Neural image compression (NIC) has become the state-of-the-art for rate-distortion performance, yet its security vulnerabilities remain significantly less understood than those of classifiers. Existing adversarial attacks on NICs are often naive adaptations of pixel-space methods, overlooking the unique, structured nature of the compression pipeline. In this work, we propose a more advanced class of vulnerabilities by introducing T-MLA, the first targeted multiscale log--exponential attack framework. Our approach crafts adversarial perturbations in the wavelet domain by directly targeting the quality of the attacked and reconstructed images. This allows for a principled, offline attack where perturbations are strategically confined to specific wavelet subbands, maximizing distortion while ensuring perceptual stealth. Extensive evaluation across multiple state-of-the-art NIC architectures on standard image compression benchmarks reveals a large drop in reconstruction quality while the perturbations remain visually imperceptible. Our findings reveal a critical security flaw at the core of generative and content delivery pipelines.

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