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Efficiency vs. Assumptions in Secure Computation Yuval Ishai Technion & UCLA

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Minicrypt Cryptomania OWF KA PRGSIGNENCPRFCOMMITZK PKEOT TDP

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More general than you might think… –encryption, commitment, ZK, coin-flipping, signatures can be captured as special cases. This talk: secure function evaluation –Two or more parties holding inputs x i –Parties wish to compute f(x 1,x 2,…) without revealing inputs to each other –Several variants Honest majority vs. two-party / no honest majority Computational vs. unconditional security Semi-honest vs. malicious parties Standalone vs. UC Secure Computation

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No honest majority –OT computationally secure MPC [Yao86,GMW87] Ideal OT Unconditional, UC MPC [Kil88,IPS08] –MPC for nontrivial f OT [CK89,KKMO94,BIM99,HNRR04] Honest majority, secure channels –Unconditional MPC [BGW88,CCD88,RB89] Feasibility Results Inputs: Alice (s 0,s 1 ) Bob c Bob outputs s c

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The Two-Party Case Alice Bob xy f(x,y) PPT PPT S Bob x,y, |x|=|y| S Bob (y) c View Bob (x,y) PPT S Alice x,y, |x|=|y| S Alice (x,f(x,y)) c View Alice (x,y)

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The Two-Party Case Alice Bob xy f(x,y) k PPT S Bob p x k,y k S Bob (1 k,y k ) c View Bob (1 k,x k,y k ) PPT S Alice p x k,y k S Alice (1 k,x k,f(x k,y k )) c View Alice (1 k,x k,y k )

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A lot of work on practical efficiency This talk: asymptotic efficiency –May also be relevant to practice –Theory beats heuristics Efficiency measures –Communication complexity –Computational complexity –Round complexity Question: given function f and security parameter k –How far can we push each efficiency measure? –Under what assumptions? Efficiency of Secure Computation

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Round Complexity Alice Bob xy f(x,y) 2-message OT necessary (for general f) Is it also sufficient? Cryptomania

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Enc(y) Randomized Encoding [Yao86,…,IK00,AIK04] g is a randomized encoding of f –Nontrivial relaxation of computing f Hope: –g can be simpler than f (meaning of simpler determined by application) –g can be used as a substitute for f xy f Enc(y)x g r decoder simulator Dec(g(x,r)) = f(x) Sim(f(x)) g(x,r)

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Notions of Simplicity Decomposable encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) x r 2-Decomposable encoding g((x,y),r)=(g x (x,r),g y (y,r)) y NC 0 encoding Output locality c Low-degree encoding Algebraic degree d over F x r

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Decomposable Encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) Application: Parallel reduction of secure 2-party computation to OT g((x,y),r)=(g 1 (x 1,r),…,g n (x n,r), g y (y,r)) Alice Bob xy r g y (y,r) f(x,y) OT x1x1 g 1 (x 1,r) g 1 (0,r) g 1 (1,r) g n (0,r) g n (1,r) xnxn g n (x n,r) More effort if Bob can be malicious

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Notions of Simplicity Decomposable encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) x r 2-Decomposable encoding g((x,y),r)=(g x (x,r),g y (y,r)) y NC 0 encoding Output locality c Low-degree encoding Algebraic degree d over F x r

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Notions of Simplicity Decomposable encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) x r 2-Decomposable encoding g((x,y),r)=(g x (x,r),g y (y,r)) y NC 0 encoding Output locality c Low-degree encoding Algebraic degree d over F x r A minimal model for secure computation [FKN94] Alice Bob xy Carol r f(x,y) g y (y,r) g x (x,r)

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Notions of Simplicity Decomposable encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) x r 2-Decomposable encoding g((x,y),r)=(g x (x,r),g y (y,r)) y NC 0 encoding Output locality c Low-degree encoding Algebraic degree d over F x r Randomizing polynomials [IK00,…] round-efficient secure multi-party computation

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Notions of Simplicity Decomposable encoding g((x 1,…,x n ),r)=(g 1 (x 1,r),…,g n (x n,r)) x r 2-Decomposable encoding g((x,y),r)=(g x (x,r),g y (y,r)) y NC 0 encoding Output locality c Low-degree encoding Algebraic degree d over F x r Cryptography in NC 0 [AIK04,…] OWF

