Amplifying privacy in privacy amplification

We study the classical problem of privacy amplification, where two parties Alice and Bob share a weak secret X of min-entropy k, and wish to agree on secret key R of length m over a public communication channel completely controlled by a computationally unbounded attacker Eve. Despite being extensively studied in the literature, the problem of designing "optimal" efficient privacy amplification protocols is still open, because there are several optimization goals. The first of them is (1) minimizing the entropy loss L = k - m. Other important considerations include (2) minimizing the number of communication rounds, (3) maintaining security even after the secret key is used (this is called post-application robustness), and (4) ensuring that the protocol P does not leak some "useful information" about the source X (this is called source privacy). Additionally, when dealing with a very long source X, as happens in the so-called Bounded Retrieval Model (BRM), extracting as long a key as possible is no longer the goal. Instead, the goals are (5) to touch as little of X as possible (for efficiency), and (6) to be able to run the protocol many times on the same X, extracting multiple secure keys. Achieving goals (1)-(4) (or (2)-(6) in BRM) simultaneously has remained open. In this work we improve upon the current state-of-the-art, by designing a variety of new privacy amplification protocols, thereby achieving the following goals for the first time: - 4-round (resp. 2-round) source-private protocol with optimal entropy loss L = O(λ), whenever k = Ω(λ²) (resp. k > n/2(1-α) for some universal constant α > 0) . - 3-round post-application-robust protocols with optimal entropy loss L = O(λ), whenever k = Ω(λ²) or k > n/2(1-α) (the latter is also source-private). - The first BRM protocol capable of extracting the optimal number Θ(k/λ) of session keys, improving upon the previously best bound Θ(k/λ²). (Additionally, our BRM protocol is post-application-robust, takes 2 rounds, and can be made source-private by increasing the number of rounds to 4.).