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Smith's SMODA factorer self-destructs

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  • Warren Smith
    My factoring algorithm SMODA has self-destructed to the extent it now seems less attractive than the quadratic sieve as a factorer. ... The problem was
    Message 1 of 1 , Mar 6, 2012
      My factoring algorithm SMODA has self-destructed to the extent it
      now seems less attractive than the quadratic sieve as a factorer.

      ----More detailed explanation:
      The problem was embarrassingly stupid and would have been noticed as soon as an
      implementation was built but Andy JENNINGS saved time by spotting the
      problem before that.

      To review, SMODA depended on the first idea that, if there were a magic oracle
      whom you could ask "O Oracle, here is B and N, now please generate for
      me a set of B-smooth numbers
      near to N or near to multiples of N... such that these numbers are (i)
      otherwise pretty random,
      (ii) come approximately as close (in terms of additive error)
      to being a multiple of N as possible existencewise,
      (iii) come at a small cost, e.g. polynomial in logN cost, per number found",
      then factoring would be possible at speeds rivalling or exceeding the
      Number Field Sieve.

      The second idea was that instead of depending on magic, it was
      actually possible to build
      the oracle by combining a certain fairly small N-independent database
      with some simple
      greedy construction algorithms. The database was hard to find, hence
      still somewhat oracle-like,
      but it seemed possible to find it fast enough that challenging world
      records might be feasible.

      Now various potential problems were pointed out, which actually did
      not seem to be problems,
      such as
      1) database too expensive to find?
      but actually just today my computer run is ending, and has constructed
      preliminary database up to 2^400 size. Jim White & I were building
      datasets of the "world's most extensive Stormer pair collection" and
      it seems clear it with feasible number of computers
      our current codes could reach 2^500 -- and likely a good deal further
      with as-yet-unimplemented algorithms.
      2) Why not use LLL lattice reduction methods to construct database?
      As far as I can tell they are an uncompetitively poor idea.

      The real problem is this. The greedoid which uses the database to
      find smooth numbers
      (call them S) near to N or near to multiples of N, works in a manner
      somewhat analogous to the way binary numbers can be constructed lying
      arbitrarily near to any real. One keeps
      on adding smaller and smaller "correction vectors" in a certain vector
      space, and proves that a certain natural vector logarithmic distance
      measure keeps getting smaller, indeed
      at least a factor 2 smaller each whack.

      That works. BUT, the problem is, when you make that natural
      logarithmic distance smaller, that does NOT imply that the actual
      difference |S - multipleofN| keeps getting smaller back in the
      ordinary world of integer numbers, rather than the vector world.
      Instead, this difference gets smaller for a while, but once |S-N| is
      of order about squareroot(N), it almost always would start to grow

      Simple example: if I had, say,
      10111 =approx= 10000 with error=111
      and I used multiplication by the rational 101/100 to try to improve
      the approximation, I'd get
      1011100 =approx= 1010000 with error=1100
      which is the opposite of an improvement in the additive error
      (although it does improve the *relative* error).

      Improving the relative error is easy and was essentially what I was doing.
      Although in principle it basically is always possible to improve the
      additive error,
      (e.g. if we use 271/274 not 100/101 then get 2740081 which is 81 mod 10000,
      or if we use 991/1002 we get 10020001 which is 1 mod 10000)
      I don't see how SMODA can gaurantee that without an unfeasibly
      enormous database
      far larger than I had in mind.
      So the net effect is that SMODA should stop working approximately at
      the point where you try to exceed the performance of the quadratic
      sieve. And it probably will be worse than QS since it is more

      This is a pity because SMODA and associated theory and data was quite big and
      aesthetic looking, but now is seen to be a "castle in the air."

      Is there any hope to salvage SMODA? I'm not sure. You'd need to
      invent a way to use
      the database that would be far more powerful than the simple greedoids
      I had in mind.
      (But it would be permissible to consume a good deal more compute-time
      than the greedoid
      consumed.) One way to try do that would be to find a rational number
      (such as 271/274
      in example above) that yields improvement (this is easy by Euclid
      alg.), but doing so in
      such a way its numerator was smooth (not so easy). For example based
      on the two good
      rationals above 271/264 and 991/1002, and 91/92 which is also good,
      we could look for smooth numbers 271a+991b+91c
      in various ways such as sieving or lattice techniques.
      Another salvage attempt would be to have stronger guarantees about
      the database
      than those I'd had in mind. For example, we believe based on both theoretical
      models and computer evidence that very small databases based on very small prime
      sets [of order only about (logN)^2 primes] should *exist* but I had given up
      on trying to construct databases that small because they asymptotically seemed
      much too hard to find. If, however, we nevertheless assumed
      SMODA had access to such a super-small super-good database, then SMODA
      might be able to become a lot more powerful due to "cancellations"
      arising from common prime factors re-occuring enough of the time.

      So, salvaging SMODA may or may not be possible. It is worth some thought.
      If anybody wants to have my SMODA parts I/II paper draft/shipwrecks and/or our
      the Smith+White datasets of Stormer pairs, let me know.

      Warren D. Smith
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