Tim Peters <t...@python.org> added the comment:
Some random notes:
- 1425089352415399815 appears to be derived from using the golden ratio to
contrive a worst case for the Euclid egcd method. Which it's good at :-) Even
so, the current code runs well over twice as fast as when replacing the
pow(that, -1, P) with pow(that, P-2, P).
- Using binary gcd on the same thing requires only 46 iterations - and, of
course, no divisions at all. So that could be a big win. There's no possible
way to get exponentiation to require less than 60 iterations, because it
requires that many squarings just to reach the high bit of P-2. However, I
never finishing working out how to extend binary gcd to return inverses too.
- All cases are "bad" for pow(whatever, P-2, P) because P-2 has 60 bits set.
So we currently need 60 multiplies to account for those, in addition to 60
squarings to reach P-2's high bit. A significant speedup could probably have
been gotten just by rewriting whatever**(P-2) as
(whatever ** 79511827903920481) ** 29
That is, express P-2 as its prime factorization. There are 28 zero bits in the
larger factor, so would save 28 multiply steps right there. Don't know how far
that may yet be from an optimal addition chain for P-2.
- The worst burden of the P-2-power method is that there's no convenient way to
exploit that % P _can_ be very cheap, because P has a very special value. The
power method actually needs to divide by P on each step. As currently coded,
the egcd method never needs to divide by P (although _in_ the egcd part it
divides narrower & narrower numbers < P).
- Something on my todo list forever was writing an internal routine for
squaring, based on that (a+b)**2 = a**2 + 2ab + b**2. That gives
Karatsuba-like O() speedup but with simpler code, enough simpler that it could
probably be profitably applied even to a relatively small argument.
Which of those do I intend to pursue? Probably none :-( But I confess to be
_annoyed_ at giving up on extending binary gcd to compute an inverse, so I may
at least do that in Python before I die ;-)
----------
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Python tracker <rep...@bugs.python.org>
<https://bugs.python.org/issue37863>
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