsect1>

<refsect1 id='output'><title>OUTPUT</title>
<variablelist remap='TP'>
  <varlistentry>
  <term><option>-q</option></term>
  <listitem>
<para>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).
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-v</option></term>
  <listitem>
<para>Verbose mode. More information is printed, more <option>-v</option> 
options increase verbosity. With one <option>-v</option>, 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 <option>-v -v</option>, the A value for ECM 
and residues at the end of step 1 and step 2 are printed. More 
<option>-v</option> print internal data for debugging.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-timestamp</option></term>
  <listitem>
<para>Print a time stamp whenever a new ECM curve or P+1 or P-1 run is 
processed.</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='modular_arithmetic_options'><title>MODULAR ARITHMETIC OPTIONS</title>
<para>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.</para>
<variablelist remap='TP'>
  <varlistentry>
  <term><option>-mpzmod</option></term>
  <listitem>
<para>Use GMP's mpz_mod function (sub-quadratic for large inputs, but induces
some overhead for small ones).</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-modmuln</option></term>
  <listitem>
<para>Use Montgomery's multiplication (quadratic version). Usually
best method for small input.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-redc</option></term>
  <listitem>
<para>Use Montgomery's multiplication (sub-quadratic version).
Theoretically optimal for large input.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-nobase2</option></term>
  <listitem>
<para>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 <option>-v</option>).</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-base2</option> <replaceable>n</replaceable></term>
  <listitem>
<para>Force use of special base-2 code, input number must divide 
2^<replaceable>n</replaceable>+1 if <replaceable>n</replaceable> &gt; 0, 
or 2^|<replaceable>n</replaceable>|-1 if <replaceable>n</replaceable> &lt; 0.
</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='file_io'><title>FILE I/O</title>
<para>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 <replaceable>B2min-B2max</replaceable> option.</para>
<variablelist remap='TP'>

  <varlistentry>
  <term><option>-inp <replaceable>file</replaceable></option></term>
  <listitem>
<para>Take input from file <replaceable>file</replaceable> instead of from
standard input.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-save <replaceable>file</replaceable></option></term>
  <listitem>
<para>Save result of step 1 in <replaceable>file</replaceable>. If 
<replaceable>file</replaceable> exists, an error is raised.
Example: to perform only step 1 with <replaceable>B1</replaceable>=1000000
on the composite number in the file "c155" and save its result in file 
"foo", use 
<programlisting>ecm -save foo 1e6 1 &lt; c155</programlisting>
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-savea <replaceable>file</replaceable></option></term>
  <listitem>
<para>Like <option>-save</option>, but appends to existing files.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-resume <replaceable>file</replaceable></option></term>
  <listitem>
<para>Resume residues from <replaceable>file</replaceable>, reads from
standard input if <replaceable>file</replaceable> is  "-".
Example: to perform step 2 following the above step 1 computation, use
<programlisting>ecm -resume foo 1e6</programlisting>
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-chkpoint <replaceable>file</replaceable></option></term>
  <listitem>
<para>Periodically write the current residue in stage 1 to 
<replaceable>file</replaceable>. In case of a power failure, etc., the
computation can be continued with the <option>-resume</option> option.
<programlisting>ecm -chkpnt foo -pm1 1e10 &lt; largenumber.txt 
</programlisting>
</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='loop_mode'><title>LOOP MODE</title>
<para>The <quote>loop mode</quote> (option <option>-c 
<replaceable>n</replaceable></option>) enables one to run several curves
on each input number. The following options control its behavior.
</para>
<variablelist remap='TP'>

