Research proof index

If Alice is correct and Bob rejects, the probability of false is low

If $d[X_1;X_2]=0$, then there exists a subgroup $H \leq G$ such that $d[X_1;U_H] = d[X_2;U_H] = 0$. Follows from the preceding claim by the triangle inequality.

If $d[X ;X]=0$, then there exists a subgroup $H \leq G$ such that $d[X ;U_H] = 0$.

Moment generating function for count

Let $H$ be a subgroup of $G \times G'$. Then there exists a subgroup $H_0$ of $G$, a subgroup $H_1$ of $G'$, and a homomorphism $\phi: G \to G'$ such that $H := \{ (x, \phi(x) + y): x \in H_0, y \in H_1 \}.$ In particular, $|H| = |H_0| |H_1|$.

The Beroulli case of Hoeffding's lemma