Proof of knowledge

One of the simplest and frequently used proofs of knowledge, the proof of knowledge of a discrete logarithm, is due to Schnorr.[3] The protocol is defined for a cyclic group ${\displaystyle G_{q}}$ of order ${\displaystyle q}$ with generator ${\displaystyle g}$.

In order to prove knowledge of ${\displaystyle x=\log _{g}y}$, the prover interacts with the verifier as follows:

1. In the first round the prover commits himself to randomness ${\displaystyle r}$; therefore the first message ${\displaystyle t=g^{r}}$ is also called commitment.
2. The verifier replies with a challenge ${\displaystyle c}$ chosen at random.
3. After receiving ${\displaystyle c}$, the prover sends the third and last message (the response) ${\displaystyle s=r+cx}$ reduced modulo the order of the group.

The verifier accepts, if ${\displaystyle g^{s}=ty^{c}}$.

We can see this is a valid proof of knowledge because it has an extractor that works as follows:

1. Simulate the prover to output ${\displaystyle t=g^{r}}$. The prover is now in state ${\displaystyle Q}$.
2. Generate random value ${\displaystyle c_{1}}$ and input it to the prover. It outputs ${\displaystyle s_{1}=r+c_{1}x}$.
3. Rewind the prover to state ${\displaystyle Q}$. Now generate a different random value ${\displaystyle c_{2}}$ and input it to the prover to get ${\displaystyle s_{2}=r+c_{2}x}$.
4. Output ${\displaystyle (s_{1}-s_{2})(c_{1}-c_{2})^{-1}}$.

Since ${\displaystyle (s_{1}-s_{2})=(r+c_{1}x)-(r+c_{2}x)=x(c_{1}-c_{2})}$, the output of the extractor is precisely ${\displaystyle x}$.

This protocol happens to be zero-knowledge, though that property is not required for a proof of knowledge.

Protocols which have the above three-move structure (commitment, challenge and response) are called sigma protocols. The Greek letter ${\displaystyle \Sigma }$ visualizes the flow of the protocol. Sigma protocols exist for proving various statements, such as those pertaining to discrete logarithms. Using these proofs, the prover can not only prove the knowledge of the discrete logarithm, but also that the discrete logarithm is of a specific form. For instance, it is possible to prove that two logarithms of ${\displaystyle y_{1}}$ and ${\displaystyle y_{2}}$ with respect to bases ${\displaystyle g_{1}}$ and ${\displaystyle g_{2}}$ are equal or fulfill some other linear relation. For a and b elements of ${\displaystyle Z_{q}}$, we say that the prover proves knowledge of ${\displaystyle x_{1}}$ and ${\displaystyle x_{2}}$ such that ${\displaystyle y_{1}=g_{1}^{x_{1}}\land y_{2}=g_{2}^{x_{2}}}$ and ${\displaystyle x_{2}=ax_{1}+b}$. Equality corresponds to the special case where a = 1 and b = 0. As ${\displaystyle x_{2}}$ can be trivially computed from ${\displaystyle x_{1}}$ this is equivalent to proving knowledge of an x such that ${\displaystyle y_{1}=g_{1}^{x}\land y_{2}={(g_{2}^{a})}^{x}g_{2}^{b}}$.

This is the intuition behind the following notation,[4] which is commonly used to express what exactly is proven by a proof of knowledge.

${\displaystyle PK\{(x):y_{1}=g_{1}^{x}\land y_{2}={(g_{2}^{a})}^{x}g_{2}^{b}\},}$

states that the prover knows an x that fulfills the relation above.