We present a new notion of public-key encryption, called multi-divisible on-line/off-line encryptions, in which partial ciphertexts can be computed and made publicly available for the recipients before the recipients' public key and/or the plaintexts are determined. We formalize its syntax and define several security notions with regard to the level of divisibility, the number of users, and the number of encryption (challenge) queries per user. Furthermore, we show implications and separations between these security notions and classify them into three categories. We also present concrete multi-divisible on-line/off-line encryption schemes. The schemes allow the computationally-restricted and/or bandwidth-restricted devices to transmit ciphertexts with low computational overhead and/or low-bandwidth network.