Asynchronous Shamir-Based Threshold FHE: Smaller Moduli and Strong Simulation Security

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We study $t$-out-of-$n$ threshold fully homomorphic encryption (ThFHE) based on Shamir secret sharing (SSS) in the asynchronous setting. With SSS, a partial decryption share is a single ring element; this allows to avoid the large number of ring elements stored per party required by prior strongly secure constructions based on $\{0,1\}$ reconstruction coefficients. However, this reduction introduces a fundamental trade-off: Lagrange reconstruction during distributed decryption severely amplifies