Function party_j_respond_proved 

pub fn party_j_respond_proved(
    i_public: &PaillierPublicKey,
    ck_i: &CommitmentKey,
    ck_j: &CommitmentKey,
    q: &BigUint,
    msg: &ProvedMessage1,
    x_j: &BigUint,
) -> Result<(ProvedMessage2, BigUint), MtaProofError>

Party j responds in the proved MtA: verify the range proof, then multiply by x_j and mask with β′.

i_public is the initiator’s Paillier public key — the exchange runs under it, so j needs NO private material (j could never open the initiator’s ciphertext anyway). ck_j is j’s own commitment key (the initiator proved against it); ck_i is party i’s key (i verifies the respondent proof j produces). Returns the response for i plus j’s own share β′.