Error Free Perfect Secrecy Systems

Shannon's fundamental bound for perfect secrecy says that the entropy of the secret message cannot be larger than the entropy of the secret key initially shared by the sender and the legitimate receiver. Massey gave an information theoretic proof of this result, however this proof does not require independence of the key and ciphertext. By further assuming independence, we obtain a tighter lower bound, namely that the key entropy is not less than the logarithm of the message sample size in any cipher achieving perfect secrecy, even if the source distribution is fixed. The same bound also applies to the entropy of the ciphertext. The bounds still hold if the secret message has been compressed before encryption. This paper also illustrates that the lower bound only gives the minimum size of the pre-shared secret key. When a cipher system is used multiple times, this is no longer a reasonable measure for the portion of key consumed in each round. Instead, this paper proposes and justifies a new measure for key consumption rate. The existence of a fundamental tradeoff between the expected key consumption and the number of channel uses for conveying a ciphertext is shown. Optimal and nearly optimal secure codes are designed.Comment: Submitted to the IEEE Trans. Info. Theor

The Kraft inequality for EPS systems

It is a well known result that the Kraft inequality is a necessary and sufficient condition for the existence of a uniquely decodable code. This paper provides an inequality which is a counterpart of the Kraft inequality in Error free Perfect Secrecy (EPS) system. Our inequality is a necessary and sufficient condition for the existence of an EPS system. It also illustrates some necessary and sufficient conditions for an EPS system to achieve the minimal expected key consumption.