In this paper, we introduce a new notion of conditional converge cast (CCC), such that we append the conditional property to converge cast.
Additionally, we generalize the three primitives with conditional property, conditional oblivious transfer (COT), conditional oblivious cast (COC), and CCC.

CCC is a three-party protocol which involves two senders S_0 and S_1 and a receiver R.
S_0 owns a secret x and a message m_0, and S_1 y and m_1.
In a CCC protocol for the predicate Q (Q-CCC), S_0 and S_1 send their messages to R in a masked form.
R obtains the message depending on the value of Q(x,y), i.e. R obtains m_0 if Q(x,y)=0 and m_1 otherwise.
Besides, the secrets x and y cannot be revealed to R or the other sender.
We propose a CCC protocol for ``equality'' predicate with an additively homomorphic encryption scheme.

Additionally, we extend 1-out-of-2 COT/COC/CCC to 1-out-of-n COT/COC/CCC.
In 1-out-of-2 protocols, a sender or senders send two messages to a receiver or receivers.
In 1-out-of-$n$ protocols, a sender or senders send n messages, where n=2^l for some l.
We provide the consecutive definitions and the concrete protocols for 1-out-of-n COT/COC/CCC protocols.
We prove that our protocols are secure under the security of 1-out-of-2 protocols.