Schnorr's Protocol

This is a Proof of Knowledge protocol to verify a users identity.

Formal Definition

1. With $P$ as the prover, $V$ as the verifier
2. With a function $G_p,p,g$ s.t $<g>=G_P,x\in G_p$ and $w\in \frac{\mathbb{Z}}{{\mathbb{Z}}_p}$ s.t $q^w=x$ shared between $P,V$
3. $P$ forms a commitment:
   1. With a power k \xleftarrow{\} \frac{\mathbb{Z}}{\mathbb{Z}_{p}}$
   2. With $t\leftarrow g^k$
4. $P$ sends $t$ to $V$
5. $V$ picks challenge c \xleftarrow{\} C \subset \frac{\mathbb{Z}}{\mathbb{Z}_{p}}$
6. $V$ send $c$ to $P$
7. $P$ computes response $r\leftarrow k+wc(\mathrm{mod}p)$
8. $P$ sends $r$ to $V$
9. $V$ outputs accept $⟺g^r=t{x}^c$