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<Chapter Label="ch:General">
<Heading>The General Factorization Routine</Heading>



<Section Label="sec:Factors">
<Heading>The method for <C>Factors</C></Heading>

The &FactInt; package provides a better method for the operation
<C>Factors</C> for integer arguments, which supersedes the one included in
the &GAP; Library:

<ManSection>
 <Meth Name="Factors" Arg="n" Label="FactInt's method, for integers"/>
 <Returns>
 a sorted list of the prime factors of <A>n</A>.
 </Returns>
 <Description>
 <Index Key="primality of the factors">primality of the factors</Index>
 The returned factors pass the built-in probabilistic primality test of
 &GAP; (<C>IsProbablyPrimeInt</C>, Baillie-PSW Primality Test; see the
 &GAP; Reference Manual).
 If the method fails to compute the prime factorization of <A>n</A>,
 an error is signalled.
 The same holds for all other factorization routines provided by this
 package.
 It follows a rough description how the factorization method works: 

 First of all, the method checks whether <M>n = b^k \pm 1</M> for some
 <M>b</M>, <M>k</M> and looks for factors corresponding to polynomial
 factors of <M>x^k \pm 1</M>.
 Provided that <M>b</M> and <M>k</M> are not too large, the factors that
 do not correspond to polynomial factors are taken from Richard P. Brent's
 Factor Tables <Cite Key="Brent04"/>.
 The code for accessing these tables has been contributed by Frank Lübeck.
 

 Then the method uses trial division and a number of cheap methods for
 various common special cases.
 After the small and other <Q>easy</Q> factors have been found this way,
 &FactInt;'s method searches for medium-sized factors using
 Pollard's Rho (by the library function FactorsRho, see the &GAP;
 Reference Manual), Pollard's p-1
 (see <Ref Func="FactorsPminus1" Label="Pollard's p-1"/>),
 Williams' p+1
 (see <Ref Func="FactorsPplus1" Label="Williams' p+1"/>)
 and the Elliptic Curves Method (ECM,
 see <Ref Func="FactorsECM" Label="Elliptic Curves Method, ECM"/>)
 in this order. 

 If there is still an unfactored part remaining after that,
 it is factored using the Multiple Polynomial Quadratic Sieve (MPQS,
 see <Ref Func="FactorsMPQS"
 Label="Multiple Polynomial Quadratic Sieve, MPQS"/>).
 

 The following options are interpreted:

 <List>
 <A>TDHints</A>
 <Item>
 A list of additional trial divisors.
 This is useful only if certain primes <M>p</M> are expected to
 divide <M>n</M> with probability significantly larger than
 <M>\frac{1}{p}</M>.
 </Item>
 <A>RhoSteps</A>
 <Item>
 The number of steps for Pollard's Rho.
 </Item>
 <A>RhoCluster</A>
 <Item>
 The number of steps between two gcd computations in Pollard's Rho.
 </Item>
 <A>Pminus1Limit1</A> / <A>Pminus1Limit2</A>
 <Item>
 The first- / second stage limit for Pollard's p-1
 (see <Ref Func="FactorsPminus1" Label="Pollard's p-1"/>).
 </Item>
 <A>Pplus1Residues</A>
 <Item>
 The number of residues to be tried by Williams' p+1
 (see <Ref Func="FactorsPplus1" Label="Williams' p+1"/>).
 </Item>
 <A>Pplus1Limit1</A> / <A>Pplus1Limit2</A>
 <Item>
 The first- / second stage limit for Williams' p+1
 (see <Ref Func="FactorsPplus1" Label="Williams' p+1"/>).
 </Item>
 <A>ECMCurves</A>
 <Item>
 The number of elliptic curves to be tried by the
 Elliptic Curves Method (ECM)
 (see <Ref Func="FactorsECM"
 Label="Elliptic Curves Method, ECM"/>).
 Also admissible: a function that takes the number <M>n</M> to be
 factored as an argument and returns the desired number
 of curves to be tried.
 </Item>
 <A>ECMLimit1</A> / <A>ECMLimit2</A>
 <Item>
 The initial first- / second stage limit for ECM
 (see <Ref Func="FactorsECM"
 Label="Elliptic Curves Method, ECM"/>).
 </Item>
 <A>ECMDelta</A>
 <Item>
 The increment per curve for the first
 stage limit in ECM. The second stage limit is adjusted appropriately
 (see <Ref Func="FactorsECM"
 Label="Elliptic Curves Method, ECM"/>).
 </Item>
 <A>ECMDeterministic</A>
 <Item>
 If true, ECM chooses its curves deterministically, i.e. repeatable
 (see <Ref Func="FactorsECM"
 Label="Elliptic Curves Method, ECM"/>).
 </Item>
 <A>FBMethod</A>
 <Item>
 Specifies which of the factor base methods should be
 used to do the <Q>hard work</Q>. Currently implemented:
 <C>"CFRAC"</C> and <C>"MPQS"</C>
 (see <Ref Func="FactorsCFRAC"
 Label="Continued Fraction Algorithm, CFRAC"/>
 and <Ref Func="FactorsMPQS"
 Label="Multiple Polynomial Quadratic Sieve, MPQS"/>,
 respectively).
 Default: <C>"MPQS"</C>.
 </Item>
 </List>

 For the use of the &GAP; Options Stack, see Chapter <E>Options Stack</E>
 in the &GAP; Reference Manual. 

