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We study the security of a variety of cryptographic tasks including traditional privacy (e.g., seeded extractors, encryptions, commitments, and secret sharing schemes) and differential privacy from the perspective of    $alpha $   -mutual information. As far as we know, encryption scheme, commitment, and differential privacy have been studied via mutual information based on the Shannon entropy. Though Bellare  et al.  in CRYPTO 2012 have got some results about encryption schemes, the upper bound of mutual information is not the tightest. Though Cuff and Yu in CCS 2016 mentioned the direction of the Renyi entropy generalization, only a few results about differential privacy were obtained, and even for Shannon entropy, the proof in that paper has some limitations. In this paper, we propose a modular and unified framework to study the relations between statistical security and mutual information security for a series of privacy schemes other than prior work that focused on a special scheme. In addition, we introduce    $alpha $   -mutual information security via the Renyi entropy for a series of privacy schemes and aim to bridge the gap between statistical security and    $alpha $   -mutual information security. By resorting to an improved upper bound on the difference between the Shannon entropy of two distributions, the convexity of a function, useful equality about statistical distance, and the absolutely homogeneous property of    $alpha $   -norm, we obtain rigorous proofs of their essential equivalence. An extra fruit is that the relations between mutual information security and statistical security of encryption and commitment schemes are improved. Hence, two fundamentally different ways of defining privacy security are connected.

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A Generalized Constraint of Privacy: &lt;inline-formula&gt; &lt;tex-math notation="LaTeX"&gt;$\alpha$ &lt;/tex-math&gt; &lt;/inline-formula&gt;-Mutual Information Security