# Which is the fundamental difference between a signature of a message and a zero-knowledge argument of a signature for that message?

Say that I have a signing/verification key pair $$(sk,vk)$$ and I generate a signature $$\sigma$$ over a message $$m$$. Then, I want to prove to someone else of this fact, which can be done in two different manners:

1. I directly send $$(\sigma,m)$$.
2. I compute a non-interactive zero knowledge argument $$\pi$$ for the statement: "I have a valid signature $$\sigma$$ for the message $$m$$". Notice that the argument does not reveal $$\sigma$$. I send $$(\pi, m)$$.

The question is: which is the fundamental difference 1 and 2? In 2, the verifiers do not learn the signature itself, but the proof $$\pi$$ can be used with the same purpose as $$\sigma$$. Are signatures and non-interactive zero knowledge arguments the very same thing?

Say that I want to prove that I have written the new Game of Thrones book and I want to authenticate it to a designated verifier. I know that 1 will not work, but what about 2?

• Related to Are digital signatures a type of zero-knowledge proof?, and to a lesser extend to Did digital signatures come from Zero Knowledge Proofs?. I have not yet made my mind about what to do about that overlap.
– fgrieu
Sep 29, 2022 at 9:12
• Proving I have possession of a signature for a particular public key on a particular message will indirectly act as a signature itself and demonstrate the same knowledge as the signature I have in my possession. So what would be the point - am I missing something? Sep 29, 2022 at 22:37
• @knaccc Yeah, that is exactly what I am asking. If the ZKP is interactive instead of non-interactive then that is not true: the proof becomes "non-reusable". In contrast, if the protocol is rendered non-interactive (via FS) then I want to know if it s really the case that both constructions are equivalent. This in fact would imply that a signature is a better option than a ZKP, since the former requires less computing power than the latter. Sep 30, 2022 at 12:46
• I think it would help if you were more specific about what you mean by "ZKP", and how that would be implemented. A very simple signature such as a Schnorr signature is a zero-knowledge proof, and can be interactive or non-interactive. You must mean something different when you say "ZKP". Sep 30, 2022 at 12:58
• @knaccc For instance, you could compute an arithmetic circuit $C$ such that $C(vk,\sigma,m) = 0$ if and only if $\sigma$ is a valid signature on $m$ with respect to $vk$. Then, you could use any SNARK that you wish to prove the correctness of the computation of such circuit $C$. This is totally different from a standard signature scheme. Sep 30, 2022 at 17:25