# First Attempt…

Fix a $(n,m)$ local leakage function $\vec{L}$ and let $\vec{\ell} \in \left(\set{0,1}^m\right)^n$ be a leakage value. Let $L_i^{-1} (\ell_i) \subseteq \mathbb{F}$ be the subset of $i$-th party’s shares such that the leakage function $L_i$ outputs $l_i \in \set{0,1}^m$.

$s_i \in L_i^{-1}(\ell_i) \iff L_i(s_i) = \ell_i$.

Therefore, for leakage function $\vec{L}$ we have leakage $\vec{\ell}$ if and only if the set of secret shares $\set{\vec{s}}$ belongs to the set

$$

\vec{L}^{-1}(\vec{\ell}) := L_1^{-1}(\ell_1) \times \cdots \times L_n^{-1}(\ell_n)

$$

Thus the probability of leakage being $\vec{\ell}$ conditioned on the secret being $s^{(0)}$ is

$$

\frac{1}{|\mathbb{F}|^k} \cdot \left| s^{(0)} \cdot \vec{v} + \langle G \rangle \cap \vec{L}^{-1}(\vec{\ell}) \right|.

$$

We can formulate the probability of leakage being $\vec{\ell}$ conditioned on the secret being $s^{(1)}$ similarly, and give the statistical distance

$$

\Delta(\vec{L}(s^{(0)}), \vec{L}(s^{(1)})) = \frac{1}{2} \cdot \frac{1}{|\mathbb{F}|^k} \sum_{\vec{\ell} \in \left(\set{0, 1}^m\right)^n} \Bigg| \left| s^{(0)} \cdot \vec{v} + \langle G \rangle \cap \vec{L}^{-1}(\vec{\ell}) \right| - \left| s^{(1)} \cdot \vec{v} + \langle G \rangle \cap \vec{L}^{-1}(\vec{\ell}) \right| \Bigg|.

$$

(nope, it is hard to bound…)