[bitcoin-dev] Schnorr signatures BIP

[Russell O'Connor]

Over chat it has been pointed out to me that computing the non-quadratic
residue is not the same cost as computing a quadratic residue.  As pointed
out in footnote 7 of the the proposed BIP, c^((p+1)/4) is always a
quadratic residue and must be negated to find the non-quadratic residue.

In light of this, I revise my proposed change to make the verification
equation

R + sG + eP = 0.

(by 0 in the equation above I mean the identity element for the (+)
operation, which is the point at infinity.)

This equation is suitable for batch verification.  This equation is clearly
written as a linear combination that doesn't use negation.  In most
implementations, equality comparison tests are implemented by subtraction
and a comparison with zero. By writing the verification equation this way,
we clearly see that only the comparison with zero test is needed.

For single signature verification the check becomes, compute Q := sG + eP.
Verify that Q isn't the point at infinity and Q.x = r.  Verify that Q.y is
*not* a quadratic residue. (While I was incorrect earlier about the costs
of computing a non-residue, it is the case the *verifying* a value is a
quadratic residue is the same cost as verifying a value is a non-residue.)

Effectively in my first email I was suggesting that the 'e' value in a
signature be negated from the current BIP proposal.  In this revision I am
effectively suggesting that the 's' value in a signature should be the one
that is negated instead.

On Sat, Aug 4, 2018 at 8:22 AM, Russell O'Connor <roconnor at blockstream.io>
wrote:

> I propose changing the verification equation from "Let *R = sG - eP*" to
>
>     Let *R = sG + eP*
>
> This allows faster verification by avoiding negating a point (or a
> coefficient).
>
>
> If, instead of directly following the literal verification specification,
> one is instead reconstructing R from r by finding a y coordinate that is a
> quadratic residue, under the existing scheme one would verify
>
>
> *sG - eP = R*
>
> which is effectively verifying
>
>   0 = *sG - eP* - R  or 0 = R - *sG + eP*
>
> Either way one needs to negate at least one point (or one coefficient)
> because of the opposite signs between sG and eP.
>
>
> Under my proposed revised verification scheme, one would instead verify
>
>   0 = sG + eP + (-R).
>
> While it seems that this requires negating R, it does not.  Rather (-R)
> can be directly constructed from r by finding a y coordinate that is *not*
> a quadratic residue, which is precisely the same amount of work that
> construction R from r was.
>
> In either verification procedure, changing the verification equation to my
> proposal removes one negation operation from the cost of doing verification.
>
>
>
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