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  1. The Complexity ofInformation-TheoreticSecure Computation Yuval Ishai Technion 2014 European School of Information Theory

  2. Information-Theoretic Cryptography • Any question in cryptography that makes sense even if everyone is computationallyunbounded • Typically: unconditional security proofs • Focus of this talk: Secure Multiparty Computation (MPC)

  3. Talk Outline • Gentle introduction to MPC • Communication complexity of MPC • PIR, LDC, and related problems • Open problems

  4. x4 x3 x5 xi x2 x6 x1 How much do we earn? Goal: compute xi without revealing anything else

  5. 0≤r<M x4 m3=m2+x3 m4=m3+x4 x3 x5 m5=m4+x5 m2=m1+x2 m6-r x2 x6 A better way? m6=m5+x6 m1=r+x1 x1 Assumption: xi<M (say, M=1010) (+ and – operations carried modulo M)

  6. x4 x3 x5 m2=m1+x2 x2 x6 A security concern m1 x1

  7. r43 r51 r25 r32 r65 r12 r16 x4 x3 x5 x2 x6 Resisting collusions x1 xi + inboxi- outboxi

  8. More generally • P1,…,Pkwant to securely compute f(x1,…,xk) • Up to t parties can collude • Should learn (essentially) nothing but the output • Questions • When is this at all possible? • How efficiently? Secure MPC protocol for f Similar feasibilityresults for security against malicious parties • Information-theoretic (unconditional) security possible when t<k/2[BGW88,CCD88,RB89] • Computational security possible for any t(under standard cryptographic assumptions) [Yao86,GMW87,CLOS02] • Or: information-theoretic security using correlated randomness [Kil88,BG89]

  9. More generally • P1,…,Pkwant to securely compute f(x1,…,xk) • Up to t parties can collude • Should learn (essentially) nothing but the output • Questions • When is this at all possible? • How efficiently? • Several efficiency measures: communication, randomness, rounds, computation • Typical assumptions for rest of talk:* t=1, k = small constant* information-theoretic security*“semi-honest” parties, secure channels

  10. Communication Complexity

  11. Fully Homomorphic Encryption Gentry ‘09 • Settles main communication complexity questions in complexity-based cryptography • Even under “nice” assumptions! [BV11] • Main open questions • Further improve assumptions • Improve practical computational overhead • FHE >> PKE >> SKE >> one-time pad

  12. One-Time Pads for MPC [IKMOP13] Alice (x) f(x,y) RA Trusted Dealer dy RB (y) Bob f(x,y) dx • Offline: • Set G[u,v] = f[u-dx, v-dy] for random dx, dy • Pick random GA,GB such that G = GA+GB • Alice gets GA,dx Bob gets GB,dy • Protocol on inputs (x,y): • Alice sends u=x+dx, Bob sends v=y+dy • Alice sends zA= RA[u,v], Bob sends zB= RB[u,v] • Both output z=zA+zB 0 1 1 0 1 2 1 0 1 0 2 0 1 2 0 0 1 1 0 1

  13. 3-Party MPC for g(x,y,z) • Define f((x,zA),(y,zB)) = g(x,y,zA+zB) RA Alice (x) zA Carol (z) g(x,y,z) zB (y) Bob RB

  14. One-Time Pads for MPC • The good: • Perfect security • Great online communication • The bad: • Exponential offline communication • Can we do better? • Yes if f has small circuit complexity • Idea: process circuit gate-by-gate • k=3, t=1: can use one-time pad approach • k>2t: use “multiplicative” (aka MPC-friendly) codes • Communication  circuit size, rounds  circuit depth

  15. MPC vs. Communication Complexity a b c

  16. MPC vs. Communication Complexity a b c

  17. MPC vs. Communication Complexity Big open question: poly(n) communication for all f ? “fully homomorphic encryption ofinformation-theoretic cryptography” a b c

  18. = communication-efficient MPC = no communication-efficient MPC Question Reformulated Is the communication complexity of MPC strongly correlated with the computational complexity of the function being computed? All functions efficiently computable functions

  19. 1990 2000 • 1995 [KT00] [IK04] MPC PIR LDC • The three problems are closely related

  20. Private Information Retrieval [Chor-Goldreich-Kushilevitz-Sudan95] database x∈{0,1}n ? ? ? “Information-Theoretic” vs. Computational Main question: minimize communication (lognvs. n) xi

