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On the Cryptographic Complexity of the Worst Functions

On the Cryptographic Complexity of the Worst Functions. Amos Beimel (BGU) Yuval Ishai ( Technion ) Ranjit Kumaresan ( Technion ) Eyal Kushilevitz ( Technion ). How Bad are the Worst Functions?. Function class F N of all functions f : [N]  [ N ]  {0,1}. Information-theoretic

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On the Cryptographic Complexity of the Worst Functions

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  1. On the Cryptographic Complexity of the Worst Functions Amos Beimel(BGU) Yuval Ishai (Technion) Ranjit Kumaresan (Technion) EyalKushilevitz(Technion)

  2. How Bad are the Worst Functions? Function class FNof all functions f: [N][N] {0,1} • Information-theoretic • Cryptography • Communication complexity • Randomness complexity • Standard Complexity Theoretic Measures • Circuit complexity • (N2/log N) [Sha48,Lup58] • 2-party communication complexity • (log N) [Yao79] This work: Cryptographic complexity of the worst functions

  3. Model • Security Model • Information-theoretic • Unbounded adversaries • Statistical/perfect security • Semi-honest adversary • No deviation from protocol • Crypto Primitives • Secure Computation • Various models • Communication/randomness • Secret Sharing • Share complexity • Functions • Function class FN: Class of all two argument functions f : [N]  [N] {0,1} • Interested in worst f  FN

  4. Secure Computation What is Known? • Information Theoretic Security • Honest majority [RB89,BGW88] • 2-party in the OT-hybrid or preprocessingmodel [Kil88,Bea95] • Impossible in plain model [Kus89] • Private Simultaneous Messages [FKN94] y x f1(x,y) f2(x,y) Can communication complexity be made logarithmicin N? • Best upper bounds linear in N • Sublinear if big honest majority [BFKR90,IK04] • Counting arguments yield weak lower bounds

  5. 2-Party Secure Computation (2PC) What is Known? • Information Theoretic Security • Impossible in plain model [Kus89] • OT-hybrid/preprocessing model • Popular protocols [GMW87, Y86] y x f1(x,y) f2(x,y) • GMW [GMW87] • Gate-by-gate evaluation of given circuit • #OTs required: Twice #AND gates • Communication cost: Twice #AND gates • Information-theoretic garbled circuits [Yao86] • Depends on circuit structure • Quadratic in formula depth • Exponential in depth overhead for circuits

  6. OT-Hybrid Model Oblivious Transfer [Rab81,EGL85] x0 , x1 b b x0 , x1 xb • Complete • Given ideal OT oracle, can get information theoretic 2-party secure computation [Kil88,GV88] xb ??? • OT Extension • Impossible in information theoretic setting [Bea97] • OT as an“atomic currency” y0 , y1 c, yc • Pre-computation • Random OT correlations can be “corrected” [Bea95] d = c  b b x0 , x1 z0 = x0yd z1 = x1y1-d zbyc

  7. OT Complexity OT Complexity of a function f Number of (bit) OTs required to securely evaluate f • LetFNbe the class of all 2-party f : [N] [N] {0,1} • What is the OT complexity of the worst function in FN? • Circuit based 2PC: • O(N2/log N) [GMW87] • Truth-table based 2PC: • O(N)via1-out-of-N OT • 1-out-of-N OT from O(N) 1-out-of-2 OTs [BCR86] y x f(x,1) f(x,2) . . f(x,N) y ??? f(x,y) This work: O(N2/3) OT complexity

  8. Preprocessing Model Correlated Randomness Offline Phase • Correlated Randomness • Independent of inputs • May depend on f rA rB Online Phase • OT Correlations • Special case • Pre-computed OTs • “Simpler” correlations • Indep. of function x y rA rB f(x,y) f(x,y)

  9. Correlated Randomness Complexity Correlated Randomness Complexity of a function f Size of correlated randomness required to securely evaluate f • LetFNbe the class of all 2-party f : [N] [N] {0,1} • Correlated randomness complexity of the worst function in FN? • O(log N) online communication [IKMOP13] • Correlated randomness: O(N2) • Truth-table based 2PC: O(N) • Via 1-out-of-N OT [BCR86] This work: 2O(log N) correlated randomness

