Function party_i_finish_proved 

pub fn party_i_finish_proved(
    i_public: &PaillierPublicKey,
    ck_i: &CommitmentKey,
    q: &BigUint,
    msg1_ciphertext: &BigUint,
    i_private: &PaillierPrivateKey,
    msg: &ProvedMessage2,
) -> Result<BigUint, MtaProofError>

Party i finishes the proved MtA: verify the respondent proof, then decrypt with i’s OWN private key.

ck_i is party i’s commitment key (the proof is addressed to i). Returns α = k_i·x_j + β′; pair with j’s β′ via α − β′. Only the initiator can run this step — the response is encrypted under i’s key — which is exactly what keeps α and β′ in different hands.