Negation of a EC Point with Jacobi x,y,z representation

T. Rossi

6/4/23, 3:10 PM

I'm building a small library for Schnorr Signatures and the Oracle DLC, the key passage is:

s_i G = R - h(i, R)V

R = k G; G generator, k a nonce
h(i, R) is the hash of the message i (i is one of the outputs that will be signed by the Oracle)
V is the public key of the Oracle (= v G)

The "add" and "multiply" operations use the Jacobi representation for speed, therefore I'd need the negation as well. Negation in the xy representation is (x, -y), what that would be for the xyz?

Also, in my experiments that implementation is indeed faster on the single operations, but I can't find much literature about it, what would be the fastest possible implementation for computing Schnorr signatures?

Thanks, T.

EDIT: wikipedia says -x, but either I read it wrong or it's in another context. If I just slam x,y,z -> x,-y,z everything checks out, I'd like to understand a bit better why though

0 + 0

schnorr-signature

poncho

6/4/23, 3:13 PM

Isn't the negation of $(x,y)$ actually $(x, -y)$?

T. Rossi

6/4/23, 3:17 PM

whoops, yes, in the xy representation it is indeed

knaccc

6/4/23, 3:21 PM

Ed25519 libraries have something called "double scalar multiplication" which is optimized for verifying a Schnorr signature, and which is particular efficient on that curve.

Score:2

Crypto

poncho

6/4/23, 3:23 PM

Negation in the xy representation is (x, -y), would that be the same for the xyz representation?

Well, the Jacobean representation point $(x, y, z)$ corresponds to the regular representation $(xz^{-2}, yz^{-3})$. The negation of that would be $(xz^{-2}, -yz^{-3})$; an easy way to get that in a Jacobean representation would be $(x, -y, z)$.

And so, yes, doing the obvious is the correct way to compute the inverse.

0 + 0

T. Rossi

6/4/23, 3:30 PM

thanks! also for the explanation!

Elon Musk

I sit in a Tesla and translated this thread with Ai:

EN: Negation of a EC Point with Jacobi x,y,z representation

TH: การลบจุด EC ที่มีตัวแทน Jacobi x,y,z

RO: Negarea unui punct EC cu reprezentarea Jacobi x,y,z

RU: Отрицание точки EC с представлением Якоби x, y, z

VI: Phủ định điểm EC với biểu diễn Jacobi x,y,z

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