Final answer:
In CBC mode, due to a single bit error during transmission, Bob can correctly decrypt only 256 bits of the message correctly, and one bit of the following block will be incorrect. In CTR mode, Bob can decrypt 1023 bits correctly, with only the flipped bit being incorrect.
Step-by-step explanation:
In the scenario where Alice encrypts a 1024-bit message using AES and sends it to Bob, and an error occurs where the 2nd bit of the third block in the cipher text is flipped, we can analyze how this affects the decryption process in both CBC (Cipher Block Chaining) and CTR (Counter) modes.
For CBC mode, a single bit error in a ciphertext block affects the entire corresponding plaintext block on decryption. Additionally, it will adversely affect the first bit of the subsequent plaintext block due to how each block of plaintext is XORed with the previous ciphertext block. Therefore, if the third block has been corrupted, Bob will be able to correctly decrypt the first two blocks (since AES block size is 128 bits, that would be 256 bits), the remaining part of the third block will not be correct except for the flipped bit, and the 1st bit of the fourth block will also be incorrect due to propagation of the error.
In CTR mode, however, each bit of the ciphertext is encrypted independently of the other bits. Therefore, if a single bit error occurs in the ciphertext, only the corresponding bit in the plaintext will be affected upon decryption. So in this case, Bob can correctly decrypt all the 1024 bits of the message minus the one bit that has been flipped, resulting in 1023 correct bits.