[FactInt] 3 The Routines for Specific Factorization Methods

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# 3 The Routines for Specific Factorization Methods

### Sections

1. Pollard's p-1
2. Williams' p+1
3. The Elliptic Curves Method (ECM)
4. The Continued Fraction Algorithm (CFRAC)
5. The Multiple Polynomial Quadratic Sieve (MPQS)

Descriptions of the factoring methods used here can be found in Bressoud89 and in Cohen93. The last book contains also descriptions of the other methods mentioned in the preface.

## 3.1 Pollard's p-1

- `FactorsPminus1(` n`[, [` a`, ]` Limit1`[,` Limit2`] ] ) F`

  `FactorsPminus1` tries to factor n using Pollard's p-1, with a as base for exponentiation (the default is a=2), Limit1 as first stage limit and Limit2 as second stage limit. `FactorsPminus1` chooses defaults for all arguments not explicitly given. If `FactorsPminus1` is called with three arguments, these arguments are regarded as n, Limit1 and Limit2. The result is returned as a list of two lists, where the first one contains the prime factors found, and the second one contains remaining unfactored parts of n, if there are any.

  Pollard's p-1 is based on the fact that exponentiation in the group of invertible residue classes (mod n) is so fast that it is possible to compute for example ak! for k large enough (e.g. 100000 or so) in a reasonable amount of time and without using much memory, and on Lagrange's Theorem, which states that ak! is congruent to 1 (mod p) (where p is a prime divisor of n) if k! is a multiple of p-1, and (a,p)=1. In this situation, if this is satisfied for no other prime factor of n, p can be determined by calculating Gcd(ak!-1,n). A prime divisor p is usually found if the largest prime factor of p-1 is not larger than Limit2, and the second-largest one is not larger than Limit1. (Compare with FactorsPplus1 and FactorsECM.)

      gap> FactorsPminus1( Factorial(158) + 1, 100000, 1000000 );
      [ [ 2879, 5227, 1452486383317, 9561906969931, 18331561438319,
            483714299709483760811581110341732950506493218122654853400674921345082310\
      906370452295654816571305041217323052879842924826121333143254713674832962773107\
      806789945715570386038565256719614524924705165110048148716160964980629081176057\
      0095669 ], [  ] ]

      # Let's see why this works:

      gap> List( last[ 1 ]{[ 3, 4, 5 ]}, p -> IntegerFactorization( p - 1 ) );
      [ [ 2, 2, 3, 3, 81937, 492413 ], [ 2, 3, 3, 3, 5, 7, 7, 1481, 488011 ],
        [ 2, 3001, 7643, 399613 ] ]

  ## 3.2 Williams' p+1

- `FactorsPplus1(` n`[, [` Residues`, ]` Limit1`[,` Limit2`] ] ) F`

  `FactorsPplus1` tries to factor n using a variant of Williams' p+1, where it tries Residues different residues (the probability of a given residue to be in principle usable is about 1/2) and uses Limit1 as first stage limit and Limit2 as second stage limit. `FactorsPplus1` chooses defaults for all arguments not explicitly given. If `FactorsPplus1` is called with three arguments, these arguments are regarded as n, Limit1 and Limit2. The result is returned as a list of two lists, where the first one contains the prime factors found, and the second one contains remaining unfactored parts of n, if there are any.

  Williams' p+1 is very similar to Pollard's p-1 (see FactorsPminus1), the only difference is that the group used here has order p+1 (instead of p-1), and that the group operation takes more time. A prime divisor p is usually found if the largest prime factor of p+1 is at most Limit2 and the second-largest one is not larger than Limit1, and if the algorithm hits a ``usable'' residue (concerning the ``unusable'' residues, it could be stated that the order of the respective group is p-1, such that the method turns into a slow p-1-algorithm in this case). (Compare also with FactorsECM.)

      gap> FactorsPplus1( Factorial(55) - 1, 10, 10000, 100000 );
      [ [ 73, 39619, 277914269, 148257413069 ],
        [ 106543529120049954955085076634537262459718863957 ] ]
      gap> List( last[ 1 ], p -> [ Factors( p - 1 ), Factors( p + 1 ) ] );
      [ [ [ 2, 2, 2, 3, 3 ], [ 2, 37 ] ],
        [ [ 2, 3, 3, 31, 71 ], [ 2, 2, 5, 7, 283 ] ],
        [ [ 2, 2, 2207, 31481 ], [ 2, 3, 5, 9263809 ] ],
        [ [ 2, 2, 47, 788603261 ], [ 2, 3, 5, 13, 37, 67, 89, 1723 ] ] ]

