Encrypt array of int for individual retrieval
As CodesInChaos comments, your basic idea seems sound, although a 64-bit block size leaves you with a rather low security margin.

Personally, I'd be more comfortable using a cipher with a 128-bit block size (like AES), zero-padding the IDs to 64 bits, appending 64 random bits and encrypting the resulting 128-bit block.  This means that:


even with the birthday paradox, an adversary must retrieve about 232 encryptions of a given ID in order to have even odds of finding one duplicate, and
the probability of a random encrypted block decrypting to a valid ID is N out of 264, where N &le; 232 is the number of valid IDs in your database.


Now, you also ask whether there is a standard cryptographic scheme for achieving your goals.  In fact, there is: it's called authenticated encryption, and can be implemented either by combining a traditional encryption mode such as CBC, CTR or OFB with a message authentication code in the Encrypt-then-MAC construction, or by using a specialized authenticated encryption mode such as EAX, OCB or GCM.

Authenticated encryption provides IND-CCA2 security, which is short for indistinguishability under an adaptive chosen-ciphertext attack.  This is a rather strong security property, and, in particular, subsumes your goals that an adversary should not be able either to guess a valid ciphertext or to determine that two ciphertexts correspond to the same plaintext.

The main disadvantage of standard authenticated encryption modes, for your purposes, is that they normally impose an overhead of at least two whole cipher blocks: one for the IV and one for the MAC.  That said, their security margin is also correspondingly larger; for a standard AE mode using a 128-bit block cipher, the number of encryptions needed for a collision is around 264, and the probability of a successful forgery is at most one in 2128.

For some modes, it may be possible to reduce this overhead by making the IV implicit.  For example, with GCM (or, say, CTR mode with HMAC), you could just transmit a random initial counter value for the first item in your list, and then use successive counter values for the rest of the list.  Also, some modes support truncating the MAC to reduce the output length at the expense of the security margin.  Thus, combining an implicit IV with, say, a 32-bit truncated MAC, you could achieve security comparable to the 128-bit single block method described above (i.e. forgery probability less than one in 232) with a 64-bit output per item.