Thanks for the write-up of that, semisol. Makes sense
I was thinking of multiplication in a different context, where it is sometimes a bit better than addition, but you're right that it doesn't help here
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Multiplication is used in ECDH.
You have private keys a and b, and public keys aG and bG. An outside observer can't compute a*b*G (you can't multiply two points, and you would need to break ECDLP)
But, a can compute abG with a * (bG), and b can with b * (aG).
I know thanks. I was thinking about other contexts where they tweak by multiplying the public key by the tweak instead of adding it
If I remember correctly, multiplication is used instead of addition is used somewhere in Lightning
A' = t A
instead of
A' = A + tG