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RE: [PrimeNumbers] Does such a factoring algorithm exist?

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  • Paul Leyland
    ... (Remark: almost all factoring algorithms have a superpolynomial dependency on the size of N, the integer being factored. I can t, off hand, think of any
    Message 1 of 7 , May 4, 2004
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      > Is there a factoring algorithm which depends on the size of
      > the factors (and
      > not the integer being factored itself), and is faster than
      > trial division,
      > yet upon failure it provides a proven lower bound on the factors?

      (Remark: almost all factoring algorithms have a superpolynomial
      dependency on the size of N, the integer being factored. I can't,
      off hand, think of any that do not. Some, such as ECM and NFS
      have subexponential dependency, others such as trial divsion and
      SQUFOF are exponential.)

      Yes, if you regard Dan Bernstein's product-tree method of finding
      small prime divisors as being different from "trial division". It's
      certainly faster asymtotically than the algorithm conventionally
      known as trial division.

      Another technique: precomputate the primorial of a prime and
      evaluate g=gcd(N,p#) repeatedly, replacing N by N/g until g=1.
      p is now your lower bound. Pull out the factors from the
      gcds with trial division (or recursive application of this
      algorithm).

      Note this is not faster than trial division for a single N,
      because of the precomputation, but will run faster if you
      amortize over many N.

      There may be other techniques.


      Paul
    • Jens Kruse Andersen
      ... I have not heard of Bernstein s product-tree before but it sounds like my TreeSieve: http://groups.yahoo.com/group/primeform/message/4231 The GMP based
      Message 2 of 7 , May 4, 2004
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        Paul Leyland wrote:
        > > Is there a factoring algorithm which depends on the size of
        > > the factors (and
        > > not the integer being factored itself), and is faster than
        > > trial division,
        > > yet upon failure it provides a proven lower bound on the factors?
        >
        > Yes, if you regard Dan Bernstein's product-tree method of finding
        > small prime divisors as being different from "trial division". It's
        > certainly faster asymtotically than the algorithm conventionally
        > known as trial division.

        I have not heard of Bernstein's product-tree before but it sounds like my
        TreeSieve:
        http://groups.yahoo.com/group/primeform/message/4231
        The GMP based implementation is 12 times faster than individual trial factoring
        by primeform for 50000-digit numbers, but slow for small numbers.
        This version can only factor to 2^32.

        --
        Jens Kruse Andersen
      • julienbenney
        The question is how it can prove a lower bound for the factors of a number. I imagine this would be easy for numbers such as repunits where factors are
        Message 3 of 7 , May 4, 2004
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          The question is how it can prove a lower bound for the factors of a
          number. I imagine this would be easy for numbers such as repunits
          where factors are restricted to special forms, but for other numbers
          it might not work very well because of the astonishing number of
          primes.

          For example, for the repunits R311 (not to be confused with R317),
          R509 and R557 - the status of which is "Composite But No Factor
          Known", this method seems very promising because what I once read
          about repunits suggests that these numbers must have two very large
          factors and if primality tests for numbers of the correct form are
          available, it might solve the problem well, especially for R311,
          which mystified me on seeing it in a table of repunits.

          The problem is with how to test the primality of the numbers of the
          appropriate form - though the effect of not doing this is most likely
          to be mere wastage of time ratehr than errors - for instance with the
          large 1187-digit cofactor of F12.
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