We consider protocols for secure multi-party computation (MPC) built from \(\textsf{FHE} \) under honest majority, i.e., for \(n =2t+1\) players of which t are corrupt, that are robust. Surprisingly there exists no robust threshold \(\textsf{FHE} \) scheme based on \(\textsf{BFV} \) to design such MPC protocols. Precisely, all existing methods for generating a common relinearization key can abort as soon as one player deviates. We address this issue, with a new relinearization key (adapted from [CDKS19, CCS’19]) which we show how to securely generate in parallel of the threshold encryption key, in the same broadcast. We thus obtain the first robust threshold \(\textsf{BFV} \) scheme, moreover using only one broadcast for the generation of keys instead of two previously. Of independent interest, as an optional alternative, we propose the first threshold \(\textsf{FHE} \) decryption enabling simultaneously: (i) robustness over asynchronous channels with honest majority; (ii) tolerating a power-of-small-prime ciphertext modulus, e.g., \(2^e\) ; and (iii) secret shares of sizes quasi-independent of \(n \) .