Entropic security and the encryption of high entropy messages

• Authors: Yevgeniy Dodis;Adam Smith
• Venue: TCC'05 Proceedings of the Second international conference on Theory of Cryptography
• Year: 2005

We study entropic security, an information-theoretic notion of security introduced by Russell and Wang [24] in the context of encryption and by Canetti et al. [5,6] in the context of hash functions. Informally, a probabilitic map $Y = \mathcal{E}(X)$ (e.g., an encryption sheme or a hash function) is entropically secure if knowledge of Y does not help predicting any predicate of X, whenever X has high min-entropy from the adversary's point of view. On one hand, we strengthen the formulation of [5,6,24] and show that entropic security in fact implies that Y does not help predicting any function of X (as opposed to a predicate), bringing this notion closer to the conventioonal notion of semantic security [10]. On the other hand, we also show that entropic security is equivalent to indistinguishability on pairs of input distributions of sufficiently high entropy, which is in turn related to randomness extraction from non-uniform distributions [21]. We then use the equivalence above, and the connection to randomness extraction, to prove several new results on entropically-secure encryption. First, we give two general frameworks for constructing entropically secure encryption schemes: one based on expander graphs and the other on XOR-universal hash functions. These schemes generalize the schemes of Russell and Wang, yielding simpler constructions and proofs, as well as improved parameters. To encrypt an n-bit message of min-entropy t while allowing at most ε-advantage to the adversary, our best schemes use a shared secret key of length $k = n - t + 2{\rm log} (\frac {1}{\epsilon})$. Second, we obtain lower bounds on the key length k for entropic security and indistinguishability. In particular, we show near tightness of our constructions: k n – t. For a large class of schemes — including all the schemes we study — the bound can be strengthened to $k \geq n - t+{\rm log} (\frac {1}{\epsilon})-O(1)$.