Improved Security Proofs in Lattice-Based Cryptography: Using the Rényi Divergence Rather Than the Statistical Distance

AbstractThe rates of quantum cryptographic protocols are usually expressed in terms of a conditional entropy minimized over a certain set of quantum states. In particular, in the device-independent setting, the minimization is over all the quantum states jointly held by the adversary and the parties that are consistent with the statistics that are seen by the parties. Here, we introduce a method to approximate such entropic quantities. Applied to the setting of device-independent randomness generation and quantum key distribution, we obtain improvements on protocol rates in various settings. In particular, we find new upper bounds on the minimal global detection efficiency required to perform device-independent quantum key distribution without additional preprocessing. Furthermore, we show that our construction can be readily combined with the entropy accumulation theorem in order to establish full finite-key security proofs for these protocols.

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Fuzzy Clustering Methods with Rényi Relative Entropy and Cluster Size

Mathematics ◽

10.3390/math9121423 ◽

2021 ◽

Vol 9 (12) ◽

pp. 1423

Author(s):

Javier Bonilla ◽

Daniel Vélez ◽

Javier Montero ◽

J. Tinguaro Rodríguez

Keyword(s):

Relative Entropy ◽

Fuzzy Clustering ◽

Cluster Size ◽

Computational Study ◽

Gaussian Kernel ◽

Clustering Methods ◽

Rényi Divergence ◽

Divergence Measures ◽

Fuzzy Clustering Methods ◽

Rényi Relative Entropy

In the last two decades, information entropy measures have been relevantly applied in fuzzy clustering problems in order to regularize solutions by avoiding the formation of partitions with excessively overlapping clusters. Following this idea, relative entropy or divergence measures have been similarly applied, particularly to enable that kind of entropy-based regularization to also take into account, as well as interact with, cluster size variables. Particularly, since Rényi divergence generalizes several other divergence measures, its application in fuzzy clustering seems promising for devising more general and potentially more effective methods. However, previous works making use of either Rényi entropy or divergence in fuzzy clustering, respectively, have not considered cluster sizes (thus applying regularization in terms of entropy, not divergence) or employed divergence without a regularization purpose. Then, the main contribution of this work is the introduction of a new regularization term based on Rényi relative entropy between membership degrees and observation ratios per cluster to penalize overlapping solutions in fuzzy clustering analysis. Specifically, such Rényi divergence-based term is added to the variance-based Fuzzy C-means objective function when allowing cluster sizes. This then leads to the development of two new fuzzy clustering methods exhibiting Rényi divergence-based regularization, the second one extending the first by considering a Gaussian kernel metric instead of the Euclidean distance. Iterative expressions for these methods are derived through the explicit application of Lagrange multipliers. An interesting feature of these expressions is that the proposed methods seem to take advantage of a greater amount of information in the updating steps for membership degrees and observations ratios per cluster. Finally, an extensive computational study is presented showing the feasibility and comparatively good performance of the proposed methods.

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Estimation of different entropies via Lidstone polynomial using Jensen-type functionals

Arabian Journal of Mathematics ◽

10.1007/s40065-020-00277-y ◽

2020 ◽

Vol 9 (3) ◽

pp. 613-631

Author(s):

Khuram Ali Khan ◽

Tasadduq Niaz ◽

Đilda Pečarić ◽

Josip Pečarić

Keyword(s):

Convex Function ◽

Shannon Entropy ◽

Rényi Divergence ◽

Lidstone Polynomial

Abstract In this work, some new functional of Jensen-type inequalities are constructed using Shannon entropy, f-divergence, and Rényi divergence, and some estimates are obtained for these new functionals. Also using the Zipf–Mandelbrot law and hybrid Zipf–Mandelbrot law, we investigate some bounds for these new functionals. Furthermore, we generalize these new functionals for m-convex function using Lidstone polynomial.

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Robust Bounds on Risk-Sensitive Functionals via Rényi Divergence

SIAM/ASA Journal on Uncertainty Quantification ◽

10.1137/130939730 ◽

2015 ◽

Vol 3 (1) ◽

pp. 18-33 ◽

Cited By ~ 18

Author(s):

Rami Atar ◽

Kenny Chowdhary ◽

Paul Dupuis

Keyword(s):

Rényi Divergence ◽

Risk Sensitive

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