  10. Private Simultaneous Messages (PSM) What is Known? • Model [FKN94] • Multiple clients • Share randomness • Single referee • Non-interactive • Referee learns only f(x,y) • No collusion f (x,y) x r r y • Why PSM? • Minimal model of secure computation [FKN94] • Applications in round-efficient protocol design [IKP10] • Connections to secret sharing! [BI05]

  11. PSM Complexity PSM Complexity of a function f Communication complexity of PSM protocol for f • What is the PSM complexity of the worst function in FN? f(x,1+s) + r1 f(x,2+s) + r2 . . f(x,N+s) + rN f(x,y) • [FKN94,IK97] • Efficient for f with small formulas, branching programs • Worst case f : O(N) • Lower bound: 3logN-4 y-s, ry-s f(x,1) f(x,2) . . f(x,N) x r r y r = s, (r1, …, rN) This work: O(N) PSM complexity

  12. Secret Sharing What is Known? • Model • External dealer + n parties • Dealer has input secret s • Sends “shares” to parties • Then, inactive • Access structure • Set of “authorized” subsets • Secret hidden from unauth. subsets • Any auth. subset can reconstruct s Share Complexity Size of each share Poly(n) share complexity for every n-party access structure? • Best upper bound: 2O(n) [BL90,Bri89,KW93] • Best lower bound: (n/log n) [Csi97]

  13. Share Complexity Forbidden Graph Access Structures • Forbidden Graph [SS97] • Graph G = (V,E) with |V| = N • Authorized subsets: • Sets {u,v} with (u,v) E • Any set of size 3 • What is the share complexity of the worst N-vertex graph? • Naïve solution: O(N) [SS97,BL90] • O(N/log N) share complexity [BDGV96,EP97,Bub86] This work: O(N) share complexity

  14. Talk Outline • Main Technical Tool – PIR • OT Complexity • Correlated Randomness Complexity • PSM Complexity • Share Complexity for Forbidden Graphs

  15. Private Information Retrieval DB DB • Model [CGKS95] • Single client • Multiple servers • Each server has same DB • Size of DB = N (bits) • DB unknown to client • Client input: index i [N] • Privately retrieve DB[ i] • No collusion among servers • Goal: min. communication q1 q2 a1 a2 i q2 z a1 q1 a2 r • Query generation • (q1, q2)  Q(i, r) • Answer generation • ak A( k, qk, DB) Best Known PIR Schemes 2-server: O(N1/3) [CGKS95] 3-server: 2O(log N)[Yek07,Efr09] • Reconstruction • z R(i, r, a1, a2)

  16. Talk Outline • Main Technical Tool – PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • PSM Complexity • Share Complexity for Forbidden Graphs 2-server PIR

  17. OT-Hybrid Model (Recap) • OT is “complete” • Pre-computation • No OT extension x0 , x1 b xb OT Complexity of a function f Number of (bit) OTs required to securely evaluate f • LetFNbe the class of all 2-party f : [N] [N] {0,1} • What is the OT complexity of the worst function in FN? • Circuit based 2PC for worst f : • O(N2/log N) [GMW87] • Truth-table based 2PC for worst f : • O(N), 1-out-of-N OT [BCR86]

  18. O(N2/3) Upper Bound on OT Complexity Via 2-server PIR Q’ = Q(x||y, r1r2) • High-level idea • Use 2 party secure computation to emulate client + 2 PIR servers • DB = truth table of f • Client query = x||y x x y y r1 r1 r2 r2 GMW(C(Q’)) GMW(C(R’)) q1 q2 a1 = A(1, q1, f ) a2 = A(2, q2, f ) • Notation • PIR Algorithms: Q, A, R • (q1, q2)  Q(i, r) • ak A( k, qk, DB) • z R(i, r, a1, a2) • Circuit for alg. B: C(B) • |C(B)|= #ANDs in C(B) R’ = R(x||y, r1r2, a1, a2) a1 a2 f(x,y) f(x,y)

