DIO_fan

[John Young <jya@pipeline.com>]

   The Sciences, March/April, 1996:

   "Beyond the Last Theorem."
   
   On Diophantine equations, by mathematician Dorian Goldfeld.

      To mathematicians, the statement and proof of the STW
      conjecture were as revolutionary as the first mingling
      of waters in the Panama Canal. Until that point, the
      mathematics of elliptic functions and the mathematics of
      rigid motions had developed in isolation from each other
      and in strikingly different ways. The study of elliptic
      curves was a branch of number theory, small, specialized
      and provincial -- not unlike the study of Diophantine
      equations. In contrast, the study of rigid motions was
      a bustling, sophisticated suburb of topology, geometry
      and analysis, with many applications to engineering and
      physics. Mathematicians had been working on rigid
      motions intensely for a hundred years and had
      accumulated a vast armamentarium of powerful
      mathematical machinery. By suggesting that the two
      fields could be linked, Shimura, Taniyama and Weil
      delivered that heavy machinery to the construction site
      of elliptic curves; by proving that the link held, Wiles
      and Taylor started the engines. The result has been a
      frenzy of productive mathematical work that has
      benefited each field and is likely to lead to solutions
      of outstanding problems in other fields as well. ...

      If the ABC conjecture yields, mathematicians will find
      themselves staring into a cornucopia of solutions to
      long-standing problems. Some of those problems are of
      more than theoretical interest. Nowadays many methods of
      ensuring the security of electronic mail and other
      computerized transactions depend heavily on number
      theory, as programmers develop ciphers based on
      time-consuming problems in arithmetic. For example, a
      highly popular technique depends on the difficulty of
      determining all the large prime factors of a very large
      number.

      In principle, it should also be straightforward to
      create a cipher based on the difficulty of solving
      problems in Diophantine analysis. The major hurdle is
      the solvability barrier: the number of variables above
      which a Diophantine equation becomes impervious to
      attack. Any cipher based on an equation with that many
      variables should be absolutely secure. But where is the
      threshold? All anyone knows is that it probably lies
      between three and nine variables. At current or
      foreseeable processing speeds, a nine-variable cipher is
      impracticably slow, even for the fastest computers. A
      four-variable Diophantine cipher, however, would be both
      practical and extremely useful.

   
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