  ## 3.3 The Elliptic Curves Method (ECM)

- `FactorsECM(` n`[,` Curves`[,` Limit1`[,` Limit2`[,` Delta`] ] ] ] ) F`

  `FactorsECM` tries to factor n using the Elliptic Curves Method (ECM), where Curves specifies the number of curves to be tried, Limit1 is the initial first stage limit, Limit2 is the initial second stage limit and Delta is the increment per curve for the first stage limit, where the second stage limit is adjusted appropriately. `FactorsECM` chooses defaults for all arguments not explicitly given. The option ECMDeterministic specifies, if set, that the choice of the curves to be tried should be deterministic, i.e. that repeated calls of `FactorsECM` yield the same curves, and hence for the same n the result after the same number of trials (this is of use mainly for testing purposes). The result is returned as a list of two lists, where the first one contains the prime factors found, and the second one contains remaining unfactored parts of n, if there are any.

  The Elliptic Curves Method is based on the fact that exponentiation in the elliptic curve groups E(a,b)/n is fast enough that it is possible to compute for example gk! for k large enough (e.g. 10000 or so) in a reasonable amount of time and without using much memory, and on Lagrange's Theorem, which states that for each elliptic curve point g, gk! is the identity element of E(a,b)/p (where p is a prime divisor of n) if k! is a multiple of |E(a,b)/p|. In this situation, under reasonable circumstances, p can be determined by taking an appropriate Gcd.

  In practice, the algorithm chooses in some sense ``better'' products Pk of small primes rather than k! as exponents, and, after reaching the first stage limit with PLimit1, it considers further products PLimit1q for primes q up to the second stage limit Limit2 (which is usually set equal to, for example, 40 times the first stage limit, or so). The prime q corresponds to the largest prime factor of the order of the group under consideration.

  A prime divisor p is usually found if the largest prime factor of the order of one of the examined elliptic curve groups E(a,b)/p is at most Limit2, and the second-largest one is at most Limit1, so trying a larger number of curves increases the chance of factoring n as well as taking a larger value for Limit1 and/or Limit2 (it turns out to be not optimal either to take a large number of curves with very small Limit1 and Limit2 or to grind on with a single curve and very large limits). (Compare with FactorsPminus1.)

  The elements of the group E(a,b)/n are the points (x,y) given by the solutions of y2 = x3 + ax + b in the residue class ring (mod n), and an additional point infty at infinity, which serves as identity element. To turn this set into a group, define the product (although elliptic curve groups are usually written additively, I prefer using the multiplicative notation here to retain the analogy to FactorsPminus1 and FactorsPplus1) of two points p1 and p2 as follows: Let l be the line through p1 and p2 if p1 neqp2, otherwise let l be the tangent to the curve C given by the above equation in the point p1 = p2. l intersects C in a third point, say p3 (if l does not intersect the curve in a third affine point, then set p3 := infty). Set p1.p2 equal to the image of p3 under the reflection across the x-axis. Define the product of any curve point p and infty by p itself. This (more or less obviously, checking associativity requires some calculation) turns the set of points on the given curve into an abelian group E(a,b)/n.

  However, the calculations are done in projective coordinates to have an explicit representation of the identity element and to avoid calculating inverses (mod n) for the group operation, which would otherwise require using an O((log n)3)-algorithm, while multiplication (mod n) is only O((log n)2). The respective equation in this case is given by bY2Z = X3 + aX2Z + XZ2 (this form allows more efficient calculations than the Weierstrass model Y2Z = X3 + aXZ2 + bZ3, which is the projective equivalent to the affine representation y2 = x3 + ax + b mentioned above). The algorithm only keeps track of two of the three coordinates, namely X and Z. The choice of curves is done in a way that ensures the order of the respective group to be divisible by 12. This increases the chance that it is smooth enough to find a factor of n. The implementation follows the description of R. P. Brent given in Brent96, pp. 5 -- 8 (in terms of this paper, for the second stage the ``improved standard continuation'' is used).

      gap> FactorsECM(2^256+1,100,10000,1000000,100);
      [ [ 1238926361552897,
            93461639715357977769163558199606896584051237541638188580280321 ], [  ] ]

  ## 3.4 The Continued Fraction Algorithm (CFRAC)

- `FactorsCFRAC(` n`) F`

  `FactorsCFRAC` tries to factor n using the Continued Fraction Algorithm (CFRAC), also known as Brillhart-Morrison Algorithm. The result is returned as a list of the prime factors of n. In case of failure an error is signalled.

