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Digital media forensics must deal with constantly increasing volumes of data. In order to efficiently scale up, outsourcing computation becomes an appealing solution. However, due to the highly sensitive nature of forensic data, its privacy must be protected when processed in an untrusted environment. This work proposes a new framework to efficiently perform an outsourced PRNU (Photoresponse Non-Uniformity) fingerprint extraction and detection on encrypted images in a fully unattended way. For this purpose, we rely on lattice-based homomorphic cryptosystems paired with advanced optimization strategies. We evaluate our solutions in terms of efficiency, security and performance for real image datasets, showing the feasibility of camera attribution in the encrypted domain.