1 / 46

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

zuri
Download Presentation

On the Cryptographic Complexity of the Worst Functions

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related