ℓ-risk (loss-based Bayes risk)

qif.measure.l_risk

A loss function \(\ell(w,x)\) gives the adversary’s loss when she takes action \(w \in \mathcal W\) and the secret is \(x\). The \(\ell\)-risk \(U_\ell(\pi) = \min_w \sum_x \pi_x\, \ell(w,x)\) is her expected loss when she takes the best action; it is an uncertainty measure, so leakage is computed as prior minus (or divided by) posterior risk.

As for g-vulnerability, a loss function is given either as a loss matrix L, with one row per action and one column per secret, or as a function l(w, x), in which case the actions are assumed to be the secrets.

Functions

prior(L: ndarray, pi: ndarray) → FloatOrRat

prior(l: Metric[int, FloatOrRat], pi: ndarray) → FloatOrRat

The \(\ell\)-risk \(U_\ell(\pi) = \min_w \sum_x \pi_x\, \ell(w,x)\).

posterior(L: ndarray, pi: ndarray, C: ndarray) → FloatOrRat

posterior(l: Metric[int, FloatOrRat], pi: ndarray, C: ndarray) → FloatOrRat

The posterior \(\ell\)-risk \(U_\ell[\pi\triangleright C] = \sum_y \min_w \sum_x \pi_x C_{x,y}\, \ell(w,x)\).

add_leakage(L: ndarray, pi: ndarray, C: ndarray) → FloatOrRat

add_leakage(l: Metric[int, FloatOrRat], pi: ndarray, C: ndarray) → FloatOrRat

The additive leakage \(U_\ell(\pi) - U_\ell[\pi\triangleright C]\).

mult_leakage(L: ndarray, pi: ndarray, C: ndarray) → FloatOrRat

mult_leakage(l: Metric[int, FloatOrRat], pi: ndarray, C: ndarray) → FloatOrRat

The multiplicative leakage \(U_\ell(\pi) / U_\ell[\pi\triangleright C]\).

strategy(L: ndarray, pi: ndarray, C: ndarray) → ndarray

strategy(l: Metric[int, FloatOrRat], pi: ndarray, C: ndarray) → ndarray

The optimal action for each observation: an array whose element \(y\) is the \(w\) minimizing \(\sum_x \pi_x C_{x,y}\, \ell(w,x)\).

add_capacity(pi: ndarray, C: ndarray, one_spanning_g: bool = False) → FloatOrRat

The additive capacity for a fixed prior, over a class of loss functions. Since the additive leakage of \(\ell\) is the same as that of the gain function \(k - \ell\) (for any constant \(k\)), this is equal to qif.measure.g_vuln.add_capacity(): the class contains the loss functions with \(\max U_\ell - \min U_\ell \le 1\), or, if one_spanning_g is true, those with \(\max \ell - \min \ell \le 1\).

loss_to_gain(n_secrets: int, n_guesses: int, l: Metric[int, FloatOrRat]) → Metric[int, FloatOrRat]

Converts a loss function into a gain function \(g(w,x) = k - \ell(w,x)\), where \(k\) is the maximum of \(\ell\) over n_guesses actions and n_secrets secrets. Then \(V_g = k - U_\ell\).

l_zero_one(type: TypeLike = def_type) → Metric[int, FloatOrRat]

The 0-1 loss function: \(\ell(w,x) = 0\) if \(w = x\), \(1\) otherwise. \(U_\ell\) is Bayes risk.

L_zero_one(n_rows: int, type: TypeLike = def_type) → ndarray

The matrix of the 0-1 loss function on n_rows secrets.