1. Secure Computation and the Combinatorics of Hidden Diversity Juan Garay ( AT&T Research) David Johnson (AT&T Research) Aggelos Kiayias (U. Athens) Moti Yung (Google)

2. Hidden Diversity and Secure Multiparty Computation Resource-based Corruptions  Adversaries corrupt parties... …for FREE! ProverVerifier

3. Hidden Diversity and Secure Multiparty Computation Resource-based Corruptions (cont’d)  How much does corruption cost? Different parties may require different “resources” to get corrupted  Can “anonymity” be used to raise those costs? Our new questions:

4. A focal point : Corruption diversity  Given that corruptions happen in different ways and based on different parameters, they can require a different amount of resources How to model corruption diversity?

5. Resource-based corruptions s1s1 s2s2 s3s3 s4s4 s5s5 Budget b (with “tokens”) Adversary’s Goal :

6. Hidden Diversity and Indistinguishability Suppose different parties require different resources for corruption but externally appear the same s2s2 s3s3 s4s4 s5s5 s1s1 ? Adversary will need to waste more resources for subverting the system!

7. A Combinatorial Game  GIVEN : Set B 1, B 2, …, B n of buckets, with bucket B i having non-negative integer size s i, and a target fraction α, 0 < α < 1.  GOAL : Fill  αn  of the buckets using as few balls as possible, where a bucket of size s i is filled if it receives s i balls. Hidden Diversity and Secure Multiparty Computation

8. n = 5, α = ½,  αn  = 3 Balls and Buckets (cont’d) Hidden Diversity and Secure Multiparty Computation

9. Only Feedback from Placing a Ball: “Bucket Now Full” or “Bucket Not Yet Full” Balls and Buckets (cont’d) Hidden Diversity and Secure Multiparty Computation How many balls?

10. Hidden Diversity and Secure Multiparty Computation In this work  Framework for realization of above abstraction Computational corruptions  Sufficient conditions for abstraction Information-Effort-Preserving (IEP) functions Hardness Indistinguishability Exact Hardness

11. Hidden Diversity and Secure Multiparty Computation Candidate Functions  Random oracle  Exponentiation f : Z q → S; q: λ- bit prime number; S: (generic) multiplicative group  Multiplication f mult : P λ x P λ → N

12. Hidden Diversity and Secure Multiparty Computation In this work  Framework for realization of above abstraction Computational corruptions  Sufficient conditions for abstraction Information-Effort-Preserving (IEP) functions Hardness Indistinguishability Exact Hardness  Much is to be gained : MPC Security : unbounded additional adversarial effort Efficiency : force corruption threshold to drop from 1/2 to 1/3, and run information-theoretic MPC protocol

13. Thanks!