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On Fair Exchange, Fair Coins and Fair Sampling Shashank Agrawal, Manoj Prabhakaran University of Illinois at Urbana-Champaign

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Fairness A secure multi-party protocol has properties like correctness, privacy of inputs. Fairness: An intuitive property desirable of secure protocols. Adversary cannot prevent honest parties from obtaining the output of computation, if he also obtains it. Ideal world: Functionality gives output to all the parties (or none of them). Finite two party functionalities. Input, output size does not depend on the security parameter.

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Motivation Wide interest in the problem of fairness. Understanding of fundamental primitives lacking. In this work, we study the relationship between Fair Exchange, Fair Coin-flipping, Fair Random-OT. Given access to a fair primitive, can we realize another fair primitive.

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AB EXCH xy yx With input Functionalities

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Sampling functionalities A B COIN b b AB R-OT Input-less AB

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Functionalities with Fair protocols COIN EXCH R-OT Any non-trivial joint distribution Cleve 1986 Zero common information X X X Our Results

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Related Work Cleve 1986: No efficient protocol for fair coin- flipping. A simple fail-stop attack. Even under computational assumptions. Any functionality of interest likely not realizable. Gordon et al. showed AND, OR, Yao’s millionaire problem have fair protocol [GHKL08] Led to a flurry of results [MNS09, BOO10, GIMOS10, GK10, BLOO11, ALR13] Landscape more complicated than unfair computation E.g. no finite complete function [GIMOS10]

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Functionalities with Fair protocols COIN EXCH R-OT X X X XOR Cleve 1986 Our Results

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COIN functionality AB COIN b b Agreement: Alice and Bob output the same bit (if nobody corrupt). Entropy: Honest party outputs a random bit.

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Alice Bob............

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XOR from COIN Assume: Alice and Bob choose inputs randomly Invalid input is substituted by a default one. Agreement: Alice and Bob output the same bit. Entropy: Honest party outputs a random bit. AB XOR Theorem: Even with access to COIN, XOR can’t be realized.

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Alice Bob COIN bb

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Functionalities with Fair protocols COIN EXCH R-OT X X X Zero common information Our Results

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Common Information AB XY Q Q

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Characteristic bipartite graph 0 0 11 COIN 00 01 10 11 R-OT Zero CI1-bit CI

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COIN from R-OT

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Graph Products 00 01 10 11 R-OT 1- sample 0…0 R-OT n-samples 0…0...... 1…1 0 0 1 1

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COIN from R-OT Lower-bounding the second eigenvalue of the Laplacian associated with the graph product. No matter how many samples from R-OT, the weight on edges going across is a constant.

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Functionalities with Fair protocols COIN EXCH R-OT X X X Conclusion

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Thank you. Questions?

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