# Group Signatures with VLR Revocation Check

I was reading the paper by Boneh et al. Link. The scheme describes what is called a Verifier Local Revocation Technique. The Scheme assumes that a Revocation List [RL] allows each verifier to check the validity of the signature. The revocation check works as follows:

• $$e(T_2/A,\hat{u}) =^{?}e(T_1,\hat{v})$$, Given $$T_1 = A_iv^\alpha,T_2 = u^\alpha$$
• $$e(A_iv^\alpha/A,\hat{u}) =^{?}e(u^\alpha,\hat{v})$$

In the case that $$A_i\in$$[RL], $$A = A_i$$, then,

• $$e(v^\alpha,\hat{u}) =^{?}e(u^\alpha,\hat{v})$$

The author then claims that these pairings are equal, The $$(\hat{u},\hat{v})$$ are generated as follows:

• $$(\hat{u},\hat{v}) \leftarrow H_0 = (gpk,M,r) \in G_2^2$$ , Given that:
• $$u\leftarrow\psi(\hat{u})$$, $$v\leftarrow\psi(\hat{v})$$,

I have three questions:

1. Why the $$(\hat{u},\hat{v}) \in G_2^2$$ not $$G_2$$.
2. What is the Hash Function that generates two parameters and is there a relation between them?
3. How the check $$e(v^\alpha,\hat{u}) =^{?}e(u^\alpha,\hat{v})$$ is valid.

1. $$(\hat{u}, \hat{v}) \in G_2^2$$ is a short version of $$(\hat{u}, \hat{v}) \in (G_2 \times G_2)$$ which is a short version of "$$\hat{u} \in G_2$$ and $$\hat{v} \in G_2$$".
• $$e$$ is a bi-linear mapping of the form $$G_1 \times G_2 \rightarrow G_3$$. I make this assumption based on the fact the second parameter is element of a group called $$G_2$$ which hints at the existence of a group $$G_1$$.
• $$\psi$$ is a linear homomorphism from $$G_2$$ to $$G_1$$. This would make sense as the first parameter of $$e$$ would be an element of $$G_1$$ under the first assumption. Also $$\psi(g_2)$$ must be $$g_1$$ for the math below to check out.
Be $$x, y \in Z_q$$ with $$\hat{u} = g_2^x$$ and $$\hat{v} = g_2^y$$, then $$u = g_1^x$$ and $$v = g_1^y$$ since $$\psi(g_2) = g_1$$. This leads to:
$$e(v^{\alpha}, \hat{u}) = e((g_1^y)^{\alpha}, g_2)^x = e(g_1^{\alpha y}, g_2)^x = e(g_1, g_2)^{(\alpha y) x} = e(g_1, g_2)^{(\alpha x) y} = e(g_1 ^ {\alpha x}, g_2^y) = e((g_1^x)^{\alpha}, g_2^y) = e(u^{\alpha}, \hat{v})$$