  <varlistentry>
  <term><option>-c <replaceable>n</replaceable></option></term>
  <listitem>
<para>Perform <replaceable>n</replaceable> runs on each input number
(default is one).
This option is mainly useful for P+1 (for example with
<replaceable>n</replaceable>=3) or for ECM, where 
<replaceable>n</replaceable> 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 <option>-resume, -sigma, 
-x0</option>. Giving <option>-c 0</option> produces an infinite loop until a 
factor is found.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-one</option></term>
  <listitem>
<para>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.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-b</option></term>
  <listitem>
<para>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 <option>-inp</option>.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-d</option></term>
  <listitem>
<para>Depth-first processing: in loop mode, run <replaceable>n</replaceable>
curves for the first number, then <replaceable>n</replaceable> curves for the
second one and so on.
This is the default mode with standard input.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-ve <replaceable>n</replaceable></option></term>
  <listitem>
<para>In loop mode, in the second and following runs,
output only expressions that have at most <replaceable>n</replaceable>
characters. Default is <option>-ve 0</option>.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-i <replaceable>n</replaceable></option></term>
  <listitem>
<para>In loop mode, increment <replaceable>B1</replaceable>
by <replaceable>n</replaceable> after each curve.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-I <replaceable>n</replaceable></option></term>
  <listitem>
<para>In loop mode, multiply <replaceable>B1</replaceable>
by a factor depending on <replaceable>n</replaceable> after each curve.
Default is one which should be optimal on one machine, while  
<option>-I 10</option> could be used when trying to factor the same number 
simultaneously on 10 identical machines.
</para>
  </listitem>
  </varlistentry>

</variablelist>
</refsect1>

<refsect1 id='shellcmd'><title>SHELL COMMAND EXECUTION</title>
<para>These optins allow for executing shell commands to supplement 
functionality to GMP-ECM.</para>
<variablelist remap='TP'>

  <varlistentry>
  <term><option>-prpcmd <replaceable>cmd</replaceable></option></term>
  <listitem>
    <para>
      Execute command <replaceable>cmd</replaceable> 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 <quote>probably prime</quote>, a non-zero exit code as 
      <quote>composite</quote>.
    </para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-faccmd <replaceable>cmd</replaceable></option></term>
  <listitem>
    <para>
      Executes command <replaceable>cmd</replaceable> 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:
<programlisting>ecm -faccmd 'mail -s <quote>$HOSTNAME found a factor</quote>
                me@myaddress.com' 11e6 &lt; cunningham.in
</programlisting>
    </para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-idlecmd <replaceable>cmd</replaceable></option></term>
  <listitem>
    <para>
      Executes command <replaceable>cmd</replaceable> before each ECM curve,
      P-1 or P+1 attempt on a number is started. If the exit status of 
      <replaceable>cmd</replaceable> is non-zero, GMP-ECM terminates 
      immediately, otherwise it continues normally. GMP-ECM is stopped while
      <replaceable>cmd</replaceable> runs, offering a way for letting GMP-ECM
      sleep for example while the system is otherwise busy.
    </para>
  </listitem>
  </varlistentry>
</variablelist>
</refsect1>

<refsect1 id='miscellaneous'><title>MISCELLANEOUS</title>
<variablelist remap='TP'>

  <varlistentry>
  <term><option>-n</option></term>
  <listitem>
  <para>Run the program in <quote>nice</quote> mode (below normal priority).
  </para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-nn</option></term>
  <listitem>
  <para>Run the program in <quote>very nice</quote> mode (idle priority).</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-B2scale <replaceable>f</replaceable></option></term>
  <listitem>
<para>Multiply the default step 2 bound <replaceable>B2</replaceable>
by the floating-point value <replaceable>f</replaceable>.
Example: <option>-B2scale 0.5</option>
divides the default <replaceable>B2</replaceable> by 2.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-stage1time <replaceable>n</replaceable></option></term>
  <listitem>
<para>Add <replaceable>n</replaceable> seconds to stage 1 time.
This is useful to get correct expected time with
<replaceable>-v</replaceable> if part of stage 1 was done in another run.
</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-cofdec</option></term>
  <listitem>
  <para>Force cofactor output in decimal (even if expressions are used).
  </para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-h</option>, <option>--help</option></term>
  <listitem>
  <para>Display a short description of ecm usage, parameters and command line
  options.</para>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term><option>-printconfig</option></term>
  <listitem>
  <para>Prints configuration parameters used for the compilation 
  and exits.</para>
  </listitem>
  </varlistentry>

</variablelist>

<!--  TeX users may be more comfortable with the \fB&lt;whatever&gt;\fP and -->
<!--  \fI&lt;whatever&gt;\fP escape sequences to invode bold face and italics,  -->
<!--  respectively. -->
</refsect1>

