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Modular Reduction
Usually, modular reduction is accomplished by long division, using the
mp_div() or mp_mod() functions. However, when performing modular
exponentiation, you spend a lot of time reducing by the same modulus
again and again. For this purpose, doing a full division for each
multiplication is quite inefficient.
For this reason, the mp_exptmod() function does not perform modular
reductions in the usual way, but instead takes advantage of an
algorithm due to Barrett, as described by Menezes, Oorschot and
VanStone in their book _Handbook of Applied Cryptography_, published
by the CRC Press (see Chapter 14 for details). This method reduces
most of the computation of reduction to efficient shifting and masking
operations, and avoids the multiple-precision division entirely.
Here is a brief synopsis of Barrett reduction, as it is implemented in
this library.
Let b denote the radix of the computation (one more than the maximum
value that can be denoted by an mp_digit). Let m be the modulus, and
let k be the number of significant digits of m. Let x be the value to
be reduced modulo m. By the Division Theorem, there exist unique
integers Q and R such that:
x = Qm + R, 0 <= R < m
Barrett reduction takes advantage of the fact that you can easily
approximate Q to within two, given a value M such that:
2k
b
M = floor( ----- )
m
Computation of M requires a full-precision division step, so if you
are only doing a single reduction by m, you gain no advantage.
However, when multiple reductions by the same m are required, this
division need only be done once, beforehand. Using this, we can use
the following equation to compute Q', an approximation of Q:
x
floor( ------ ) M
k-1
b
Q' = floor( ----------------- )
k+1
b
The divisions by b^(k-1) and b^(k+1) and the floor() functions can be
efficiently implemented with shifts and masks, leaving only a single
multiplication to be performed to get this approximation. It can be
shown that Q - 2 <= Q' <= Q, so in the worst case, we can get out with
two additional subtractions to bring the value into line with the
actual value of Q.
Once we've got Q', we basically multiply that by m and subtract from
x, yielding:
x - Q'm = Qm + R - Q'm
Since we know the constraint on Q', this is one of:
R
m + R
2m + R
Since R < m by the Division Theorem, we can simply subtract off m
until we get a value in the correct range, which will happen with no
more than 2 subtractions:
v = x - Q'm
while(v >= m)
v = v - m
endwhile
In random performance trials, modular exponentiation using this method
of reduction gave around a 40% speedup over using the division for
reduction.
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