GAP (FactInt) - Chapter 2: The General Factorization Routine

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2 The General Factorization Routine
2.1 The method for `Factors`
2.1-1 Factors
2.1-2 FactInt
2.2 Getting information about the factoring process
2.2-1 InfoFactInt

### 2 The General Factorization Routine

#### 2.1 The method for `Factors`

The FactInt package provides a better method for the operation `Factors` for integer arguments, which supersedes the one included in the GAP Library:

##### 2.1-1 Factors

`‣ Factors`( n ) | ( method )

Returns: a sorted list of the prime factors of n.

The returned factors pass the built-in probabilistic primality test of GAP (`IsProbablyPrimeInt`, Baillie-PSW Primality Test; see the GAP Reference Manual). If the method fails to compute the prime factorization of n, an error is signalled. The same holds for all other factorization routines provided by this package. The following is a rough description of how the factorization method works:

First of all, the method checks whether \(n = b^k \pm 1\) for some \(b\), \(k\) and looks for factors corresponding to polynomial factors of \(x^k \pm 1\). Provided that \(b\) and \(k\) are not too large, the factors that do not correspond to polynomial factors are taken from Richard P. Brent's Factor Tables [Bre04]. The code for accessing these tables has been contributed by Frank Lübeck.

These tables are nowadays maintained by Jonathan Crombie, and have grown far too large to be distributed in full. The subset shipped with FactInt is determined as follows: for a base \(b\) and an exponent \(k\), all currently known factors of \(b^k \pm 1\) are included if \(b \leq 100\), or if \(b\) is a prime less than 1000, or if the pair \((b,k)\) was already covered by the tables distributed with FactInt 1.6.3. In particular, everything that was available in earlier versions of FactInt is still available, in many cases with additional factors.

Users who need factors beyond this range can replace the distributed tables by the full collection using the undocumented function `FetchBrentFactors`; see the comments in `lib/general.gi`. This requires write access to the FactInt installation directory as well as the external programs `curl` and `gzip`.

Then the method uses trial division and a number of cheap methods for various common special cases. After the small and other easy 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 `FactorsPminus1` (3.2-1)), Williams' \(p+1\) (see `FactorsPplus1` (3.3-1)) and the Elliptic Curves Method (ECM, see `FactorsECM` (3.4-1)) in this order.

If there is still an unfactored part remaining after that, it is factored using the Multiple Polynomial Quadratic Sieve (MPQS, see `FactorsMPQS` (3.6-1)).

The following options are interpreted:

TDHints

  A list of additional trial divisors. This is useful only if certain primes \(p\) are expected to divide \(n\) with probability significantly larger than \(\frac{1}{p}\).

RhoSteps

  The number of steps for Pollard's Rho.

RhoCluster

  The number of steps between two gcd computations in Pollard's Rho.

Pminus1Limit1 / Pminus1Limit2

  The first- / second stage limit for Pollard's \(p-1\) (see `FactorsPminus1` (3.2-1)).

Pplus1Residues

  The number of residues to be tried by Williams' \(p+1\) (see `FactorsPplus1` (3.3-1)).

Pplus1Limit1 / Pplus1Limit2

  The first- / second stage limit for Williams' \(p+1\) (see `FactorsPplus1` (3.3-1)).

ECMCurves

  The number of elliptic curves to be tried by the Elliptic Curves Method (ECM) (see `FactorsECM` (3.4-1)). Also admissible: a function that takes the number \(n\) to be factored as an argument and returns the desired number of curves to be tried.

ECMLimit1 / ECMLimit2

  The initial first- / second stage limit for ECM (see `FactorsECM` (3.4-1)).

ECMDelta

  The increment per curve for the first stage limit in ECM. The second stage limit is adjusted appropriately (see `FactorsECM` (3.4-1)).

ECMDeterministic

  If true, ECM chooses its curves deterministically, i.e. repeatable (see `FactorsECM` (3.4-1)).

FBMethod

  Specifies which of the factor base methods should be used to do the hard work. Currently implemented: `"CFRAC"` and `"MPQS"` (see `FactorsCFRAC` (3.5-1) and `FactorsMPQS` (3.6-1), respectively). Default: `"MPQS"`.

For the use of the GAP Options Stack, see Chapter Options Stack in the GAP Reference Manual.

Setting RhoSteps, Pminus1Limit1, Pplus1Residues, Pplus1Limit1, ECMCurves or ECMLimit1 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.

    gap> Factors( Factorial(44) + 1 );
    [ 694763, 9245226412016162109253, 413852053257739876455072359 ]
    gap> Factors( 2^997 - 1 );
    [ 167560816514084819488737767976263150405095191554732902607,
      79934306053602222928609369601238840619880168466272137576868879760059\
    3002563860297371289151859287894468775962208410650878341385577817736702\
    2158878920741413700868182301410439178049533828082651513160945607018874\
    830040978453228378816647358334681553 ]

The above method for `Factors` calls the following function, which is the actual workhorse of this package:

##### 2.1-2 FactInt

`‣ FactInt`( n ) | ( function )

Returns: a list of two lists, where the first list contains the determined prime factors of n and the second list contains the remaining unfactored parts of n, if there are any.

This function interprets all options which are interpreted by the method for `Factors` described above. In addition, it interprets the options cheap and FactIntPartial. If the option cheap is set, only usually cheap factorization attempts are made. If the option FactIntPartial 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 MPQSLimit or CFRACLimit, respectively.

`Factors(n)` is equivalent to `FactInt(n:cheap:=false, FactIntPartial:=false)[1]`.

    gap> FactInt( Factorial(300) + 1 : cheap );
    [ [ 461, 259856122109, 995121825812791, 3909669044842609,
          4220826953750952739, 14841043839896940772689086214475144339 ],
      [ 104831288231765723173983836560438594053336296629073932563520618687\
    9287645058010688827246061541065631119345674081834085960064144597037243\
    9235869682208979384309498719255615067943353399357029226058930732298505\
    5816977495398426741656633461747046623641451042655247093315505417820370\
    9451745871701742000546384614472756584182478531880962594857275869690727\
    9733563594352516014206081210368516157890709802912711149521530885498556\
    1244667790208245620301404499928532222524585946881528337257061789593197\
    99211283640357942345263781351 ] ]

#### 2.2 Getting information about the factoring process

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

##### 2.2-1 InfoFactInt

`‣ InfoFactInt` | ( info class )

`‣ FactIntInfo`( level ) | ( function )

This Info class makes it possible to monitor what happens during the factoring process.

If `InfoLevel(InfoFactInt) = 1`, then basic information about the factoring techniques used is displayed. If this InfoLevel has value 2, then additionally all relevant 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 `FactIntInfo(level)` to set the `InfoLevel` of `InfoFactInt` to the positive integer level. The call `FactIntInfo(level);` is equivalent to `SetInfoLevel(InfoFactInt,level);`.

The informational output is usually not literally the same in each factorization attempt for a given integer with given parameters. For a description of the Info mechanism, see Section Info Functions in the GAP Reference Manual.

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