Revisiting Conditional Rényi Entropies and Generalizing Shannon’s Bounds in Information Theoretically Secure Encryption

Information theoretic cryptography is discussed based on conditional Rényi entropies. Our discussion focuses not only on cryptography but also on the definitions of conditional Rényi entropies and the related information theoretic inequalities. First, we revisit conditional Rényi entropies, and clarify what kind of properties are required and actually satisfied. Then, we propose security criteria based on Rényi entropies, which suggests us deep relations between (conditional) Rényi entropies and error probabilities by using several guessing strategies. Based on these results, unified proof of impossibility, namely, the lower bounds of key sizes is derived based on conditional Rényi entropies. Our model and lower bounds include the Shannon’s perfect secrecy, and the min-entropy based encryption presented by Dodis, and Alimomeni and Safavi-Naini. Finally, a new optimal symmetric key encryption is proposed which achieve our lower bounds.