  19. O(N2/3) Upper Bound on OT Complexity Via 2-server PIR Q’ = Q(x||y, r1r2) x x y y r1 r1 r2 r2 • Privacy • Privacy of GMW • Privacy of 2-server PIR • Query does not leak additional info GMW(C(Q’)) GMW(C(R’)) q1 q2 a1 = A(1, q1, f ) a2 = A(2, q2, f ) • Efficiency • 2-server PIR [CGKS95] • |C(Q)|=|C(R)|= O(N2/3) • By property of GMW: • O(N2/3) OT comp. • O(N2/3) communication R’ = R(x||y, r1r2, a1, a2) a1 a2 f(x,y) f(x,y)

  20. More Applications • Honest majority secure computation • Efficient in circuit size [RB89,BGW88] • Specific setting: n = 3 parties with at most 1 corruption • Communication 2O(log N)via 3-server PIR • “ - Secure Sampling” from joint distribution D [PP12] • Protocol lets Alice & Bob to sample (x,y) from D • Alice knows nothing about y (over what is implied by D) • Bob knows nothing about x (over what is implied by D) • Rate of secure sampling D[N] [N]from OT • New upper bound: O(N2/3 poly(log N, 1/))

  21. Talk Outline • Main Technical Tool – PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Share Complexity for Forbidden Graphs 2-server PIR 3-server PIR

  22. Preprocessing Model (Recap) Correlated Randomness Offline Phase • Correlated Randomness • Independent of inputs • May depend on f • OT correlations special case Correlated Randomness Complexity of a function f Size of correlated randomness required to securely evaluate f rA rB Online Phase Correlated randomness complexity of the worst function in FN? x y rA rB • Truth-table based 2PC: O(N) • Via 1-out-of-N OT [BCR86] f(x,y) f(x,y)

  23. Correlated Randomness Complexity: 2O(log N) Upper Bound Via 3-server PIR Offline Phase • High-level idea • Use 2 party secure computation to emulate client + 3 PIR servers • DB = truth table of f • Client query = x||y r1 r2 q3=Q3(r1  r2) a3 = A(3, q3, f ) a3= a3,1a3,2 a3,1 a3,2 OTA OTB • Key Observation • Individual PIR query independent of input • Q = (Q1,2 , Q3) • (q1, q2)  Q1,2(i, r) • q3 Q3 (r) r1 a3,1 OTA OTB a3,2 r2

  24. Correlated Randomness Complexity: 2O(log N) Upper Bound Q’ = Q1,2(x||y, r1r2) x x y y Online Phase r1 r2 GMW(C(Q’)) GMW(C(R’)) • Correlated Randomness • Shares of randomness for PIR query generation alg. • Shares of answer to third PIR query • OT correlations for GMW q1 q2 a1 = A(1, q1, f ) a2 = A(2, q2, f ) R’ = R(x||y, r1r2, a1, a2, a3,1a3,1) r1 a1 a3,1 a3,2 a2 r2 • Notation • PIR Algorithms: Q, A, R • Circuit for alg. B: C(B) • |C(B)|= #ANDs in C(B) f(x,y) f(x,y)

  25. Correlated Randomness Complexity: 2O(log N) Upper Bound Q’ = Q1,2(x||y, r1r2) x x y y • Privacy • Additive secret sharing • Privacy of GMW • Privacy of 3-server PIR • Query does not leak additional info r1 a3,1 a3,2 r2 GMW(C(Q’)) GMW(C(R’)) q1 q2 a1 = A(1, q1, f ) a2 = A(2, q2, f ) • Efficiency • 3-server PIR [Efr09] • |C(Q)|=|C(R)|=2O(log N) • By property of GMW: • 2O(log N)OT correlations • 2O(log N) communication • Correlated rand.: 2O(log N) R’ = R(x||y, r1r2, a1, a2, a3,1a3,1) r1 a1 a3,1 a3,2 a2 r2 f(x,y) f(x,y)