<refsect1 id='syntax'><title>INPUT SYNTAX</title>
<para>The input numbers can have several forms:</para>
<para>Raw decimal numbers like 123456789.</para>
<para>Comments can be placed in the file: everything after <quote>//</quote>
is ignored, up to the end of line.</para>
<para>Line continuation. If a line ends with a backslash character
    <quote>\</quote>, it is considered to continue on the next line.</para>
<para>Common arithmetic expressions can be used. Example:
     <replaceable>3*5+2^10</replaceable>.</para>
<para>Factorial: example <replaceable>53!</replaceable>.</para>
<para>Multi-factorial: example <replaceable>15!3</replaceable>
means 15*12*9*6*3.</para>
<para>Primorial: example <replaceable>11#</replaceable> means
2*3*5*7*11.</para>
<para>Reduced primorial: example <replaceable>17#5</replaceable> means
5*7*11*13*17.</para>
<para>Functions: currently, the only available function is 
<replaceable>Phi(x,n)</replaceable>.</para>
</refsect1>

<refsect1 id='exitstatus'><title>EXIT STATUS</title>
<para>
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:
</para>
<variablelist remap='TP'>
  <varlistentry>
  <term>Bit 0</term>
  <listitem>
    <simpara>0 if normal program termination, 1 if error occurred</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>Bit 1</term>
  <listitem>
    <simpara>0 if no proper factor was found, 1 otherwise</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>Bit 2</term>
  <listitem>
    <simpara>0 if factor is composite, 1 if factor is a probable prime</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>Bit 3</term>
  <listitem>
    <simpara>0 if cofactor is composite, 1 if cofactor is a probable prime</simpara>
  </listitem>
  </varlistentry>
</variablelist>

<para>Thus, the following exit status values may occur:</para>

<variablelist>
  <varlistentry>
    <term>0</term>
    <listitem>
      <simpara>Normal program termination, no factor found</simpara>
    </listitem>
  </varlistentry>

  <varlistentry>
    <term>1</term>
    <listitem>
      <simpara>Error</simpara>
    </listitem>
  </varlistentry>

  <varlistentry>
  <term>2</term>
  <listitem>
    <simpara>Composite factor found, cofactor is composite</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>6</term>
  <listitem>
    <simpara>Probable prime factor found, cofactor is composite</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>8</term>
  <listitem>
    <simpara>Input number found</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>10</term>
  <listitem>
    <simpara>Composite factor found, cofactor is a probable prime</simpara>
  </listitem>
  </varlistentry>

  <varlistentry>
  <term>14</term>
  <listitem>
    <simpara>Probable prime factor found, cofactor is a probable prime</simpara>
  </listitem>
  </varlistentry>
</variablelist>

</refsect1>

<refsect1 id='bugs'><title>BUGS</title>
<para>
Report bugs to &lt;ecm-discuss@lists.gforge.inria.fr&gt;, after checking
&lt;http://www.loria.fr/~zimmerma/records/ecmnet.html&gt; for bug fixes
or new versions.
</para>
</refsect1>

<refsect1 id='author'><title>AUTHORS</title>
<para>Pierrick Gaudry &lt;gaudry at lix dot polytechnique dot fr&gt;
contributed efficient assembly code for combined mul/redc;</para>
<para>Jim Fougeron &lt;jfoug at cox dot net&gt; contributed the expression
parser and several command-line options;</para>
<para>Laurent Fousse &lt;laurent at komite dot net&gt; contributed the middle 
product code, the autoconf/automake tools, and is the maintainer of the 
Debian package;</para>
<para>Alexander Kruppa &lt;(lastname)al@loria.fr&gt; 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;</para>
<para>Dave Newman &lt;david.(lastname)@jesus.ox.ac.uk&gt;
contributed the Kronecker-Schoenhage and NTT multiplication code;</para>
<para>Jason S. Papadopoulos contributed a speedup of the NTT code</para>
<para>Paul Zimmermann &lt;zimmerma at loria dot fr&gt; is the author of the
first version of the program and chief maintainer of GMP-ECM.</para>
<para>Note: email addresses have been obscured, the required substitutions
should be obvious.
</para>
</refsect1>
</refentry>