 Setting <A>RhoSteps</A>, <A>Pminus1Limit1</A>, <A>Pplus1Residues</A>,
 <A>Pplus1Limit1</A>, <A>ECMCurves</A> or <A>ECMLimit1</A> equal to zero
 switches the respective method off. The method chooses defaults for all
 option values that are not explicitly set by the user. The option values
 are also interpreted by the routines for the particular factorization
 methods described in the next chapter.
<Example>
<![CDATA[
gap> Factors( Factorial(44) + 1 );
[ 694763, 9245226412016162109253, 413852053257739876455072359 ]
gap> Factors( 2^997 - 1 );
[ 167560816514084819488737767976263150405095191554732902607,
 79934306053602222928609369601238840619880168466272137576868879760059\
3002563860297371289151859287894468775962208410650878341385577817736702\
2158878920741413700868182301410439178049533828082651513160945607018874\
830040978453228378816647358334681553 ]
]]>
</Example>
 </Description>
</ManSection>

The above method for <C>Factors</C> calls the following function,
which is the actual <Q>working horse</Q> of this package:

<ManSection>
 <Func Name="FactInt" Arg="n" Label="factorization of an integer"/>
 <Returns>
 a list of two lists, where the first list contains the determined
 prime factors of <A>n</A> and the second list contains the
 remaining unfactored parts of <A>n</A>, if there are any.
 </Returns>
 <Description>
 This function interprets all options which are interpreted by the
 method for <C>Factors</C> described above. In addition, it interprets
 the options <A>cheap</A> and <A>FactIntPartial</A>.
 If the option <A>cheap</A> is set, only usually
 cheap factorization attempts are made.
 If the option <A>FactIntPartial</A> is set, the factorization process
 is stopped before invoking the (usually time-consuming) MPQS or CFRAC,
 if the number of digits of the remaining unfactored part exceeds the
 bound passed as option value <A>MPQSLimit</A> or <A>CFRACLimit</A>,
 respectively. 

 <C>Factors(<A>n</A>)</C> is equivalent to
 <C>FactInt(<A>n</A>:<A>cheap</A>:=false,
 <A>FactIntPartial</A>:=false)[1]</C>.
<Example>
<![CDATA[
gap> FactInt( Factorial(300) + 1 : cheap );
[ [ 461, 259856122109, 995121825812791, 3909669044842609,
 4220826953750952739, 14841043839896940772689086214475144339 ],
 [ 104831288231765723173983836560438594053336296629073932563520618687\
9287645058010688827246061541065631119345674081834085960064144597037243\
9235869682208979384309498719255615067943353399357029226058930732298505\
5816977495398426741656633461747046623641451042655247093315505417820370\
9451745871701742000546384614472756584182478531880962594857275869690727\
9733563594352516014206081210368516157890709802912711149521530885498556\
1244667790208245620301404499928532222524585946881528337257061789593197\
99211283640357942345263781351 ] ]
]]>
</Example>
 </Description>
</ManSection>

</Section>



<Section Label="sec:Info">
<Heading>Getting information about the factoring process</Heading>

<Index Key="information about factoring process">
 information about factoring process
</Index>

Optionally, the &FactInt; package prints information on the progress of
the factorization process:

<ManSection>
 <InfoClass Name="InfoFactInt"
 Label="FactInt's Info class"/>
 <Func Name="FactIntInfo" Arg="level"
 Label="setting the InfoLevel of InfoFactInt"/>
 <Description>
 This Info class allows to monitor what happens during the factoring
 process. 

 If <C>InfoLevel(InfoFactInt) = 1</C>, then basic information
 about the factoring techniques used is displayed. If this InfoLevel has
 value 2, then additionally all <Q>relevant</Q> steps in the
 factoring algorithms are mentioned. If it is set equal to 3, then
 large amounts of details of the progress of the factoring process are
 shown. 

 Enter <C>FactIntInfo(<A>level</A>)</C> to set the
 <C>InfoLevel</C> of <C>InfoFactInt</C> to the positive
 integer <A>level</A>. The call <C>FactIntInfo(<A>level</A>);</C>
 is equivalent to <C>SetInfoLevel(InfoFactInt,<A>level</A>);</C>. 

 The informational output is usually not literally the same in
 each factorization attempt to a given integer with given parameters.
 For a description of the Info mechanism, see Section
 <E>Info Functions</E> in the &GAP; Reference Manual.
 </Description>
</ManSection>

</Section>



</Chapter>

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