PPSS is a central primitive introduced by Bagherzandi et al. [2] which allows a user to store a secret among n servers such that the user can later reconstruct the secret with the sole possession of a single password by contacting t + 1 (t < n) servers. At the same time, an attacker breaking into t of these servers - and controlling all communication channels - learns nothing about the secret (or the password). Thus, PPSS schemes are ideal for on-line storing of valuable secrets when retrieval solely relies on a memorizable password. We show the most efficient Password-Protected Secret Sharing (PPSS) to date (and its implied Threshold-PAKE scheme), which is optimal in round communication as in Jarecki et al. [10] but which improves computation and communication complexity over that scheme requiring a single per-server exponentiation for the client and a single exponentiation for the server. As with the schemes from [10] and Camenisch et al. [4] we do not require secure channels or PKI other than in the initialization stage. We prove the security of our PPSS scheme in the Universally Composable (UC) model. For this we present a UC definition of PPSS that relaxes the UC formalism of [4] in a way that enables more efficient PPSS schemes (by dispensing with the need to extract the user's password in the simulation) and present a UC-based definition of Oblivious PRF (OPRF) that is more general than the (Verifiable) OPRF definition from [10] and is also crucial for enabling our performance optimization.
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