  21. i q1 q2 a1=X·q1 a2=X·q2 q1+ q2 = ei a1+a2=X·ei A Simple I.T. PIR Protocol n1/2 X S1 S2 n1/2 i  2-server PIR with O(n1/2) communication

  22. A Simple Computational PIR Protocol [Kushilevitz-Ostrovsky97] a b a+b  = Tool: (linear) homomorphic encryption Protocol: • Client sends E(ei) • E(0) E(0) E(1) E(0) (=c1 c2 c3 c4) • Server replies with E(X·ei) • c2c3 • c1 c2c3 • c1c2 • c4 • Client recoversith column of X n1/2 0 1 1 0 1 1 1 0 1 1 0 0 0 0 0 1 n1/2 X= i  1-server CPIR with ~ O(n1/2) communication

  23. Cons: Requires multiple servers Privacy against limited collusions Worse asymptotic complexity (with const. k):2(logn)^[Yekhanin07,Efremenko09] vs. polylog(n)[Cachin-Micali-Stadler99, Lipmaa05, Gilboa-I14] Why Information-Theoretic PIR? • Pros: • Interesting theoretical question • Unconditional privacy • Better “real-life” efficiency • Allows for very short (logarithmic) queries or very short (constant-size) answers • Closely related to locally decodable codes & friends

  24. i Locally Decodable Codes Requirements: • High robustness • Local decoding x y  If < 1% of y is corrupted, xi is recovered w/prob > 0.51 Question: how large should m(n) be in a k-query LDC? k=2: 2(n) k=3: 22^O~(sqrt(logn)) (n2)

  25. From I.T. PIR to LDC [Katz-Trevisan00] Simplifying assumptions: • Servers compute same function of (x,q) • Each query is uniform over its support set • Uniform PIR queries  “smooth” LDC decoder  robustness • Arrows can be reversed k-server PIR with -bit queries and -bit answers k-query LDC of length 2over ={0,1} y[q]=Answer(x,q) • Binary LDC  PIR with one answer bit per server

  26. Applications of Local Decoding • Coding • LDC, Locally Recoverable Codes (robustness) • Batch Codes (load balancing) • Cryptography • Instance Hiding, PIR (secrecy) • Efficient MPC for “worst” functions • Complexity theory • Locally random reductions, PCPs • Worst-case to average-case reductions, hardness amplification

  27. Complexity of PIR: Total Communication • Mainly interesting for k=2 • Upper bound (k=2): O(n1/3) [CGKS95] • Tight in a restricted model [RY07] • Lower bound (k=2): 5logn [Man98,…,WW05] • No natural coding analogue

  28. Complexity of PIR: Short Answers • Short answers = O(1) bit from each server • Closely related to k-query binary LDCs • k=2 • Simple O(n) upper bound [CGKS05] • PIR analogue of Hadamard code • Ω(n)lower bound [GKST02, KdW04] • k > logn / loglogn • Simple polylog(n) upper bound [BF90,CGKS05] • PIR analogue of RM code • Binary LDCs of length poly(n) and k=polylog(n) queries

  29. Complexity of PIR: Short Answers • k=3 • Lower bound • [KdW04,…,Woo07] 2logn • Upper bounds • [CGKS95] O(n1/2) • [Yekhanin07] nO(1/loglogn) • [Efremenko09] nO~(1/sqrt(logn)) Assuming infinitely many Mersenne primes More practical variant [BIKO12]

  30. Complexity of PIR: Short Answers • k=4,5,6,… • Lower bound • [KdW04,…,Woo07] c(k).logn • Upper bounds • [CGKS95] O(n1/k-1) • [Yekhanin07] nO(1/loglogn) • [Efremenko09] nO~(1/(logn)^c’(k)) Assuming infinitely many Mersenne primes

  31. Complexity of PIR: Short Queries • Short queries = O(logn) bit to each server • Closely related to poly(n)-length LDCs over large Σ • Application: PIR with preprocessing [BIM00] • k=2,3,4,… • Answer length = O(n1/k+ε) [BI01] • Lower bounds: ???

