Privacy-Preserving Semantic Caching

• Privacy-Preserving Semantic Caching is an information-theoretic framework that employs functional representation techniques to balance semantic utility and privacy constraints.
• It uses a structured two-phase design—placement and delivery—ensuring lossless semantic recovery under strict cache capacity, transmission rate, and privacy leakage bounds.
• Explicit constructions like EFRL and ESFRL enable precise privacy-utility trade-offs, making the approach practical for secure semantic caching applications.

Privacy-preserving semantic caching constitutes an information-theoretic framework for maximizing the semantic utility of cached content under explicit privacy constraints. In this paradigm, a cache server encodes semantic information about requested data, such that a user can efficiently retrieve the desired semantic goal, yet the leakage concerning any sensitive (private) variables correlated with the raw data remains rigorously bounded in the information-theoretic sense. Recent advances employ functional representation lemmas and their extensions to achieve tight privacy-utility tradeoffs and low-complexity, constructive code designs for semantic caching applications (Zamani et al., 2024).

1. System Model and Formalization

The system comprises a private variable $X$ (sensitive data) with values in $\mathcal{X}$, and a raw data or file request $Y$ in $\mathcal{Y}$, jointly distributed as $P_{X,Y}$. The desired semantic "goal" $T = h(Y)$, representing the minimal sufficient content that the user aims to recover, resides in $\mathcal{T}$.

Semantic caching unfolds in two phases:

• Placement Phase: The cache encoder produces a static cache entry

$Z = f_{\sf c}(Y, W_{\sf c}),$

where $W_{\sf c}$ is cache-server-side randomness. $Z$ (of entropy at most $M$ bits) is stored at the user.

• Delivery Phase: Upon demand, after realizing $Y$, the server sends

$X_{\sf d} = f_{\sf d}(Y, Z, W_{\sf d}),$

with fresh randomness $W_{\sf d}$, at transmission rate $R$ bits. The user, presented with $(Z, X_{\sf d})$, must perfectly recover $T$:

$H(T \mid Z, X_{\sf d}) = 0.$

Privacy is quantified by bounding the mutual information

$I(X; Z, X_{\sf d}) \leq \epsilon,$

where $\epsilon$ is a pre-specified privacy budget. Utility is characterized by the informativeness about $T$: $$I(T; Z, X_{\sf d}) = H(T) \quad \text{(for perfect recovery)}.$$ The design objective is to construct the mappings $(f_{\sf c}, f_{\sf d})$ to optimize utility (semantic recovery), satisfy memory ($H(Z)\leq M$) and delivery rate ($H(X_{\sf d}\mid Z)\leq R$), and enforce the privacy constraint.

2. Functional Representation Lemmas in Privacy Mechanism Design

Classical and extended versions of the Functional Representation Lemma (FRL) underpin the constructive code designs for privacy-preserving semantic caching:

• Classical FRL identifies an auxiliary variable $V$ (independent of $S$) such that $T$ is a deterministic function of $(S, V)$, with a cardinality bound $$|\mathcal{V}| \leq |\mathcal{S}|(|\mathcal{T}| - 1) + 1$$.
• Strong FRL (SFRL) ensures, under similar construction, that $I(S; V \mid T) \leq \log(I(S; T) + 1) + 4$.
• Extended FRL (EFRL) constructs the composite auxiliary variable $U = (V, W)$, where

$U = \phi(S, V, W),$

with $W$ a Bernoulli randomizer, such that \begin{align*} I(S; U) &= \epsilon, \ H(T \mid S, U) &= 0, \end{align*} and marginal $P_{S,U}$ emerges from a randomized-response mechanism—injecting precisely $\epsilon$ bits of leakage. The construction preserves lossless semantic recovery, achieves tight privacy-utility tradeoffs, and maintains explicit cardinality bounds.

• Extended Strong FRL (ESFRL) further quantifies "excess" leakage conditional on $T$:

$$I(S; U \mid T) \leq \frac{\epsilon}{H(S)} H(S \mid T) + \bigl(1 - \epsilon/H(S)\bigr)\bigl[\log(I(S;T) + 1) + 4\bigr].$$

These lemmas provide the mathematical means for mapping information-theoretic privacy requirements into explicit constructions for encoding and delivery in semantic caching systems (Zamani et al., 2024).

