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Gate Evaluation Secret Sharing and Secure Two-Party Computation Vladimir Kolesnikov University of Toronto

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Input: x 2 D 1 Input: y 2 D 2 Secure Function Evaluation f: D 1 £ D 2 D 3 f(x,y) f(x,y) One-Round … ?

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SFE Models Semi-honest Both players follow the protocol Observe communication, try to learn additional info Malicious Players can freely cheat Solutions can be obtained by “compilation” of a semi- honest protocol

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Approaches to SFE SFE for specific functions Greater Than, Auctions, Voting SFE for arbitrary functions Functions given as a circuit, branching program, etc. This work: SFE of any boolean formula

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Input: b Input: secrets s 0, s 1 Learn: Learn: nothing Oblivious Transfer (OT) sbsb

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Reduction of SFE to OT OT is a fundamental primitive Rabin ’81, Kilian ‘88 Unconditional reductions are possible OT is implementable under a variety of computational and physical assumptions

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Previous Work Yao’s Garbled circuit Sander, Young and Yung ’99 Kilian ’88 + Cleve ’90 (also CFIK ’03) Based on Permutation Branching Programs Ishai and Kushilevitz ’00, ’02 Based on Branching Programs

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Secure Gate Evaluation x 2 {0,1}y 2 {0,1} G(x,y)? G:{0,1} 2 {0,1} s 0 ’,s 0 ’’ G(0,0) s 0 ’,s 1 ’’ G(0,1) s 1 ’,s 0 ’’ G(1,0) s 1 ’,s 1 ’’ G(1,1) s y ’’ OT (x, (s 0 ’,s 1 ’)) G(x,y) s x ’,s y ’’ ?

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Composition x 2 {0,1}y 2 {0,1} …… … s 0 3,s 0 4 s’ G 1 (0,0) s 00 s 0 3,s 1 4 s’ G 1 (0,1) s 01 s 1 3,s 0 4 s’ G 1 (1,0) s 10 s 1 3,s 1 4 s’ G 1 (1,1) s 11 Gate Evaluation Secret Sharing (GESS ) s 00 s 01 s 10 s 11 I

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GESS for Gates with Binary Inputs s 00 s 01 s 10 s 11 R0R0 R1R1 R 0 © s 00 R 0 © s 01 R 1 © s 10 R 1 © s 11 Wire 1Wire 2Output wire b b 2 R {0,1} :b:b Permute if b=1 Reconstruction: (c r, r 0 r 1 ) r © r c For OR and AND gates either left or right columns of wire 2 are equal! Exponential growth with depth 0 1

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GESS for AND/OR gates Key: view secrets as being equal, except for one column of blocks. share column-wise. 2 R ( {1..n+1} {1..n+1}) 1) 2) 3) 4) n blocks of size k example: n = 3 Shares have the same block equality properties

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GESS Performance Given a boolean formula F Cost ¼ d i 2 ( d i – depth of leaf i) F is balanced quazilinear in |F| Rebalance F to log depth (Bonet-Buss, Spira) Previous best exponential in depth directly for circuits quadratic in |F| via Branching Programs

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GESS Performance Cost of SFE of boolean NC 1 circuit of depth d This workO(2 d d 2 ) Previous best (2 d 2 d 1/2 ) (Kilian-Cleve, Cramer-Fehr-Ishai-Kushilevitz ‘03)

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Other results Lower Bounds New Efficient Protocol for GT Generalization of Yao’s Garbled Circuit

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Lower Bounds S 00 S 01 S 10 S 11 Wire 1Wire 2Output wire 0 1 A0A0 A1A1 B0B0 B1B1 When secrets are independent H(A i ) + H(B j ) ¸ 3 H(S)

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