Some quick comments:

Signing

To sign:
Let k = int(hash(bytes(d) || m)) mod n^[8].
Let R = kG.
If jacobi(y(R)) ≠ 1, let k = n - k.
Let e = int(hash(bytes(x(R)) || bytes(dG) || m)) mod n.
The signature is bytes(x(R)) || bytes(k + ex mod n).

Can we avoid mutable variables in these specification? I know this is commonly done in RFCs, but I think it is fairly confusing to have `k` defined in two different ways within a single specification.

Let's let k' = k when jacobi(y(R)) = 1 and let k' = n - k when jacobi(y(R)) = -1. Note that this ensures that jacobi(y(k'G)) = 1.

Also you've sort of left it undefined what to do when k = 0. According to the current specification, you will produce an invalid signature. The expected result is that you should win a 1000 BTC prize.

One solution is to let k = 1 + int(hash(bytes(d) || m)) mod (n-1). Alternatively you could let k' = 1 when k = 0. Or you could just make a note that signature generation fails with this message and private key pair when this happens.

Let e = int(hash(bytes(x(R)) || bytes(dG) || m)) mod n.

P = dG should probably be noted somewhere in the text. I.e. this signature is generated for the public key P = dG.

If the inputs to hash were reordered as hash(bytes(dG) || bytes(x(R)) || m) then there is an opportunity for SHA256 expander to be partially prefilled for a fixed public key. This could provide a little benefit, especially when multiple signatures for a single public key need to be generated and/or verified. If all things are otherwise equal, perhaps this alternate order is better.

The signature is bytes(x(R)) || bytes(k + ex mod n).

You haven't defined `x`. I'm guessing you mean `d` instead.

Optimizations
Jacobian coordinates
oncurve(P) can be implemented as y² = x³ + 7z⁶ mod p.

oncurve(P) requires that `P` be on the curve and not infinity. You need another condition here to ensure that `P` is not infinity.

On Fri, Jul 6, 2018 at 2:08 PM, Pieter Wuille via bitcoin-dev <bitcoin-dev@lists.linuxfoundation.org> wrote:

Hello everyone,

Here is a proposed BIP for 64-byte elliptic curve Schnorr signatures,
over the same curve as is currently used in ECDSA:
https://github.com/sipa/bips/blob/bip-schnorr/bip-schnorr.mediawiki

It is simply a draft specification of the signature scheme itself. It
does not concern consensus rules, aggregation, or any other
integration into Bitcoin - those things are left for other proposals,
which can refer to this scheme if desirable. Standardizing the
signature scheme is a first step towards that, and as it may be useful
in other contexts to have a common Schnorr scheme available, it is its
own informational BIP.

If accepted, we'll work on more production-ready reference
implementations and tests.

This is joint work with several people listed in the document.

Cheers,

--
Pieter
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