Computational information theory

What is information? In a fundamental sense, Shannon’s definition of entropy captures the notion of information in situations where unlimited computing power is always available. As a result, in applications such as cryptography, where computational cost plays a central role, the classical information theory does not provide a totally satisfactory framework. In recent years, after Diffie and Hellman proposed the use of trapdoor function as the cornerstone for a new genre of cryptography, this deficiency is particularly dramatized; a ciphertext contains all the Shannon information about the plaintext, yet this information is ‘inaccessible’, i.e., it cannot be efficiently computed. This begs the challenging question ‘what is accessible information?’ Can we combine two very successful theories, namely, Information Theory and Computational Complexity Theory, to capture the notion of accessible information? In this chapter, we will give an exposition of a new information theory along this line and examine its applications in cryptography.