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fast modular reduction
During the Crypto' 95 Rump Session, Josh Benaloh of Microsoft Corp.
presented a new modular reduction algorithm that he and I developed. It
is faster than the Montgomery method by about 10 to 15%, and is more
general and easier to understand. The central idea is that it is easy to
reduce a number to an equivalent one that's just one "block" (machine
word) longer than the modulus, by repeatedly subtracting off the highest
block, and adding back something that's equivalent, but smaller.
In the following pseudocode, B is the radix in which the numbers are
represented (2^32 for a 32-bit machine), n is the length of modulus in
blocks, U is B^(n+1) mod the modulus, X is the number to be reduced, k+1
is the length of X, and Y is the result.
1. Y = X
2. For i from k down to n+1, repeat steps 3 and 4
3. Y = Y - Y[i] * B^i + Y[i] * U * B^(i-n-1)
4. If Y >= B^i, then Y = Y - B^i + U * B^(i-n-1)
Tricks can be used to eliminate step 4, and to reduce Y to n blocks using
one single precision division, and n more single precision
multiplications. The algorithm will hopefully be written up more