Hi Brian:
With ECDSA, one has R:=(1/s)(eG+rQ), where e:=h(m), and r= x(R) mod n.
If R=(X, Y, Z) in Jacobian coordinates, then x(R)=X/(Z^2), where
computations are over GFp.
One has x(R) Z^2 = X, which is equivalent to r Z^2 = X only if the
modular reduction mod n does not do anything. For secp256k1, one has
n<p, so for the tiny fraction of x(R)'s in the interval [n,p-1], this
yields the wrong result.
The equation is always correct, had ECDSA been defined with r=x(R),
i.e., without the mod n reduction step to compute r.
Please note that if x(R) in the interval [n,p-1], then r=x(R) mod n is
in the interval [0,p-n-1], so one could still apply the trick in the
vast majority of cases, by simply incorporating a test on whether r >
p-n-1 and applying the trick if so.
Best regards, Rene
On 3/23/2016 8:16 AM, Brian Smith wrote:
Hi,
[I am not sure if boring topics like ECDSA are appropriate for this
list. I hope this is interesting enough.]
ECDSA signature verification is quite expensive. A big part of why it
is expensive is the two inversions--one mod q, one mod n--that are
typically used.
A while back I stumbled across an interesting optimization [1] in
libsecp256k1. The optimization completely avoids the second inversion
during verification.
The comments in the code explain how, but here's a rough summary:
Normally we convert the Jacobian coordinates (X, Y, Z) of the point
multiplication result to affine (X, Y) so that the affine X coordinate
can be compared to the signature's R component. The conversion to
affine coordinates requires the inversion of Z. But, instead of doing
that, we can simply multiply the signature's R component by Z**2 and
then compare it with the *Jacobian* X coordinate, avoiding any inversion.
I asked Greg Maxwell, the author of that code, about it and he didn't
know of anybody else using this optimization.
The optimization has two important properties:
1. It make verification notably (but not hugely) faster.
2. It reduces the amount of code required by an enjoyable amount, if
one is writing prime- specific specialized inversion routines.
Two questions:
1. Does anybody know of prior published software or papers documenting
this?
2. Does anybody know why it would be a bad idea to use this technique?
I.e. am I overlooking some reason why it doesn't actually work?
[1]
https://github.com/bitcoin/secp256k1/blob/269d4227038b188128353235a272a8f030c307b1/src/ecdsa_impl.h#L225-L253
(shortened: https://git.io/vad3K)
Thanks,
Brian
--
https://briansmith.org/
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