1. Efficient Multiparty Protocols via Log-Depth Threshold Formulae Ron Rothblum Weizmann Institute Joint work with Gil Cohen, Ivan Damgard, Yuval Ishai, Jonas Kolker, Peter Bro Miltersen and Ran Raz

2. Secure Multiparty Computation (MPC) [Yao86,GMW87] An adversary that controls a (limited) subset of the parties learns nothing more than the inputs and outputs of the parties it controls.

3. Feasibility Results: Perfect Security [BGW88,CCD88]

4. Our Contribution Huge body of work on secure MPC but protocols are fairly complicated. We suggest a conceptually simple and flexible approach to designing efficient MPC protocols. Building blocks: 1.Player emulation - builds on Hirt-Maurer [HM00] but with a different motivation. 2.Simple constant-party MPC protocols. 3.Threshold formulae composed of threshold gates.

5. Applications 1.Conceptually simple protocols for perfectly secure MPC – obtaining passive/active security. 2.New results on feasibility of MPC in a variety of settings, e.g., secure MPC over algebraic structures such as non-Abelian groups. 3.Distributed computing – broadcast/Byzantine agreement.

6. MPC via Player Emulation [HM00]

7. MPC via Player Emulation Minimal number of parties needed for security against one passive party

8. MPC with a Trusted Party 5 4 3 2 1

9. 5 4 3 2 1

10. 5 4 3 2 1 MPC via Player Emulation

11. 5 4 3 2 1

12. 5 4 3 2 1 The output is sent back to the parties.

13. MPC via Player Emulation

14. 5 4 3 2 1

15. 5 4 3 2 1

16. 5 4 3 2 1

17. 5 4 3 2 1

18. 5 4 3 2 1

19. Associate wires with parties and place 1 on input wires that the adversary controls. If output is 0 then the protocol is secure against this adversary.

20. MPC via Player Emulation

21. Comparison with [HM00]

22. MPC via Player Emulation

24. 3-Party Protocols Can use BGW restricted to 3 parties or better yet use the MPC made simple protocol of [Maurer02]. Maurers protocol is simple and elegant but exponential in the number of parties. For 3 parties– not an issue!

25. MPC via Player Emulation

26. Majority from Majorities

27. Active Security Minimal number of parties for security against one active party

28. Applications Simplifications*: 1.MPC over fields ([BGW88], [CCD88], [AL13]). 2.MPC over rings ([CFIK03]). 3.MPC over groups ([DPSW07,DPS+12a,DPS12+b]). 4.Distributed computing: broadcast, broadcast from 2-cast ([FM00]). * Caveats: non-optimal threshold and higher polynomial complexity.

29. Applications

30. Conclusions and Open Questions MPC methodology: 1.Design simple constant-party protocols. 2.Prove player emulation theorem. Intriguing connections to open questions in complexity-theory: 1.Explicit exact majority-from-majorities formula. 2.Exact threshold-from-thresholds formula (even non-explicit).

31. Thank you!