A Convex Analysis Approach to Computational Entropy

This paper studies the notion of computational entropy. Using techniques from convex optimization, we investigate the following problems: 1. Can we derandomize the computational entropy? More precisely, for the computational entropy, what is the real difference in security defined using the three important classes of circuits: deterministic boolean, deterministic real valued, or (the most powerful) randomized ones? 2. How large the difference in the computational entropy for an unbounded versus efficient adversary can be? 3. Can we obtain useful, simpler characterizations for the computational entropy? The first question was answered affirmatively for the most important notion of HILL entropy but was open for the metric-type computational entropy, widely used in the leakage-resilent cryptography. In this case, we show that the answer depends on what is the underlying variant of the information-theoretic entropy in the definition of the metric entropy. More precisely, the answer is negative for the commonly used min-entropy based computational entropy. Surprisingly, we show that for all other Renyi entropies the answer is positive security given by unbounded deterministic circuits can be still much worse than that guaranteed by efficient randomized circuits. In the second problem, we obtain some lower-bound type results. Especially, considering conditional computational entropy for two random variables X ∈ {0, 1} and Z ∈ {0, 1}, we show that even if the security parameters are exponential in n+m, the ammount of entropy can be still noticeably higher than that seen by unbounded adversary. Also, for a fixed distribution, decreasing the security parameters by a factor 2 , can result in increasing the entropy by C bits, which agrees with intuition. Studying the third problem, we derive a series of lemmas giving a characterization of the metric entropy for various definitions. As an example of application, we give extremely simple proofs of leakage lemmas, being a central tool in the leakage-resilent cryptography.