  26. Improving the Bounds? • (OT + communication) complexity of 2PC • Bounded by communication complexity of 2-server PIR • Client shares its input, then acts as OT oracle • (Cor. Rand. + communication) complexity of 2PC • Bounded by communication comp. of 3-server PIR [IKM+13] • 3rd server provides correlated randomness to servers 1 & 2

  27. Summary • Main Technical Tool – PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Upper bound: O(N) • Share Complexity for Forbidden Graphs • Upper bound: O(N) 2-server PIR 3-server PIR 4-server PIR Using PSM above

  28. Thank You! Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf Slides: www.cs.umd.edu/~ranjit/BIKK.pptx

  29. Talk Outline • Main Technical Tool – PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Upper bound: O(N) • Share Complexity for Forbidden Graphs • Upper bound: O(N) 2-server PIR 3-server PIR 4-server PIR Using PSM above

  30. Share Complexity (Recap) Forbidden Graph Access Structures • Model • External dealer + n parties • Dealer inactive after sending “shares” • Access structure: “authorized” subsets • Forbidden Graph [SS97] • Graph G = (V,E) with |V| = N • Authorized subsets: • Sets {u,v} with (u,v) E • Any set of size 3 Share Complexity Size of each share • What is the share complexity of the worst N-vertex graph? • O(N/log N) share complexity [DPGV96,EP97,B86]

  31. Bipartite Case • Forbidden Bipartite Graph • Graph G = (L,R,E) with |L| = |R| = N • Authorized subsets: • {x,y} with x L, y  R, (x,y) E • Any set of size 3 • G associated with f :[N][N] {0,1} • Secret Sharing • Share s using 3-out-of-2N Shamir secret sharing • Also secret share s = sLsRs’ • Send sL to x  L • Send sR to y  R • How to share s’ ?

  32. PSM & Secret Sharing • High-level Idea • Shares : • PSM messages • Reconstruction : • PSM reconstruction r x L y  R Bf (y,r) Af(x,r) • Secret Sharing Scheme for s’ • If dealer input s’ = 0 • x L : Af(x0,r) • y R : Bf(y0,r) • If dealer input s’ = 1 • x L : Af(x,r) • y R : Bf(y,r) • PSM Notation • Shared rand. : r • Alice with input x • Message: Af(x,r) • Bob with input y • Message: Bf (y,r) Good for s’ = 1 For s’ = 0 Pick some x0, y0s.tf (x0 , y0) = 0

  33. Forbidden Graph Access Structures • From Bipartite to General Graphs • Decomposed into log N bipartite graphs • Apply standard techniques [BL90,Sti94] • Forbidden graph access structures • O(N) share complexity • Via O(N) PSM • Scheme is non-linear (?) • Matches best known lower bound for linear schemes: (N) [Min12]

  34. Summary • Cryptographic complexity of worst functions • Main Technical Tool - PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Upper bound: O(N) • Share Complexity for Forbidden Graphs • Upper bound: O(N) 2-server PIR 3-server PIR 4-server PIR Using PSM above

  35. Thank You! Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf Slides: www.cs.umd.edu/~ranjit/BIKK.pptx

  36. Talk Outline • Main Technical Tool – PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Upper bound: O(N) • Share Complexity for Forbidden Graphs 2-server PIR 3-server PIR 4-server PIR

  37. PIR Examples [CGKS95] 2d server PIR with O(N1/d) communication DB DB • PIR Queries • T1R [N] • T2 = T1 i T c T{c}, if c T T \{c}, if c T    T1 T2 A(1,T1) A(2,T2) PIR Answers DB[ j ] j  T i T1 T2 z = A(1,T1) A(2,T2) • Efficiency • Client  Server j : O(N) bits • Server j  Client : 1 bit

  38. PIR Examples [CGKS95] 2d server PIR with O(N1/d) communication • DB as d-dim. hypercube • Indexi (i1, … , id) • Binary rep of (i-1) DB DB    T00...0 T11…1 d S2 d A(1, T00...0) A(2d,T11…1) i S1    z = A(1,T00..0)  A(2d,T11..1 ) • PIR Queries • Pick (T1 , … , Td) R [N1/d]d • Server k : Query T • (T1(k1i1), … ,Td(kdid)) where k (k1,…, kd) PIR Answers DB[k1,…, kd] k1T1’,…,kdTd’ • Efficiency • Client  Server j : O(dN1/d) bits • Server j  Client : 1 bit k1 , … ,kd