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Basic Facts If we dont care about efficiency, every f has a perfect, decomposable encoding g with –degree 3 over F 2 (generalizes to arbitrary rings) –output locality 4 Negative result: degree 3 is optimal over finite fields, assuming perfect privacy [IK00] –Big fields can be tricky: g(x,r)= ( 2 i x i + c) r 2 mod p Open –degree 2 with statistical or computational privacy? 2-round MPC with t

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Degree-3 Encoding for Branching Programs BP(x)=det(L(x)), where L is a degree-1 mapping which outputs matrices of a special form. Encoding: 1 $ $ $ 0 1 $ $ 0 0 1 $ 0 0 0 1 * * * * -1 * * * 0 -1 * * 0 0 -1 * 1 0 0 $ 0 1 0 $ 0 0 1 $ 0 0 0 1 g(x,r 1,r 2 )= R 1 (r 1 ) L(x) R 2 (r 2 )

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Complexity of Randomized Encoding Computational privacy –OWFs exist Decomposable encoding for a circuit C of length O(k |C|) Yaos garbled circuit technique [Yao86] Yields 2-message secure protocols from 2-message OT –Easy PRG (say, PRG in NC 1 ) NC 0 encoding of length |C| poly(k) [AIK05] Assumption implied by factoring, discrete log, lattice assumptions Primitive X exists X exists in NC 0 under Easy PRG assumption Perfect privacy –Efficient NC 0 encodings for formulas, branching programs [Kil88,FKN94,IK00,AIK04,…] –Capture complexity classes NC 1, NL/poly, L/poly

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Open Complexity Questions No nontrivial lower bounds… Computational privacy –OWF efficient NC 0 encoding for circuits? Crypto implies crypto in NC 0 ! –Decomposable encoding of size O(|C|)? –Arithmetic garbled circuit? Perfect / statistical privacy –Efficient encoding for circuits? Constant-round unconditionally secure MPC for P? [BMR90] Relation with other questions? –Great LDC poly-communication protocols for unbounded parties –Better overhead for concrete representations

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Back to Secure Computation Recap: Two-message secure protocol for f(x,y) –Assumes 2-message OT –O(k |C|) communication –poly(k) |C| computation Better assumption? No Better rounds? No Better computation? –PRG G:{0,1} n {0,1} n^2 in NC 0 constant overhead [IKOS08] –Not implied by standard assumptions –Semi-explicit candidate in [MST03] Better communication? –Rest of talk

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Life After the Bomb Gentry 09: fully homomorphic encryption scheme –Enc pk (x), C Enc(C(x)) –Size of encrypted output independent of |C|,|x|! –Can hide C,x (even given sk) –Can make encrypted input size |x|+poly(k) –Corollaries Secure evaluation of f(x,y) with |input|+|output|·poly(k) bits General protocol compiler with poly(k) communication overhead –poly-time version of [NN01] –Big poly(k) computational overhead What is left to be done? –Assumptions –Better communication complexity?

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Communication Complexity Sometimes life is a long sequence of finite tasks… –Circuit size = O(|output|) –In this case, still need poly(k) bits per gate [IKOS08]: –O(1) communication (and computation) per gate –Under exotic crypto in NC 0 assumption [IKOS09]: –O(1) communication, poly(k) computation per gate –Under -Hiding Assumption [CMS99,GR05] Allows generating (G,g) such that m | ord(g) but m is hidden

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Assumptions Weaker results under weaker assumptions? –Beat circuit size bound for useful function classes? General problem: compute a program P on an encrypted input c Enc(x) Two sources of non-triviality –Encrypted output hides P –Encrypted output is shorter than |P| Good solutions for useful classes of P –Linear functions: standard homomorphic encryption –Truth tables: PIR [CGKS95,KO97,CMS99,…] –Degree-2 polynomials [BGN05] –Length-bounded branching programs [NN01,IP07]

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Observation –most natural candidates for average-case hard problems imply one-way functions –most natural candidates for one-way functions imply public-key encryption typically shown in an ad-hoc way –Are we just lucky? Thesis –Hardness + structure world upgrade –Concrete instantiation inspired by [KO97,BIKM99,DMO00,IKO05,HN06] Defined via communication complexity of secure computation Relevance to Impagliazzos Worlds

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Most instances of f,X,Y are hard. What if Alice can send Bob c R Enc(x) for free? Bob computationally bounded, Alice bounded or unbounded. Efficiency of secure computation with security against Bob –Generalizes PIR, homomorphic encryption Communication Complexity Alice Bob x Xy Y f(x,y) How many bits should be communicated to compute f whp?

