Function party_i_init_proved 

pub fn party_i_init_proved(
    i_public: &PaillierPublicKey,
    ck_j: &CommitmentKey,
    q: &BigUint,
    k_i: &BigUint,
) -> Result<ProvedMessage1, MtaProofError>

Party i initiates the proved MtA: encrypt k_i under i’s OWN public key and prove it is in range.

ck_j is party j’s commitment key (j verifies this proof); q is the ECDSA group order. Party j will respond using only public material — it can never open this ciphertext.