## Simple and Tight Bounds for Information Reconciliation and Privacy Amplification

### Renato Renner and Stefan Wolf

Shannon entropy is a useful and important measure in information processing, for instance, data compression or randomness extraction, under the assumptionwhich can typically safely be made in communication theorythat a certain random experiment is independently repeated many times. In cryptography, however, where a systems working has to be proven with respect to a malicious adversary, this assumption usually translates to a restriction on the latters knowledge or behavior and is generally not satisfied. An example is quantum key agreement, where the adversary can attack each particle sent through the quantum channel differently or even carry out coherent attacks, combining a number of particles together. In information-theoretic key agreement, the central functionalities of information reconciliation and privacy amplification have, therefore, been extensively studied in the scenario of general distributions: Partial solutions have been given, but the obtained bounds are arbitrarily far from tight, and a full analysis appeared to be rather involved to do. We show that, actually, the general case is not more difficult than the scenario of independent repetitionsin fact, given our new point of view, even simpler. When one analyzes the possible efficiency of data compression and randomness extraction in the case of independent repetitions, then Shannon entropy H is the answer. We show that H can, in these two contexts, be generalized to two very simple quantities–called smooth Rényi entropies-which are tight bounds for data compression (hence, information reconciliation) and randomness extraction (privacy amplification), respectively. It is shown that the two new quantities, and related notions, do not only extend Shannon entropy in the described contexts, but they also share central properties of the latter such as the chain rule as well as sub-additivity and monotonicity.

## BibTeX Citation

@inproceedings{RenWol05, author = {Renato Renner and Stefan Wolf}, title = {Simple and Tight Bounds for Information Reconciliation and Privacy Amplification}, editor = {Bimal Roy}, booktitle = {Advances in Cryptology --- ASIACRYPT 2005}, pages = {199--216}, series = {Lecture Notes in Computer Science}, volume = {3788}, year = {2005}, month = {12}, publisher = {Springer-Verlag}, }