  32. Complexity of PIR: Low Storage • Different servers may store different functions of x • Goal: minimize communication subject to storage rate=1-ε • Corresponds to binary LDCs with rate 1-ε • Rate =1-ε, k=O(nε),1-bit answers • Multiplicity codes [DGY11] • Lifting of affine-invariant codes [GKS13] • Expander codes [HOW13]

  33. Best 2-Server PIR[CGKS95,BI01] • Reduce to private polynomial evaluation over F2 • Servers: x  p = degree-3 polynomial in m≈n1/3 vars. • Client: i z∈ F2m • Local mappings must satisfy px(zi)=xi for all x,i • Simple implementation: z(i) = i-th weight-3 binary vector • Privately evaluate p(z) • Client: • splits z into z=a+b, where a,b are random • sends a to S1 and b to S2 • Servers: • write p(z)=p(a+b) as pa(b)+pb(a) where deg(pa),deg(pb) ≤ 1, pa known to S1, and pb known to S2 • Send descriptions of pa,pb to Client, who outputs pa(b)+pb(a) • d=O(logn)  O(logn)-bit queries, O(n1/2+ε)-bit answers

  34. Tool: Secret Sharing • Randomized mapping of secret s to shares (s1,s2,…,sk) • Linear secret sharing: shares = L(s,r1,…,rm) • Access structure: subset A of 2[k] specifying authorized sets • Sets of shares not in A should reveal nothing about s • Optimal share complexity for given A is wide open • Here: k=3, each share hides s, all shares determine s • Useful examples for linear schemes • Additive sharing: s=s1+s2+s3 • Shamir’s secret sharing: si=p(i) where p(x)=s+rx • CNF secret sharing: s=r1+r2+r3, s1=(r2,r3), s2=(r1,r3), s3=(r2,r3) • CNF is “maximal”, Additive is “minimal” • For any linear scheme: [v], x  [<v,x>] (without interaction) • PIR with short answers reduces to client sharing [ei] while hiding i • Enough to share a multiple of [ei]

  35. Tool: Matching Vectors[Yek07,Efr09, DGY10] • Vectors u1,…,un in Zmh are S-matching if: • <ui,ui> = 0 • <ui,uj> ∈ S (0∉S) • Surprising fact: super-polynomial n(h) when m is a composite • For instance, n=hO(logh) for m=6, S={1,3,4} • Based on large set systems with restricted intersections modulo m [BF80, Gro00] • Matching vectors can be used to compress “negated” shared unit vector • [v] = [<ui,u1>, <ui,u2>, …,<ui,un>] • v is 0 only in i-th entry • Apply local share conversion to obtain shares of [v’], where v’ is nonzero only in i-th entry • Efremenko09: share conversion from Shamir’ to additive, requires large m • Beimel-I-Kushilevitz-Orlov12: share conversions from CNF to additive, m=6,15,…

  36. Matching Vectors & Circuits  • Actual dimension wide open; related to size of: • Set systems with restricted intersections[BF80, Gro00] • Matching vector sets [Yek07,Efr09, DGY10] • Degree of representing “OR” modulo m [BBR92] mod6 mod6 mod6 mod6 mod6 mod6 x1 x2 x3 xh 2h^logh< VC-dim << 22^h

  37. Share Conversion Given: CNF shares of s mod 6 s=0  s’0 s0  s’=0 s=1,3,4

  38. Big Set System with Limited mod-6 Intersections • Goal: find N subsets Ti of [h] such that: • |Ti|1 (mod 6) • |TiTj|  {0,3,4} (mod 6) • h = query length; N = database size • [Frankl83]: h=, N= • h  7N1/4 • Better asymptotic constructions exist

  39. Big Set System with Limited mod-6 Intersections r-clique 11 11 11 3 h= N=|Ti|==551 (mod 6) • |TiTj|=, 3t10  {0,3,4} (mod 6)

  40. PIR  MPC • Arbitrary polylogarithmic 3-server PIR  MPC with poly(|input|) communication [IK04] • Applications of computationally efficient PIR [BIKK14] • 2-server PIR  OT-complexity of secure 2-party computation • 3-server PIR  Correlated randomness complexity • Applications of “decomposable” PIR [BIKK14] • Private simultaneous messages protocols • Secret-sharing for graph access structures

  41. Open Problems: PIR and LDC • Understand limitations of current techniques • Better bounds on matching vectors? • More powerful share conversions? • t-private PIR with no(1)communication • Known with 3t servers [Barkol-I-Weinreb08] • Related to locally correctable codes • Any savings for (classes) of polynomial-timef:{0,1}n{0,1} ? • Barriers for strong lower bounds? • [Dvir10]: strong lower bounds for locally correctable codes imply explicit rigid matrices and size-depth lower bounds.

  42. Open Problems: MPC • High end: understand complexity of “worst” f • O(2n^) vs. (n) • Closely related to PIR and LDC • Mid range: nontrivial savings for “moderately hard” f? • Low end: bounds on amortized rate of finite f • In honest-majority setting • Given noisy channels