Common input $x\in L$
Completeness:
$\forall x \in L, Pr\{p ↔ V \rm makes V accept\} > \dfrac{2}{3}$ (some random constant)
Soundness:
$\forall x \in L, \forall P^, Pr\{p^ ↔ \rm V makes V accept\} < 1/3$ (random constant < 2/3)
Probability is over V’s coin flips
$V\in \rm PPT$ (probabilistic polytime)
$\forall x \in L, \forall V^\in \mathrm{PPT}, \exists \mathrm{expected\,polytime\, simulator}\, S\,s.t. [S_{V^}(x)] = [P \leftrightarrow V^*(x)]$
With probability $(1-\dfrac{1}{2^k})$ runs in 2 steps
With probability $\dfrac{1}{2^k}$, runs in $2^k$ steps
(This is actually polytime since the cost is amortized)
(Graph isomorphism)
$P, V$
V: Chooses some new $\pi$ and $b$. Computes $H \leftarrow \pi(G_b)$.
H: Receives $H$. Calculates some $b’$. Sends over new permutation $\pi’$.
Repeat this $k$ times; accept if exists at least one $b\neq b’$.