  To CFRAC, the same warning applies as to the Quadratic Sieve (see FactorsMPQS).

  The Continued Fraction Algorithm tries to find integers x, y, such that x2 equivy2 (mod n), but not pmx equivpmy (mod n). In this situation, taking Gcd(x - y,n) yields a non-trivial factor of n. For determining such a pair (x,y), the algorithm uses the continued fraction expansion of the square root of n (because n is usually very large, it is impossible to compute the whole period of it, but a much smaller number of terms is enough in this case). If xi/yi is a continued fraction approximation for the square root of n, then ci := xi2 - nyi2 has size only about a small constant times the square root of n. The algorithm tries to find as many ci as possible which factor completely over a chosen factor base (a list of small primes) or with only one factor not in the factor base. The latter ones are usable only if a second ci with the same ``large factor'' is found Then, Gaussian Elimination over GF(2) is used to determine which of the congruences xi2 equivci (mod n) have to be multiplied together to get a congruence of the desired form x2 equivy2 (mod n), where the involved matrix M is given by Mij = 1, if an odd power of the j-th element of the factor base divides the i-th usable factored value, and Mij = 0 otherwise. For this purpose it is necessary that the number of factored ci is larger than the rank of M, which is approximately given by the size of the factor base. (Compare with FactorsMPQS.)

      gap> FactorsCFRAC( Factorial(34) - 1 );
      [ 10398560889846739639, 28391697867333973241 ]

  ## 3.5 The Multiple Polynomial Quadratic Sieve (MPQS)

- `FactorsMPQS(` n`) F`

  `FactorsMPQS` tries to factor n using the Single Large Prime Variation of the Multiple Polynomial Quadratic Sieve (MPQS). The result is returned as a list of the prime factors of n. In case of failure an error is signalled.

  The intermediate results of a computation could be saved by interrupting the calculation with [Ctrl][C] and calling `Pause();` from the break loop. `FactorsMPQS` then pushes all data important for resuming the computation again as a record MPQSTmp on the options stack. When called again, it will look whether there is such an appropriate record on the options stack and get the previously computed factorization data, if so. For continuing the factorization process in another session, you will have to write this record to a file. This is done by the function `SaveMPQSTmp(` filename`)`, the file written by this function is readable by the standard `Read`-function of GAP.

  Caution: The runtime of `FactorsMPQS` depends only on the size of n, not on the size of its factors, so if a small factor is not found during the preprocessing which is done before invoking the sieving process, you will have to wait as long as if n would have two prime factors of roughly equal size.

  The Multiple Polynomial Quadratic Sieve tries to find integers x, y, such that x2 equivy2 (mod n), but not pmx equivpmy (mod n). In this situation, taking Gcd(x - y,n) yields a non-trivial factor of n. For determining such a pair (x,y), the algorithm chooses polynomials fa of the form fa(r) = ar2 + 2br + c with suitably chosen coefficients a, b and c, satisfying b2 equivn (mod a) and c = (b2 - n)/a. The identity a timesfa(r) = (ar + b)2 - n yields a congruence (mod n) with a perfect square on one side and a timesfa(r) on the other. The algorithm searches for factorizations of polynomial values fa(r) over a chosen factor base (a list of small primes), either complete or with a single large factor which is not in the factor base (where the latter ones are only usable if a second fa(r) with the same ``large factor'' is found) using a sieving technique over an appropriately chosen sieving interval (similar to the Sieve of Eratosthenes). Taking more polynomials and hence shorter sieving intervals gives the advantage of having smaller fa(r) to factor over the factor base. Then, Gaussian Elimination over GF(2) is used to determine the congruences which have to be multiplied together to get a congruence of the desired form x2 equivy2 (mod n), where the involved matrix M is given by Mij = 1, if an odd power of the j-th element of the factor base divides the i-th usable factored value, and Mij = 0 otherwise. For this purpose it is necessary that the number of factored fa(r) is larger than the rank of M, which is approximately given by the size of the factor base. (Compare with FactorsCFRAC.)

      gap> FactorsMPQS( Factorial(38) + 1 );
      [ 14029308060317546154181, 37280713718589679646221 ]

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