What does signal-to-noise ratio mean in the context of differential cryptanalysis, and how can one use it to derive the number of required plaintext pairs to carry out a successful attack?
The point of differential cryptanalysis is to find a differential characteristic with sufficiently high probability. You go through plaintexts until you find pairs that give you "enough" of a difference. You know only plaintext and ciphertext difference and you do not know difference between individual rounds, because cipher is a black box. (this is called filtration btw)
You use these pairs as a start for "key-recovery" by exploring the probabilities into possible subkeys in order to try to correlate the pairs. Every pair suggests several subkeys. Good pair suggests exactly one good subkey and few wrong subkeys. How many times the correct subkey is more frequent than other subkeys is signal to noise (S/N) ratio.