1. Secure Multi-party Computations (MPC) A useful tool to cryptographic applications Vassilis Zikas

2. Secure Multi-party Computations (MPC) The problem: There is given a set of parties (players, computers, authorites...) who want to do a joint computation but may not trust eachother!!! Example (The millionair ‘s problem): There are 2 millionairs who want to find out how is richer (without of course revealing eachother the exact ammount of money they own).

3. Secure Multi-party Computations (MPC) Obvious solution: Existence of a fully Trusted Party(TP) All players send their values to the TP The TP does the computation and sends each player what he is supposed to know Goal of MPC Simulate the TP (when such dosn‘t exist) via a protocol among the parties. 1

4. Secure Multi-party Computations (MPC) 1

5. Special case of MPC: Secure function evaluation(SFE): n players want to compute a function of their inputs whithout giving them away (actualy the function can output n values of which only the i-th should be known to the i-th player). e.g. a.e-voting (f=sum of votes) b.f:N n ! N n where p n learns only f n (x 1, ,x n )

6. Secure Multi-party Computations (MPC)

8. Difficulty??? Dishonest players (adversary)!!! Adversary types: 1.Pasive: All the corrupted players follow the protocol but the aversary can see averything they see. 2.Fail: The corrupted player might stop sending messages at some point of the execution. 3.Active: (Most general) The adversary can see what the corrupted players see, and he can force them to misbehave arbitrarily.

9. Secure Multi-party Computations (MPC)

10. Categories (according to the communication channels and the resources of the adversary) 1.Secure Channels Model: The parties communicate via secure authenticated channels Perfect (information-theoretic) security. Unconditional security (small error-probability) 1.Cryptographic model

11. Secure Multi-party Computations (MPC)

12. Not good when p 1 is corrupted

13. Secure Multi-party Computations (MPC)

19. Broadcast (definition): input: x 1, outputs: y 1, ,y n 1.(consistency): All honest players have the same output y. sender is honest all the honest players 2.(validity): If the sender is honest then all the honest players output x 1. 3.(termination): Every player ends with an output.

20. Secure Multi-party Computations (MPC) Consensus (Agreement) (definition): input: x 1, ,x n, outputs: y 1, ,y n 1.(consistency): All honest players have the same output y. all honest players have input x all the honest players 2.(validity): If the all honest players have input x then all the honest players output y=x. 3.(termination): Every player ends with an output.

21. Secure Multi-party Computations (MPC)

22. Secret sharing (thresshold case): Player p wants to share a secret s to players p 1, , p n in a way that the shares of any t players (put alltogether) give no information about s, the shares of t+1 players uniquely define s

23. Secure Multi-party Computations (MPC) Shamir ‘s secret sharing: Vector (a 1, ,a n ) is publicly known. Sharing phase: p chooses a random polynomial q( ¢ ) of degree t where the constant term is s (i.e. q(0)=s). p sends q(a i ) to player p i. Reconstruction phase: In order for p i to learn the secret s all player send him their shares and he applies Lagrange’s interpolation:

24. Secure Multi-party Computations (MPC)

26. MPC (secure channels - passive case) INVARIANT: The inputs and the results of the computations remain shared to the players throughout the protocol. 1. Inputs Sharing: Every player p i shares his input (Shamir’s SS Scheme) using a random polynomial q i ( ¢ ). 2. Computation: i.Addition: Can be done without interaction locally. ii.Multiplication: (BOARD) 3. Reconstruction (towards p j ) All players send their shares of the output to p j and he does the reconstruction

27. Secure Multi-party Computations (MPC) When active adversaries are considered SS is not enough (why?) we need Verifiable SS!!! Difference: The dealer is committed to the value he shares (therefore verifiable) All players are committed to the values they ‘ve recieved

28. Secure Multi-party Computations (MPC) Mixed (Active+Passive+Fail) Model: There is an MPC protocol for any spacification iff 3t a +2t p +t f <n

29. Secure Multi-party Computations (MPC) General Adversaries: Adversary structure Z={(A i,P i,F i )} A i ={set of players that can be actively corrupted by adversary Z i } P i, F i similar defined Z is a monotone set Z can be characterized by the class of maximal sets (Base of Z ( )). We will consider on Active + Passive corruption for the general adversaries

30. Secure Multi-party Computations (MPC)

31. Results for General Adversaries: (secure channels model) MPC (Perfect security)Q (3,2) MPC (Unconditional security) BC is given Q (2,2) MPC (Unconditional security) Q (2,2) Æ Q (3,0)