Final answer:
The similarities between two ElGamal ciphertexts B1 = B2 and B1 = g · B2 indicate the use of the same or sequentially incrementing ephemeral keys respectively during the encryption process, although it does not reveal the exact plain-texts M1 and M2.
Step-by-step explanation:
The question deals with observations made in the context of the ElGamal encryption system. Specifically, it focuses on what can be inferred when certain similarities are identified between two sets of ciphertexts.
(a) B1 = B2
If you observe that B1 = B2, you can infer that the same random ephemeral key k was used in both encryption processes for M1 and M2. This is because B is calculated as g^k mod p where g is the generator and p is the prime modulus. Despite using the same k, without the private key, you still cannot determine the exact plain-texts M1 and M2 unless additional information is provided.
(b) B1 = g · B2
If B1 = g · B2, this suggests that the ephemeral key k used for the second ciphertext pair is exactly one more than that used for the first (k1 = k2 + 1). This is again because B is created by raising g to the power of the ephemeral key, so multiplying by g suggests an increment by one in the power. However, without additional information or the private key, you cannot determine more about M1 and M2.