Paper 2026/1238

On the Additive Sensitivity of LZ77 Under Consecutive Edits

Seunghoon Lee, University of Waterloo

Abstract

We revisit the problem of mitigating information leakage in the widely used but insecure compress-then-encrypt paradigm. While encryption hides message contents, the ciphertext length is directly related to the length of the compressed message, which may, in turn, leak information about the {\em content} of the message itself. Recent work of Blocki et al. (TCC~2025) proposed an $(\varepsilon,\delta)$-differentially private approach that adds randomized padding calibrated to the global sensitivity of the compression algorithm, and showed that LZ77 has global sensitivity $O(W^{2/3}\log n)$ for input length $n$ and sliding window size $W$. However, prior analysis focused only on sensitivity with respect to single-character edits, which leads to limited privacy guarantees when protecting longer substrings such as passwords, passphrases, cookies, or confidential user records. A natural attempt to handle longer secrets is to appeal to group privacy, but for approximate differential privacy, this leads to very poor parameter degradation: in particular, the effective value of $\delta$ can grow exponentially with the group size $g$. In this work, we introduce and study the sensitivity of compression schemes under block edits. Specifically, we define two strings to be $g$-neighbors if they differ only within a contiguous interval of length $g$. Our main technical contribution is a nearly tight characterization of the $g$-consecutive sensitivity of LZ77. We show that the $g$-consecutive sensitivity of LZ77, both with and without self-referencing, is at most $O\!\left((W^{2/3}+g+\sqrt{Wg})\log n\right)$. In particular, when $g \leq W^{1/3}$, the bound simplifies to $O(W^{2/3}\log n)$, matching the known bound for single-character edits. Combined with the framework of Blocki et al., our bound yields $(\varepsilon,\delta)$-differential privacy for $g$-neighbors with the same asymptotic padding scale as for single-character edits. We provide matching lower bounds to demonstrate that our upper bound is tight, e.g., when $n=W=\Theta(g^2)$, the $g$-consecutive sensitivity of LZ77 is at least $\tilde{\Omega}(g^{3/2})$, matching the $\sqrt{Wg}=\Theta(g^{3/2})$ term from our upper bound up to a logarithmic factor.