  39. Reducing the #Servers [CGKS95] Key Observation Any server can emulate d other servers with cost O(N1/d) Query T for Server k (T1(k1i1), … ,Td(kdid)) where k  ( k1,…, kd) k1 , … ,kd Example: 2-server O(N1/3) PIR Server 1: Query T000 = (T1 , T2 , T3) List “potential” queries for T100: (T1t, T2 , T3) for t [N1/3] Similarly for T010: (T1, T2t, T3) & T001: (T1, T2, T3t) Answer query & 3N1/3 “potential” queries Server 2: Query T111 =(T1  i1, T2 i2, T3  i3) List “potential” queries for T011 ,T101, T110 Answer query & 3N1/3 “potential” queries Clientpicks correct answer in each answer list and XORs them

  40. Private Simultaneous Messages (Recap) • Model [FKN94] • Single referee • Two (or more) clients • Non-interactive • Referee learns only f(x,y) • Clients share randomness • Unknown to referee • All parties know f • No collusion f(x,y) x r r y PSM Complexity of a function f Communication complexity of PSM protocol for f • What is the PSM complexity of the worst function in FN? Efficient for small-depth formulae Worst case f : O(N) [FKN94]

  41. O(N)Upper Bound on PSM Complexity Via 4-server PIR • High-level idea • Clients use shared randomness & referee’s help to emulate client + 3 PIR servers in 4-server PIR scheme of [CGKS95] • DB = truth table of f • Client query i = x||y f(x,y) x r r y • Key Observation • Index i(i1 , i2 , i3, i4) • Input x specifies i1, i2 • Input y specifies i3, i4 • 15 of 16 servers emulated by clients 4-server PIR [CGKS95] Obtained by collapsing basic 16-server O(N1/4) PIR scheme

  42. O(N)Upper Bound on PSM Complexity Via 4-server PIR • Query + Answer Generation • Alice knows T1 i1 , T2 i2 • Answers for T**00 • “Potential” answers for T**01, T**10 • Bob knows T3 i3 , T4 i4 • Answers for T00** • “Potential” answers for T01**, T10** • Missing query T1111 equals • (T1 i1 , T2 i2, T3 i3 , T4 i4) • Answer to T1111 computed by referee Query T for Server k (T1(k1i1), … ,T4(k4i4)) where k  ( k1,…, k4) k1 , … ,kd • Key Observation • i(i1 , i2 , i3, i4) • x specifies i1, i2 • y specifies i3, i4 T1111 i1 i2 i3 i4 x T0000=(T1,…,T4) y T1 i1 T2 i2 T**00 T00** T3 i3 T4 i4 T**01 T**10 T01** T10**

  43. O(N)Upper Bound on PSM Complexity Via 4-server PIR • Query + Answer Generation • Answers for T**00,T00** • “Potential” answers for T**01, T**10 ,T01**, T10** • Referee answers T1111 • Reconstruction • Selecting from “potential” answer list • Use known PSM (small-depth circuit) • PSM outputs XOR of these 15 answers • Remaining answer computed by referee • Finally, XORs this with PSM output Referee’s reconstruction function is “non-universal”

  44. Summary • Cryptographic complexity of worst functions • Main Technical Tool - PIR • OT Complexity • Upper bound: O(N2/3) • Correlated Randomness Complexity • Upper bound: 2O( log N) • PSM Complexity • Upper bound: O(N) • Share Complexity for Forbidden Graphs • Upper bound: O(N) 2-server PIR 3-server PIR 4-server PIR Using PSM above

  45. Thank You! Preliminary Version: www.cs.umd.edu/~ranjit/BIKK.pdf Slides: www.cs.umd.edu/~ranjit/BIKK.pptx

  46. The research leading to these results has received funding from the European Union's Seventh Framework Programme(FP7/2007-2013)under grant agreement no. 259426 – ERC – Cryptography and Complexity

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