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Cryptomania x c x Minicrypt x c x Pessiland ? c x Algorithmica x c x Types of Encryption samplable pksk

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How to Get an Upgrade Need: poly-time computable f(x,y) and input distributions X,Y such that: –f has high communication complexity on X Y Low communication error > 1/poly(n) –f has lower communication complexity when c R Enc(x) is created by Alice and given to Bob. Possibly with small error Then Enc can be upgraded Weak homomorphic property

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Candidate f,X,Y f(x,y)= x i y i mod 2 –X,Y uniform on {0,1} n –Hard for interactive protocols with n-O(1) communication [Yao,Vaz,CG] f(x,y)= x i y i –Y uniform on {0,1} n, X uniform of weight 1 –Hard for non-interactive Bob Alice protocols with n-1 bits of communication

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Minicrypt Cryptomania+ Given: –symmetric encryption (Gen,Enc,Dec) –weakly homomorphic for (f,X,Y) with bounded Alice Goal: Build public-key encryption (Gen,Enc,Dec) Alice Bob x Xy Y f(x,y) c=Enc sk (x) d=Bob(c,y) Alice(sk,d,x) sk Gen Multi-round protocol KA

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Minicrypt Cryptomania+ Gen –sk Gen; x X; c Enc sk (x) –pk = (c,x) Enc pk (b) –y Y –Output (Bob(c,y), b f(x,y)) Dec sk (d,e) –Recover f(x,y) from (d,sk) using Alices algorithm –Output e f(x,y) Security: using hybrid game with c Enc sk (x) –Predicting f(x,y) from (c,x,Bob(c,y)) is impossible unconditionally –Hybrid game computationally indistinguishable from real game Implies 2-message OT with statistical security for Sender

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Example: Kids Encryption PKE Let p = public k-bit prime –sk R Z p –Enc sk (b)= (2r+b) sk mod p r R [0, p/(4k)] –Dec sk (c) = ((c sk -1 ) mod p) mod 2 –Enc sk (x)=Enc sk (x 1 ) … Enc sk (x n ) Weak homomorphism: –Let x,y {0,1} 2k –Given c=(c 1,…,c 2k ) Enc sk (x) and y, Bob(c,y)= y i c i allows Alice to decode x i c i

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Example: LWE PKE Decisional LWE: (M,Mr+e) is pseudorandom –M,x random over Z q –e random with small entries Symmetric encryption: –sk = random r –Enc sk (x)=(M,Mx+e+ q/2 x) Weak homomorphism –By adding rows, as long as e i << q

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Pessiland Minicrypt+ Given: –Pessiland Encryption Enc –Enc is weakly homomorphic for (f,X,Y) with unbounded Alice –(f,X,Y) is nontrivial: for any distinct y,y, Pr x X f(x,y)=f(x,y)<1-1/poly Goal: Build a collision-resistant hash function Construction –Key generation: c Enc –Hashing: h c (y)=Bob(c,y) –Collision resistance: h c (y)=h c (y) f(x,y)=f(x,y) for x=Dec(c) nontrivial info on x

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Failed Attempt: LPN CRHF Assumption: (M,Mr+e) is pseudorandom –M,r random over Z 2, e random with low Hamming weight –Similar to LWE but over binary field –Follows from hardness of search problem Implies symmetric encryption n 1/2- -noise LPN implies PKE [Ale03] –Also 2-message OT Not known to imply CRHF Explanation –Homomorphism limited by dimension –In case of LWE, field size gives extra degree of freedom

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Summary Under standard assumptions –Constant rounds –poly(k) communication and computation per gate Pushing communication to an extreme –Fully homomorphic encryption Secure communication poly(k) insecure communication Same round complexity – -hiding assumption O(1) communication per gate O(depth) rounds –Both expensive in computation Pushing computation to an extreme –poly-stretch PRG in NC 0 O(1) computation per gate O(depth) rounds

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Concluding Remarks Ambitious goals call for nonstandard assumptions. –especially when no heuristics are available Does nonstandard mean more risky? –Factoring requires super-polynomial time vs. –A random NC 0 function is exponentially hard to invert

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