Final answer:
Galois/Counter Mode (GCM) is the block cipher mode that meets the requirements of using a synchronous counter and adding additional authenticated data (AAD) for both encryption and data integrity purposes.
Step-by-step explanation:
The block cipher mode of operation you are looking for is the Galois/Counter Mode (GCM). This mode combines the counter (CTR) mode of operation with a Galois field multiplication operation to provide both confidentiality and data integrity. In GCM, the synchronous counter provides the sequence of counters used with the block cipher to encrypt the plaintext, and the additional authenticated data (AAD) is included in the integrity check but not encrypted, allowing for the authentication of system-level information along with message content.
GCM requires both the sender and the receiver to have access to the same counter values during encryption and decryption, so they must be synchronized. The AAD input is part of what makes GCM unique, providing integrity not just for the ciphertext, but for additional data that needs to remain in the clear. Out of all the options given, GCM is the only mode that fits the requirements specified in the question.