Compression with Privacy-Preserving Random Access

TL;DR

This work resolves whether privacy constraints in locally decodable compression reduce the fundamental rate—answering in the negative for memoryless sources: for any 0<p≤1/2 and rate R>H(p), there exist private locally decodable codes with vanishing error. The authors develop a graph-based encoder and a block-marginal polytope framework to satisfy privacy and reliability despite overlaps in codeword access. A perturbation-based construction shows that the desired joint distribution of codeword marginals is attainable, and small distortion is upgraded to zero distortion via a residual-error coding step, yielding rates arbitrarily close to the entropy. The approach clarifies how to reconcile locality and privacy in lossless compression and provides tools (block-marginal polytope, perturbations) that may apply to broader marginal-problem variants.

Abstract

It is shown that an i.i.d. binary source sequence $X_1, \ldots, X_n$ can be losslessly compressed at any rate above entropy such that the individual decoding of any $X_i$ reveals \emph{no} information about the other bits $\{X_j : j \neq i\}$.

Figures (5)

• Figure 1: Illustration of a graph-based compression scheme with $n=6$ and $R=2/3$, as described in Section \ref{['sec:scheme']}. For a given source sequence ${\bm{x}}$, the encoder outputs a random codeword ${\bm{C}}\sim P_{{\bm{C}}|{\bm{x}}}$. The decoding is based on a bipartite graph $\mathcal{G}$, here with constant right-degree $b$ equal to two, and a set of local decoding functions $\{f_i\}_{i\in [n]}$. The $i$'th local decoder outputs $f_{i}(C_{\mathcal{I}_{i}})$---here $\widehat{X}_{1}=f_{1}(C_{1},C_{2})$.
• Figure 2: An example to illustrate the block-marginal polytope.
• Figure 3: Illustration of the idea behind the proof of Proposition \ref{['lemma:phi_NI_generated']}. Recall that $\mathcal{P}_\mathcal{G}$ denotes the set of block-marginal vectors that can be generated. This is a convex subset of the set of block-marginal vectors $\mathcal{P}$, which is itself convex. The vector $\phi_U$ belongs to $\mathcal{P}_\mathcal{G}$. Lemma \ref{['lemma:eta_small_existence']} says that a broad class of vectors obtained through perturbations of $\phi_U$ can also be generated. This family can be expressed as $\phi_U-n \eta$ with perturbation $\eta \in \mathcal{E}_\mathcal{G}$ and is illustrated as an oval region around $\phi_U$. Furthermore, Lemma \ref{['lemma:eta_good_existence']} guarantees that with high probability, there exists $\eta \in \mathcal{E}_\mathcal{G}$ such that $\phi_{I, \bm{x} }+\eta$ can be generated. Consequently, since $\phi_{A, \bm{x} }$ can be written as a convex combination of $\phi_U-n \eta$ and $\phi_{I, \bm{x} }+\eta$ for some $\eta$, it follows that $\phi_{A, \bm{x} }$ can also be generated.
• Figure 4: Illustration of the proof of Lemma \ref{['lemma:eta_small_existence']}. Each region represents the set of distributions consistent with an increasing number of marginal constraints derived from $\{\rho_{\mathcal{I}_i}\}_{i\in [n]}$: the outermost region enforces the bit marginals, the intermediate region enforces the bit-pair marginals, and the innermost region enforces the full block marginals.
• Figure 5: The coupled–concatenated scheme of Proposition \ref{['lemma:exp_decay_pe_step4aa']} illustrated for a toy example with $k=5$. In the $i$th row, the red rectangles indicate the codewords whose Hamming distortion with respect to the $i$th source block ${\bm{x}}(i)$ exceeds $\delta n$. For the realization of the random variable $S$ shown in this example, the encoder selects the sixth, fifth, second, eleventh, and eighth codewords from the respective lists of blocks $1$--$5$.

Theorems & Definitions (29)

• Definition 1: Locally decodable code
• Definition 2: Error probability and privacy
• Theorem 1
• Definition 3: Valid codewords
• Remark 1
• Remark 2
• Definition 4: Block-marginal polytope
• Example 1
• Definition 5: $\mathcal{P}_\mathcal{G}$
• Lemma 1