Correlated random variables are a key tool in cryptographic applications like secure multi-party computation. We investigate the power of a class of correlations that we term group correlations: A group correlation is a uniform distribution over pairs (x,y) ∈ G² such that x+y ∈ S, where G is a (possibly non-abelian) group and S is a subset of G. We also introduce bi-affine correlation{s}, and show how they relate to group correlations. We present several structural results, new protocols and applications of these correlations. The new applications include a completeness result for black box group computation, perfectly secure protocols for evaluating a broad class of black box "mixed-groups" circuits with bi-affine homomorphisms, and new information-theoretic results. Finally, we uncover a striking structure underlying OLE: In particular, we show that OLE over 𝔽_{2ⁿ}, is isomorphic to a group correlation over ℤ_4^n.
@InProceedings{policharla_et_al:LIPIcs.ITC.2021.1, author = {Policharla, Guru-Vamsi and Prabhakaran, Manoj and Raghunath, Rajeev and Vyas, Parjanya}, title = {{Group Structure in Correlations and Its Applications in Cryptography}}, booktitle = {2nd Conference on Information-Theoretic Cryptography (ITC 2021)}, pages = {1:1--1:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-197-9}, ISSN = {1868-8969}, year = {2021}, volume = {199}, editor = {Tessaro, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITC.2021.1}, URN = {urn:nbn:de:0030-drops-143208}, doi = {10.4230/LIPIcs.ITC.2021.1}, annote = {Keywords: Group correlations, bi-affine correlations, secure computation} }
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