Overcoming Weak Expectations via the Rényi Entropy and the Expanded Computational Entropy

In the ideal world, cryptographic models take for granted that the secret sources (e.g. secret keys and other secret randomness) are derived from uniform distribution. However, in reality, we may only obtain some ‘weak’ random sources guaranteed with high unpredictability (e.g. biometric data, physical sources, and secrets with partial leakage). Formally, the security of cryptographic models is measured by the expectation of some function, called ‘perfect’ expectation in the ideal model and ‘weak’ expectation in the real model respectively. We propose some elementary inequalities which show that the ‘weak’ expectation is not much worse than the ‘perfect’ expectation. Instead of discussing the results based on the min-entropy and collision entropy by Dodis and Yu [TCC 2013], we present how to overcome weak expectations dependent on the Rényi entropy and the expanded computational entropy. We achieve these results via employing the discrete form of the Hölder inequality. We also use some techniques to guarantee that the expanded computational entropy is useful in the security model. Thus our results are more general, and we also obtain some results from a computational perspective. The results apply to all ‘unpredictability’ applications and some indistinguishability applications including CPA-secure symmetric-key encryption schemes, weak Pseudorandom Functions andWeaker Computational Extractors.