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The only remaining use case for direct reduction (outside of the unit
tests) is in calculating the constant R^2 mod N used during Montgomery
multiplication.
The current implementation of direct reduction requires a writable
copy of the modulus (to allow for shifting), and both the modulus and
the result buffer must be padded to be large enough to hold (R^2 - N),
which is twice the size of the actual values involved.
For the special case of reducing R^2 mod N (or any power of two mod
N), we can run the same algorithm without needing either a writable
copy of the modulus or a padded result buffer. The working state
required is only two bits larger than the result buffer, and these
additional bits may be held in local variables instead.
Rewrite bigint_reduce() to handle only this use case, and remove the
no longer necessary uses of double-sized big integers.
Signed-off-by: Michael Brown <mcb30@ipxe.org>
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The NIST elliptic curves are Weierstrass curves and have the form
y^2 = x^3 + ax + b
with each curve defined by its field prime, the constants "a" and "b",
and a generator base point.
Implement a constant-time algorithm for point addition, based upon
Algorithm 1 from "Complete addition formulas for prime order elliptic
curves" (Joost Renes, Craig Costello, and Lejla Batina), and use this
as a Montgomery ladder commutative operation to perform constant-time
point multiplication.
The code for point addition is implemented using a custom bytecode
interpreter with 16-bit instructions, since this results in
substantially smaller code than compiling the somewhat lengthy
sequence of arithmetic operations directly. Values are calculated
modulo small multiples of the field prime in order to allow for the
use of relaxed Montgomery reduction.
Signed-off-by: Michael Brown <mcb30@ipxe.org>
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