General Cryptographic Protocols (aka secure multi-party computation) Oded Goldreich Weizmann Institute of Science

(and me) Joachim (and Claus)

A general framework (for casting crypto problems) An m-ary (randomized) functionality (desired process) F:({0,1} n ) m → ({0,1} n ) m (where m  2 denotes the # of parties). P 1 P 2 P m x 1 x 2 x m (local inputs) y 1 y 2 y m (local outputs) (y 1,y 2,…,y m ) = F(x 1,x 2,…,x m ) Desired solution: delivery of outputs as if the operation was performed by a trusted party.

Secure Multi-Party Computation (Crypto Protocols) A secure protocol obtains the same effect as the operation of a trusted party. Thus, mutually distrustful parties emulate the effect of a trusted party.

On the feasibility of General Secure MPC Meta-THM: General Secure MPC is possible under a variety of natural assumptions. Assuming an honest majority + TDP Allowing abort + TDP (i.e., not considering early termination as breach of security) [reflected in the ideal model] Assuming a 2/3-majority + private channels. TDP == Trapdoor Permutations (which exist, e.g., assuming the intractability of factoring integers).

Two-Step construction of General Secure MPC E.g., assuming an honest majority + TDP 1.Constructing protocols that are secure wrt semi-honest (“honest-but-curious”) adversaries. [“privacy only”] 2.Enforcing semi-honest behavior via ZK proofs (+commit) T = public information (transcript) Sender (secret input s) Receiver Supposed to send y = f(T,s) y’ Idea: provide a ZK proof that  s’ s.t y’=f(T,s’) Step 2:enforcing

Secure (private) MPC in the semi-honest model. We assume a TDP (trapdoor permutation). Reduce to deterministic functionalities with same outputs. Let C be a GF(2) circuit for computing the m-ary function. Idea: The parties propagate shares of the values of all wires in C from the input wires of C to its output wires. xy z = z 1 +z 2 +z 3 +… +z m x 1 x 2 x 3 x m y 1 y 2 y 3 y m z 1 z 2 z 3 z m (x = x 1 +x 2 +x 3 +… +x m y = y 1 +y 2 +y 3 +… +y m )

Secure (private) MPC of the gate functionality. xy z = z 1 +z 2 +z 3 +… +z m x 1 x 2 x 3 x m y 1 y 2 y 3 y m z 1 z 2 z 3 z m (x = x 1 +x 2 +x 3 +… +x m y = y 1 +y 2 +y 3 +… +y m ) Easy case – addition gate: Set z i  x i +y i (local computation). Similarly for negation: z i  x i +1 if i=1 and z i  x i o.w. Hard case – multiplication gate: we wish z 1 +z 2 +… +z m = (x 1 +x 2 +… +x m ) ∙ (y 1 +y 2 +… +y m ) (use algebra) (x 1 +x 2 +… +x m ) ∙ (y 1 +y 2 +… +y m ) = ∑ i x i y i + ∑ i≠j (x i y j +x j y i ) local 2PC The parties need to propagate shares of the values through each gate. (Shares with subscript i belong to party i.)

Secure 2-PC of s.t. Recall: General secure MPC “reduces” to secure 2PC of ((x 1,y 1 ),(y 2,x 2 )) → (z 1,z 2 ), where (z 1,z 2 ) is random subject to z 1 +z 2 = x 1 x 2 +y 2 y 1. Sender Receiver Inputs : s 0,s 1 c Outputs : - s c 1st 2nd Inputs : x 1,y 1 x 2,y 2 Outputs : r r+x 1 x 2 +y 1 y 2 1st 2nd Inputs : x,z y Outputs : - z+xy In the i-th invocation use inputs (x i,r i ) and y i, where r i is a random bit. Each party sets its final output = sum of both intermediate outputs. (OT) Sender sets s y = z+yx.

Implementing OT (OT = Oblivious Transfer) Sender Receiver Inputs : s 0,s 1 c Outputs : - s c Background: assuming a collection of TDP {f i :D i →D i } Sender Receiver Inputs: s 0,s 1 c desired outputs: - s c selects an index i select x c,y 1-c  D i compute y c =f i (x c ) find the f i -preimages of both: z 0, z 1, and send b(z 0 )+s 0, b(z 1 )+s 1 y 0, y 1

Conclusion: General Secure MPC is feasible Meta-THM: General Secure MPC (i.e., secure emulation of trusted parties) is possible under a variety of natural assumptions. MPC for an honest majority, assuming TDP Similar ideas (+more) yield MPC wo honest majority, but when “allowing abort” (i.e., not considering early termination as breach of security). ( Also assuming TDP). Assuming a 2/3-majority + private channels.

The End The slides of this talk are available at A related survey is available at

Zero-Knowledge Proofs A secure protocol (i.e., ZK proof) obtains the same effect as the operation of a trusted party. Thus, mutually distrustful parties emulate the effect of a trusted party.

Secure 2-PC of the Inner Product mod 2 of two vectors Recall: General secure MPC “reduces” to secure 2PC of the inner product mod 2 of two input vectors held by the two parties. (For us n=2 suffices.) Sender Receiver Inputs : s 0,s 1 c Outputs : - s c 1st 2nd Inputs : x 1,…,x n y 1,…,y n Outputs : r r+∑ i x i y i 1st 2nd Inputs : x,z y Outputs : - z+xy In the ith invocation use inputs (xi,ri) and yi, where ri is a random bit. Final output = sum of all n outputs.