How to prove element equality in G1 using Groth-Sahai proofs

Question

In the BeleniosRF e-voting scheme, Groth-Sahai proofs with the "Instantiation based on the SXDH assumption" are used (see https://eprint.iacr.org/2007/155, version 20160411:065033, page 24).

In the BeleniosRF paper on page 8, Figure 2, a Groth-Sahai proof is used to prove $g_1^\overline{r} = c_1$ with $g_1$ being the generator of the first group of a pairing. Using the notation of the Groth-Sahai paper, an equality of two elements in $A_1 = G_1$ must be proven. How can this be done?

This is most likely a special case of one of the described equation types

1. pairing product equation
2. multi-scalar equation in $G_1$
3. multi-scalar equation in $G_2$
4. quadratic equation

but I'm not seeing how it fits into one of those.

Answer 1

First of all, the idea of proving element equality does not make much sense, since that would reveal the secret element. $prove(X == T)$ with secret $X \in G_1$ and public constant $T \in G_1$ reveals the value of $X$. What I was looking for instead is $prove(g_1^x == T)$ with public constants $g_1, T \in G_1$ and secret scalar $x$.

It tuned out that type 2. multi-scalar equation in $G_1$ was what I was looking for. Since both left hand side and right hand side of those equations are elements in $G_1$, this is exactly what I needed. The special case to use was the liniear equation:

$A^y = T_1$

with public constants $A, T_1 \in G_1$ and private scalar $y$.

Note: Here I used the multiplicative notation for the operation in $G_1$ as it is done in the BeleniosRF paper; the GS paper uses the additive notation.