Error-free perfect-secrecy systems

Shannon´s fundamental bound for perfect secrecy says that the entropy of the secret message U cannot be larger than the entropy of the secret key R shared by the sender and the legitimate receiver. Massey gave an information theoretic proof of this result and the proof does not require U and R to be independent. By adding an extra assumption that I(U; R) = 0, we show a tighter lower bound on H(R) by proving that the logarithm of the message sample size cannot be larger than the entropy of the secret key. Then we consider that a perfect secrecy system is used multiple times. A new parameter, namely effective key consumption, is defined and justified. This paper shows the existence of a fundamental tradeoff between the effective key consumption and the number of channel uses for transmitting a ciphertext.