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Oblivious Transfer and Secure Multiparty Computation

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Oblivious Transfer and Secure Multiparty Computation Oblivious Transfer and Secure Multiparty Computation Brett Hemenway September 11th 2013 Introduction Oblivious Transfer Circuit Garbling Secure Computation from Secret Sharing Applications of MPC The Satellite Problem Conjunction Analysis: Overview Specific Calculations Integration Doing The Computation Securely Extras Cryptographic Assumptions Constructing Oblivious Transfer The Millionaire's Problem An example of secure two-party computation I Two Millionaires want to determine who is richer, without revealing their wealth [Yao82] I They want a secure computation of a comparison (<) gate I More generally, how can two (or more) parties compute a function while keeping their inputs private? Solving the Millionaire's Problem I If there is a family of one-way trapdoor permutations, F, f : X ! Y for f 2 F e.g. f (x) = xe mod N (the RSA function) I and a hash function H : X ! Z I Then there is a simple solution to the Millionaire's Problem I We assume Alice's wealth is a, Bob's is b, and there is an a priori upper bound m > max(a; b). BobAlice fr gm−1 $ i fi=0 −1 x X y a xi = f (y + i) for i = 0;:::; m − 1 b y = f (x) − b x0 ::: xb ::: xm−1 if H(x) = rb then a > b $ −1 x X−1 xb = f (y + b) = f ((f (x) − b) + b) = x if H(xy)=6=fr(bxthen) − ba ≤ b H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 y fri gi=0 r0 ::: rb ::: rm−1 a > b ) rb = H(x) Solving the Millionaire's Problem Alice a f ; f −1 $ F f AliceBob fr gm−1 $ i i=0 −1 x X y a xi = f (y + i) for i = 0;:::; m − 1 b y = f (x) − b x0 ::: xb ::: xm−1 if H(x) = rb then a > b −1 −−11 $ xb = f (y + b)f =; ff ((fF(x) − b) + b) = x if H(x) 6= rb then a ≤ b H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 fri gfi=0 r0 ::: rb ::: rm−1 a > b ) rb = H(x) Solving the Millionaire's Problem Bob f b x $ X y = f (x) − b y AliceBob m−1 $ fri gfi=0 a x X b y = f (x) − b if H(x) = rb then a > b $ −1 x −X−11 $ xb = f (y + b)f =; ff ((fF(x) − b) + b) = x if H(xy)=6=fr(bxthen) − ba ≤ b H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 y fri gfi=0 r0 ::: rb ::: rm−1 a > b ) rb = H(x) Solving the Millionaire's Problem Alice −1 y a xi = f (y + i) for i = 0;:::; m − 1 x0 ::: xb ::: xm−1 AliceBob m−1 $ fri gfi=0 a x X b y = f (x) − b if H(x) = rb then a > b $ xf ; f −X1 $ F if H(xy)=6=fr(bxthen) − ba ≤ b H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 y fri gfi=0 r0 ::: rb ::: rm−1 a > b ) rb = H(x) Solving the Millionaire's Problem Alice −1 y a xi = f (y + i) for i = 0;:::; m − 1 x0 ::: xb ::: xm−1 −1 −1 xb = f (y + b) = f ((f (x) − b) + b) = x AliceBob m−1 $ fri gfi=0 a x X b y = f (x) − b if H(x) = rb then a > b $ −1 x −X−11 $ xb = f (y + b)f =; ff ((fF(x) − b) + b) = x if H(xy)=6=fr(bxthen) − ba ≤ b m−1 y fri gfi=0 Solving the Millionaire's Problem Alice −1 y a xi = f (y + i) for i = 0;:::; m − 1 x0 ::: xb ::: xm−1 H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 r0 ::: rb ::: rm−1 a > b ) rb = H(x) AliceBob m−1 $ fri gfi=0 a x X b y = f (x) − b if H(x) = rb then a > b $ −1 x −X−11 $ xb = f (y + b)f =; ff ((fF(x) − b) + b) = x if H(xy)=6=fr(bxthen) − ba ≤ b y f a > b ) rb = H(x) Solving the Millionaire's Problem Alice −1 y a xi = f (y + i) for i = 0;:::; m − 1 x0 ::: xb ::: xm−1 H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 fri gi=0 r0 ::: rb ::: rm−1 AliceBob f −1 y a xi = f (y + i) for i = 0;:::; m − 1 b x0 ::: xb ::: xm−1 $ −1 x −X−11 $ xb = f (y + b)f =; ff ((fF(x) − b) + b) = x y = f (x) − b H(xi + 1) for i = 0;:::; a ri = H(xi ) for i = a + 1;:::; m − 1 m−1 y fri gfi=0 r0 ::: rb ::: rm−1 a > b ) rb = H(x) Solving the Millionaire's Problem Bob m−1 fri g $ i=0 x X b y = f (x) − b if H(x) = rb then a > b if H(x) 6= rb then a ≤ b Secure Multiparty Computation I Cryptographic tools exist that allow a group of participants to securely calculate any function of their joint inputs. I Cryptography removes the need for a trusted third party Privacy is defined in a simulation paradigm I A protocol is secure if there is a simulator that, when given only the output of the protocol can simulate an execution of the protocol that is indistinguishable from the real protocol. I This ensures that nothing beyond the output of the protocol is learned I In the Millionaire's Problem revealing whose salary is higher leaks information. A secure protocol should leak nothing more. Standard technique: first build protocols in the Semi-Honest model. Then use standard tools (e.g. Zero-Knowledge proofs) to force participants to follow the protocol Security Models Definition (Semi-Honest Adversaries) Semi-Honest (Honest-But-Curious) adversaries I always follow whatever protocol they are asked to perform I always send well-formed messages I try to learn other participants' secrets by looking at their own transcript in the protocol Definition (Malicious Adversaries) Malicious adversaries are: I allowed to deviate from the protocol I allowed to send mal-formed messages I allowed to behave in any way Security Models Definition (Semi-Honest Adversaries) Semi-Honest (Honest-But-Curious) adversaries I always follow whatever protocol they are asked to perform I always send well-formed messages I try to learn other participants' secrets by looking at their own transcript in the protocol Definition (Malicious Adversaries) Malicious adversaries are: I allowed to deviate from the protocol I allowed to send mal-formed messages I allowed to behave in any way Standard technique: first build protocols in the Semi-Honest model. Then use standard tools (e.g. Zero-Knowledge proofs) to force participants to follow the protocol Methods of Secure MPC Protocol Assumption Players Reference Yao's Garbled Circuit OT 2 [Yao82, Yao86] GMW OT 2+ [GMW87] BGW/CCD Honest Majority 3+ [BOGW88, CCD88] FHE Lattice Problems 2+ [Gen09] Methods of Secure MPC Protocol Assumption Players Reference Yao's Garbled Circuit OT 2 [Yao82, Yao86] GMW OT 2+ [GMW87] BGW/CCD Honest Majority 3+ [BOGW88, CCD88] FHE Lattice Problems 2+ [Gen09] Introduction Oblivious Transfer Circuit Garbling Secure Computation from Secret Sharing Applications of MPC The Satellite Problem Conjunction Analysis: Overview Specific Calculations Integration Doing The Computation Securely Extras Cryptographic Assumptions Constructing Oblivious Transfer I S learns nothing about b I R learns nothing about X1−b Oblivious Transfer Sender Receiver