The particular instance here is integer factorization. A quick look through intfact.spad reveals that integer factorization is implemented by trial division, followed by Pollard's rho method - nothing very sophisticated. The file itself claims: "We should also employ Lenstra's elliptic curve method." Yes, we should. And the various modern sieving methods (quadratic sieve, number field sieve, and so on). Now, I'm no expert in this, and I doubt I could program such things myself, but they do exist, either stand-alone (as in Zimmermann et al's GMP-ECM), or as part of GPL software such as LiDIA or Pari/GP.
Is there any system for importing such things into Axiom? It seems to me that if we have integer factorization, we should use the best methods available, so how can it be done? -Alasdair P.S. I've decided for the coming semester to trial Axiom with my cryptography students; it seems that Axiom is admirably suited for the number theory, and string and list handling which cryptography (at least as I teach it) requires. So I'm having fun developing worksheets.
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