(* Copyright (c) 2011-2020 Stefan Krah. All rights reserved. *)
==========================================================================
Calculate (a * b) % p using special primes
==========================================================================
A description of the algorithm can be found in the apfloat manual by
Tommila [1].
Definitions:
------------
In the whole document, "==" stands for "is congruent with".
Result of a * b in terms of high/low words:
(1) hi * 2**64 + lo = a * b
Special primes:
(2) p = 2**64 - z + 1, where z = 2**n
Single step modular reduction:
(3) R(hi, lo) = hi * z - hi + lo
Strategy:
---------
a) Set (hi, lo) to the result of a * b.
b) Set (hi', lo') to the result of R(hi, lo).
c) Repeat step b) until 0 <= hi' * 2**64 + lo' < 2*p.
d) If the result is less than p, return lo'. Otherwise return lo' - p.
The reduction step b) preserves congruence:
-------------------------------------------
hi * 2**64 + lo == hi * z - hi + lo (mod p)
Proof:
~~~~~~
hi * 2**64 + lo = (2**64 - z + 1) * hi + z * hi - hi + lo
= p * hi + z * hi - hi + lo
== z * hi - hi + lo (mod p)
Maximum numbers of step b):
---------------------------
# To avoid unnecessary formalism, define:
def R(hi, lo, z):
return divmod(hi * z - hi + lo, 2**64)
# For simplicity, assume hi=2**64-1, lo=2**64-1 after the
# initial multiplication a * b. This is of course impossible
# but certainly covers all cases.
# Then, for p1:
hi=2**64-1; lo=2**64-1; z=2**32
p1 = 2**64 - z + 1
hi, lo = R(hi, lo, z) # First reduction
hi, lo = R(hi, lo, z) # Second reduction
hi * 2**64 + lo < 2 * p1 # True
# For p2:
hi=2**64-1; lo=2**64-1; z=2**34
p2 = 2**64 - z + 1
hi, lo = R(hi, lo, z) # First reduction
hi, lo = R(hi, lo, z) # Second reduction
hi, lo = R(hi, lo, z) # Third reduction
hi * 2**64 + lo < 2 * p2 # True
# For p3:
hi=2**64-1; lo=2**64-1; z=2**40
p3 = 2**64 - z + 1
hi, lo = R(hi, lo, z) # First reduction
hi, lo = R(hi, lo, z) # Second reduction
hi, lo = R(hi, lo, z) # Third reduction
hi * 2**64 + lo < 2 * p3 # True
Step d) preserves congruence and yields a result < p:
-----------------------------------------------------
Case hi = 0:
Case lo < p: trivial.
Case lo >= p:
lo == lo - p (mod p) # result is congruent
p <= lo < 2*p -> 0 <= lo - p < p # result is in the correct range
Case hi = 1:
p < 2**64 /\ 2**64 + lo < 2*p -> lo < p # lo is always less than p
2**64 + lo == 2**64 + (lo - p) (mod p) # result is congruent
= lo - p # exactly the same value as the previous RHS
# in uint64_t arithmetic.
p < 2**64 + lo < 2*p -> 0 < 2**64 + (lo - p) < p # correct range
[1] http://www.apfloat.org/apfloat/2.40/apfloat.pdf
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