3. Privacy–Utility Trade-off Characterization

The essential optimization considers, with $T$ as the semantic goal and $S$ (shorthand for $X$) as the secret,

$\max I(T; U)$

subject to

• $I(S; U) \leq \epsilon$,
• lossless semantic recovery: $H(T \mid U, S) = 0$,
• cardinality: $|\mathcal{U}|$ bounded.

Via the chain rule

$$I(T; U) = I(S; U) + H(T \mid S) - H(T \mid U, S) - I(S; U \mid T),$$

and given $H(T \mid U, S) = 0$, bounds for the achievable privacy-utility region are

$$H(T \mid S) \leq \sup I(T; U) \leq H(T \mid S) + \epsilon.$$

EFRL-based design attains the lower bound exactly: $I(T; U_{\rm EFRL}) = H(T \mid S) + \epsilon,$ with explicit cardinality bounds. ESFRL yields the tighter lower bound

$$I(T; U_{\rm ESFRL}) \geq H(T \mid S) + \epsilon - I(S; U \mid T),$$

the penalty for excess leakage being as in the previous section.

The privacy–utility trade-off curve $F(\epsilon)$ thus resolves to linear dependence in regimes where the "common information" between $T$ and $S$ matches their mutual information, notably when one is a deterministic function of the other.

4. Privacy-Preserving Semantic Caching Code Construction

Semantic caching code designs proceed in two structured phases:

• Placement: Apply EFRL to joint law $(S, T)$, generating $(V, W)$ with $I(S; V, W) = \epsilon_{\sf c} \leq \epsilon$. Set $Z := (V, W)$, which the server stores in the user cache, subject to capacity $M \geq H(Z)$.
• Delivery: For a realized demand $Y$ (and thus $T$), the server computes the residual

$X_{\sf d} = \phi(S, T, V, W),$

ensuring $(Z, X_{\sf d})$ suffices for lossless $T$ recovery. ESFRL affords minimal delivery rate $R \geq H(X_{\sf d} \mid Z)$.

The construction guarantees

• Memory:

$$H(Z) \leq \log(|\mathcal{S}|(|\mathcal{T}| - 1) + 2)$$

• Delivery rate:

$R \approx H(T \mid S) - \epsilon_{\sf c}$

• Privacy:

$$I(S; Z, X_{\sf d}) \leq \epsilon_{\sf c} \leq \epsilon$$

• Semantic utility: perfect recovery ($H(T \mid Z, X_{\sf d}) = 0$)

The linear memory–privacy–utility trade-off,

$M + R = H(T \mid S) + \epsilon$

is achieved for $$0 \leq M \leq \log(|\mathcal{S}|(|\mathcal{T}| - 1) + 2)$$, and the entire trade-off region can be swept via time-sharing (Zamani et al., 2024).

5. Algorithmic Instantiations and Examples

An explicit implementation for binary-valued semantic and sensitive data illustrates the construction's practicality. For $S, T \in \{0,1\}$ related via a Binary Symmetric Channel with crossover $p < 1/2$, $H(T \mid S) = h(p)$, and the (binary entropy function) trade-off curve is

$$F(\epsilon) = h(p) + \epsilon, \quad 0 \leq \epsilon \leq 1.$$

A high-level pseudocode for the cache mechanism follows:

For cache of size $M = 1$ bit ($|\mathcal{V}| = 2$), the delivery rate is $R = h(p)$, with linear trade-off $M + R = h(p) + \epsilon$.

6. Cardinality Bounds, Constructiveness, and Extensions

EFRL delivers explicit cardinality bounds for the auxiliary variables, with $$|\mathcal{Z}| \leq |\mathcal{S}|(|\mathcal{T}| - 1) + 2$$. The mechanism is constructive, involving only randomized encoding steps and elementary operations.

A key implication is the possibility of extending the methodology to settings where privacy sensitivities are heterogeneous (i.e., portions of the private attribute are of differing privacy priorities). This modular framework allows tailored trade-offs between privacy, cache memory, and update bandwidth across diverse semantic information access tasks.

7. Summary and Research Directions

The extended functional representation lemma framework provides a low-complexity, constructive approach for deploying semantic caches with rigorous information-theoretic privacy guarantees and optimal utility. The theory precisely characterizes the boundary of achievable privacy–utility trade-offs and enables algorithmic instantiations with explicit guarantees. These principles generalize to semantic communication, content delivery networks, and compression design where privacy leakage is a first-class constraint (Zamani et al., 2024). A plausible implication is applicability in systems where "semantic" goals can be formally specified, privacy budgets are explicit, and efficient, provably secure semantic retrieval is demanded.