k -Round Multiparty Computation from k -Round Oblivious Transfer via Garbled Interactive Circuits Fabrice Benhamouda Huijia (Rachel) Lin IBM Research / Columbia University, US University of California, Santa Barbara, US Eurocrypt 2018, May 1, 2018
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation (MPC) x 1 Auction Seller P 1 (no input) Buyer P i bids x i USD x 5 x 2 P 1 P 5 P 2 x 4 x 3 P 4 P 3 Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 2 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation (MPC) x 1 Auction Seller P 1 (no input) Buyer P i bids x i USD x 5 x 2 y 1 P 1 Seller P 1 gets y 1 = ( max bid, buyer ) Buyer P i gets � y 5 y 2 P 5 P 2 1 if winner y i = x 4 x 3 0 else y 4 y 3 P 4 P 3 Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 2 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation (MPC) x 1 Auction Seller P 1 (no input) Buyer P i bids x i USD x 5 x 2 y 1 P 1 Seller P 1 gets y 1 = ( max bid, buyer ) Buyer P i gets � y 5 y 2 P 5 P 2 1 if winner y i = x 4 x 3 0 else Question: How many rounds? y 4 y 3 P 4 P 3 Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 2 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation Adversarial Model Adversary can corrupt any party at the beginning semi-honest: corrupted parties behave honestly Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 3 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation Adversarial Model Adversary can corrupt any party at the beginning semi-honest: corrupted parties behave honestly malicious: corrupted parties can behave arbitrarily Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 3 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation Adversarial Model Adversary can corrupt any party at the beginning semi-honest: corrupted parties behave honestly semi-malicious: corrupted parties behave honestly but adaptively choose input and randomness malicious: corrupted parties can behave arbitrarily Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 3 / 23
Introduction Overview Round Collapsing via GIC FC with WE Secure Multiparty Computation Adversarial Model Adversary can corrupt any party at the beginning semi-honest: corrupted parties behave honestly semi-malicious: corrupted parties behave honestly but adaptively choose input and randomness malicious: corrupted parties can behave arbitrarily k -round semi-malicious MPC + NIZK ⇒ k -round malicious MPC Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 3 / 23
Introduction Overview Round Collapsing via GIC FC with WE Oblivious Transfer (OT) b ∈ { 0 , 1 } x 0 , x 1 . . . receiver sender Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 4 / 23
Introduction Overview Round Collapsing via GIC FC with WE Oblivious Transfer (OT) b ∈ { 0 , 1 } x 0 , x 1 . . . receiver sender x b ⊥ Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 4 / 23
Introduction Overview Round Collapsing via GIC FC with WE Oblivious Transfer (OT) b ∈ { 0 , 1 } x 0 , x 1 . . . receiver sender x b ⊥ k -round OT k -round MPC Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 4 / 23
Introduction Overview Round Collapsing via GIC FC with WE Oblivious Transfer (OT) b ∈ { 0 , 1 } x 0 , x 1 . . . receiver sender x b ⊥ k -round OT k -round MPC ? Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 4 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Semi-Honest Setting N : number of parties; L : number of rounds N L Assumptions [Yao82, Yao86] 2 k k -round OT Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 5 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Semi-Honest Setting N : number of parties; L : number of rounds N L Assumptions [Yao82, Yao86] 2 k k -round OT N O ( d ) O ( 1 ) -round OT [GMW87] O ( 1 ) O ( 1 ) -round OT [BMR90] N . . . Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 5 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Semi-Honest Setting N : number of parties; L : number of rounds N L Assumptions [Yao82, Yao86] 2 k k -round OT N O ( d ) O ( 1 ) -round OT [GMW87] O ( 1 ) O ( 1 ) -round OT [BMR90] N . . . N 2 CRS/... + LWE [AJLTVW12, MW16, CM15, BP16, PS16] [BGI16, BGI17, BGILT18] N 2 PKI + DDH 2 iO or WE [GGHR14, GP15, CGP15, DKR15, GLS15] N Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 5 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Semi-Honest Setting N : number of parties; L : number of rounds N L Assumptions [Yao82, Yao86] 2 k k -round OT N O ( d ) O ( 1 ) -round OT [GMW87] O ( 1 ) O ( 1 ) -round OT [BMR90] N . . . N 2 CRS/... + LWE [AJLTVW12, MW16, CM15, BP16, PS16] [BGI16, BGI17, BGILT18] N 2 PKI + DDH 2 iO or WE [GGHR14, GP15, CGP15, DKR15, GLS15] N [GS17a] N 2 bilinear group Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 5 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Semi-Honest Setting N : number of parties; L : number of rounds N L Assumptions [Yao82, Yao86] 2 k k -round OT N O ( d ) O ( 1 ) -round OT [GMW87] O ( 1 ) O ( 1 ) -round OT [BMR90] N . . . N 2 CRS/... + LWE [AJLTVW12, MW16, CM15, BP16, PS16] [BGI16, BGI17, BGILT18] N 2 PKI + DDH 2 iO or WE [GGHR14, GP15, CGP15, DKR15, GLS15] N [GS17a] N 2 bilinear group N k k -round OT [GS17b] k -round OT ours N k Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 5 / 23
Introduction Overview Round Collapsing via GIC FC with WE Results Theorem k -round MPC ‡ k -round OT ⇔ Corollary � � semi-honest semi-honest 2 -round MPC ‡ 2 -round OT ⇔ semi-malicious semi-malicious Corollary (using [AJLTVW12]) semi-malicious k -round OT + NIZK ⇒ malicious k -round MPC ‡ ∗ delayed semi-malicious security is sufficient; † for k ≥ 5 ‡ simultaneous messages, broadcast channel, static corruptions, with abort Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 6 / 23
Introduction Overview Round Collapsing via GIC FC with WE Results Theorem semi-honest semi-honest k -round MPC ‡ k -round OT ⇔ semi-malicious semi-malicious malicious ∗ malicious † Corollary � � semi-honest semi-honest 2 -round MPC ‡ 2 -round OT ⇔ semi-malicious semi-malicious Corollary (using [AJLTVW12]) semi-malicious k -round OT + NIZK ⇒ malicious k -round MPC ‡ ∗ delayed semi-malicious security is sufficient; † for k ≥ 5 ‡ simultaneous messages, broadcast channel, static corruptions, with abort Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 6 / 23
Introduction Overview Round Collapsing via GIC FC with WE Results Theorem semi-honest semi-honest k -round MPC ‡ k -round OT ⇔ semi-malicious semi-malicious malicious ∗ malicious † Corollary � � semi-honest semi-honest 2 -round MPC ‡ 2 -round OT ⇔ semi-malicious semi-malicious Corollary (using [AJLTVW12]) semi-malicious k -round OT + NIZK ⇒ malicious k -round MPC ‡ ∗ delayed semi-malicious security is sufficient; † for k ≥ 5 ‡ simultaneous messages, broadcast channel, static corruptions, with abort Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 6 / 23
Introduction Overview Round Collapsing via GIC FC with WE Results Theorem semi-honest semi-honest k -round MPC ‡ k -round OT ⇔ semi-malicious semi-malicious malicious ∗ malicious † Corollary � � semi-honest semi-honest 2 -round MPC ‡ 2 -round OT ⇔ semi-malicious semi-malicious Corollary (using [AJLTVW12]) semi-malicious k -round OT + NIZK ⇒ malicious k -round MPC ‡ ∗ delayed semi-malicious security is sufficient; † for k ≥ 5 ‡ simultaneous messages, broadcast channel, static corruptions, with abort Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 6 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Malicious Setting in the Plain Model N : number of parties; L : number of rounds Blackbox lower-bound: L ≥ 4 N L Assumptions [ACJ17] N 5 DDH N 4 subexp DDH [ACJ17] [BHP17] N 4 subexp LWE + adp. com. 4 ETDP + DDH/LWE or QR [HHPV17] N [BGJKKS17] N 4 DDH or QR or N-th res N k ≥ 5 k -round OT ours Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 7 / 23
Introduction Overview Round Collapsing via GIC FC with WE Previous Results Malicious Setting in the Plain Model N : number of parties; L : number of rounds Blackbox lower-bound: L ≥ 4 N L Assumptions [ACJ17] N 5 DDH N 4 subexp DDH [ACJ17] [BHP17] N 4 subexp LWE + adp. com. 4 ETDP + DDH/LWE or QR [HHPV17] N [BGJKKS17] N 4 DDH or QR or N-th res N k ≥ 5 k -round OT ours Open problem: 4-round MPC from 4-round OT Fabrice Benhamouda (IBM) k -Round MPC from k -Round OT Eurocrypt 2018 7 / 23
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