x0 b OT x1 xb Oblivious Transfer Sender Receiver x0 b OT x1 xb I S learns nothing about b I R learns nothing about X1−b Facts About OT I Introduced by Rabin [Rab81], Even, Goldreich and Lempel [EGL85] I OT is equivalent to random OT [Cr´e88] I OT is symmetric [WW06] I OT is \complete" for secure multiparty computation [Kil88, IPS08] I Black-box construction of OT from one-way permutations implies P 6= NP [IR89] I Perfect OT cannot be constructed using quantum mechanics [Lo97] I OTs can be extended under computational assumptions[IKNP03] I OTs cannot be extended using quantum mechanics [SSS09, WW10] + I OT impies PKE, but not vice-versa [GKM 00] I Constructions: I PIR [CMO00] I DDH [NP01] I Projective hash proofs[Kal05, HK07] I Blind signatures (requires RO) [CNS07] I Bilinear assumptions [GH07] I Dual-mode encryption [PVW08] + I Noisy Channels [IKO 11] 8 w $ [j j] > G (β0; β1) < w βb = g $ 9 > k0; k1 [jGj] : β1−b = r > k0 > α0 = g > > k1 = ((α0; γ0); (α1; γ1)) α1 = g k0 x0 > γ0 = h0 g > > k1 x1 > γ1 = h g ; 1 −w xb = 1 iff γb · αb = g OT From The DDH Assumption Blind Generation of El-Gamal Public-keys a b ab a b c (g; g ; g ; g ) ≈c (g; g ; g ; g ) x0; x1 b Sender Receiver $ 9 k0; k1 [jGj] > k0 > α0 = g > > k1 = ((α0; γ0); (α1; γ1)) α1 = g k0 x0 > γ0 = h0 g > > k1 x1 > γ1 = h g ; 1 −w xb = 1 iff γb · αb = g OT From The DDH Assumption Blind Generation of El-Gamal Public-keys a b ab a b c (g; g ; g ; g ) ≈c (g; g ; g ; g ) x0; x1 b Sender Receiver 8 w $ [j j] > G (β0; β1) < w βb = g > : β1−b = r −w xb = 1 iff γb · αb = g OT From The DDH Assumption Blind Generation of El-Gamal Public-keys a b ab a b c (g; g ; g ; g ) ≈c (g; g ; g ; g ) x0; x1 b Sender Receiver 8 w $ [j j] > G (β0; β1) < w βb = g $ 9 > k0; k1 [jGj] : β1−b = r > k0 > α0 = g > > k1 = ((α0; γ0); (α1; γ1)) α1 = g k0 x0 > γ0 = h0 g > > k1 x1 ;> γ1 = h1 g OT From The DDH Assumption Blind Generation of El-Gamal Public-keys a b ab a b c (g; g ; g ; g ) ≈c (g; g ; g ; g ) x0; x1 b Sender Receiver 8 w $ [j j] > G (β0; β1) < w βb = g $ 9 > k0; k1 [jGj] : β1−b = r > k0 > α0 = g > > k1 = ((α0; γ0); (α1; γ1)) α1 = g k0 x0 > γ0 = h0 g > > k1 x1 > γ1 = h g ; 1 −w xb = 1 iff γb · αb = g Oblivious Transfer Sender Receiver x0 b OT x1 xb I S learns nothing about b I R learns nothing about X1−b Random Oblivious Transfer Sender Receiver x0 b OT x1 xb I S learns nothing about b I R learns nothing about X1−b Sender Receiver y0 c ROT y1 yc d b ⊕ c Idea: use Random OT on z0 = x0 ⊕ yrandomd values(z0; z as1) a one-time z1 = x1 ⊕ y1−dpad to blind real OT zb ⊕ yc y0; y1 generated at random if b = 0 then d = c, so receiver knows yd if b = 1 then d = 1 − c, so receiver knows y1−d Random OT + 3 Bits Communication = OT Precomputing OT Sender Receiver b x0 x1 OT xb Sender Receiver y0 c ROT y1 yc d b ⊕ c z0 = x0 ⊕ yd (z0; z1) z1 = x1 ⊕ y1−d zb ⊕ yc y0; y1 generated at random if b = 0 then d = c, so receiver knows yd if b = 1 then d = 1 − c, so receiver knows y1−d Random OT + 3 Bits Communication = OT Precomputing OT Sender Receiver b x0 x1 OT Idea: use Random OT on random values as a one-time pad to blind real OT xb OT d b ⊕ c Idea: use Random OT on z0 = x0 ⊕ yrandomd values(z0; z as1) a one-time z1 = x1 ⊕ y1−dpad to blind
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