Error Free Perfect Secrecy Systems

TL;DR

This paper refines the bounds on key entropy for perfect secrecy systems, introduces a new measure for key consumption, and designs optimal secure coding schemes considering multiple uses of the cipher.

Contribution

It provides a tighter lower bound on key entropy under independence assumptions, and proposes a new key consumption rate measure for repeated cipher use.

Findings

Lower bound on key entropy equals log of message sample size.

Introduction of a key consumption rate measure.

Design of optimal and nearly optimal secure codes.

Abstract

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.