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Round-Optimal Secure Two-Party Computation

Round-Optimal Secure Two-Party Computation. Jonathan Katz U. Maryland. Rafail Ostrovsky U.C.L.A. Motivation. Round complexity is a central measure of protocol efficiency. Minimizing the number of rounds is often important in practice.

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Round-Optimal Secure Two-Party Computation

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  1. Round-Optimal Secure Two-Party Computation Jonathan Katz U. Maryland Rafail Ostrovsky U.C.L.A. 1/48

  2. Motivation • Round complexity is a central measure of protocol efficiency. • Minimizing the number of rounds is often important in practice. • Lower and upperbounds have deepened our understanding of various tasks… 2/48

  3. For example… • ZK [FS89, GO94, GK96a, GK96b, BLV03, etc.], NIZK [BFM88, etc.], WI [FS89,DN00,BOV03] • Concurrent ZK [DNS98, KPR01, CKPR01, PRS02] • Commitment, identification schemes, … • … • 2-party and multi-party computation [BMR90, IK00, GIKR01, L01, KOS03, etc.] 3/48

  4. This work • We concentrate on secure two-party computation • Encompasses many functionalities of independent interest (e.g., ZK) • Important “special case” of MPC without honest majority • Interestingly, exact round complexity of 2PC was not previously known! 4/48

  5. This work (1) • We exactly characterize black-box round complexity of secure 2PC! • THM1: Impossibility result for any black-box 4-round coin-tossing (also XOR, other functionalities…) 5/48

  6. This work (2) • THM2: 5-round secure 2PC protocol for any functionality, based on trapdoor perms* (e.g. RSA, Rabin) or Homomorphic Encryption (e.g. DDH). 6/48

  7. This work (3) • THM3: 5-round secure 2PC protocol an adaptive adversary corrupting any one party without erasure in 5 rounds. 7/48

  8. Prior work (2PC) • Honest-but-curious setting • 4 rounds using trapdoor perms. [Yao86] • 3 rounds using number-theoretic assumptions (optimal) [Folklore] • Malicious case • “Compiler” for any protocol secure in honest-but-curious setting [GMW87] • Round complexity? 8/48

  9. Round complexity of 2PC? • Upper bounds • O(k) rounds [GMW87] • O(1) rounds [Lindell01] • Unspecified, but roughly 20-30 rounds • Lower bounds (black-box) • No 3-round ZK [GK96] • No 3-round coin-tossing [Lindell01] 9/48

  10. Security definition • We use the standard definitions of [GMW87, GL90, MR91, Ca00] • This will be an informal review, focusing on a static adversary 10/48

  11. Set-up • Functionality F = (F1, F2), possibly randomized; player Pi gets Fi(x, y) • In real world, players execute a protocol to compute F • In ideal world, a trusted party computes F for the players 11/48

  12. Ideal model • Players send x, y to TTP • Malicious player can send any value it likes; honest party sends its input value • If no value sent, a default value is used • TTP chooses uniformly-random r; sends v1 = F1(x, y; r) to P1 • If P1 aborts, TTP sends v2 =  to P2 • Else, TTP sends v2 = F2(x, y; r) to P2 12/48

  13. Ideal model • Let Viewi denote the view of Pi • Let (B1, B2) be strategies • Define IDEAL = (B1(View1), B2(View2)) • Note: for Bi honest, Bi(Viewi) = vi 13/48

  14. Real model • Players execute protocol… • Let (A1, A2) be strategies • Define REAL = (A1(View1), A2(View2)) • Again, if Ai honest, then Ai(Viewi) = vi 14/48

  15. Security… • A pair of strategies is admissible if at least one is honest • Protocol is secure if for all admissible PPT (A1, A2) in the real world, there exist admissible expected poly-time (B1, B2) in ideal world such that REAL and IDEAL are comp. indistinguishable • Even with auxiliary inputs… 15/48

  16. Black-box security • The definition of security requires:  (malicious) Ai,  (malicious) Bi, s.t. Bi satisfies the condition…. • Black-box security imposes stronger requirement: (S1, S2),  (malicious) Ai, (malicious) Bi =SiAi satisfies the condition… 16/48

  17. More formally… • For malicious A1, define B1 as follows: • B1(x, z; r, r’) = S1A1(x, z; r)(x; r’) • S1 not given auxiliary input z • Exp. running time of S1 is a fixed polynomial, independent of A1 • But running time of B1 depends on A1 • The above formulation avoids some technical problems… 17/48

  18. Lower bound 18/48

  19. Theorem 1 • No secure (black-box) 4-round protocol for flipping (log k) coins • This rules out 4-round protocols for other functionalities as well (e.g., XOR) • (Note: 3-round protocols for O(log k) coins do exist [Bl82, GMW87]) • Details: (next) 19/48

  20. Intuition • W.l.o.g., P2 sends the first message • No way to simulate for a malicious P1 who aborts “very often” • Sending different msg1 doesn’t help • P1 starts over with “new randomness” [GK] • Sending different msg3 doesn’t help • P1 anyway aborts “very often” 20/48

  21. Proof details I • Let s() be the expected r.t. of S1 • Define A1 as follows: • Use msg1 to define random string for an “honest” execution of the protocol (using O(s)-wise independent hash function) • After msg3, compute coin c; abort unless first (3log s) bits of c are 0 • Note: here we use |c| = (log k) 21/48

  22. Proof details II • REAL is “non-aborting” with noticeable probability 1/s3 • Thus, IDEAL must be “non-aborting” with roughly the same probability • Conditioned on “good” coin from TTP, S1 must “force” A1 not to abort with probability essentially 1 22/48

  23. Proof details III • Run S1 for at most 2s steps • Now, strict poly-time • Conditioned on “good” coin from TTP, “forces” A1 not to abort with probability essentially 1/2 23/48

  24. Proof details IV • Define A2 as follows: • Feed “good” coin to S1; guess i, j • Send ith query of S1 to P1 as msg1, return msg2 to S1 • Send jth query of S1 to P1 as msg2 • Answer other queries of S1 internally, by either aborting or playing the role of A1 24/48

  25. Proof details V • Analysis: • Conditioned on “correct” guesses of i, j, honest player P1 outputs “good” coin with probability essentially 1/2 • Probability of correct guess > 1/4s2 • So probability that honest P1 outputs “good” coin is at least 1/8s2 > 1/s3 • A2 noticeably biases the coin! 25/48

  26. Implications • No 4-round (black-box) protocol for general secure computation • Note: Could also derive from [GK]… • …but our techniques rule out 4-round protocols for wider class of functions 26/48

  27. THM2: A 5-round protocol for secure two-party computation (for malicious adversary) We construct a 5-round protocol where we “force”’ good behavior on both sides and can “simulate” malicious Adv view from both sides… 27/48

  28. Somewhat easier task • [folklore]: k-round with one player learning the output  (k+1)-round with both players learning the outputs • the output in the kth round includes encrypted and MAC’ed output for other player. • SO: we need a 4-round protocol where, say, player 1 gets the output. 28/48

  29. observation It suffices to consider deterministic functionalities. Rest of the talk: we show a 4-round protocol tolerating malicious players where player 1 learns the output. 29/48

  30. Rest of the talk • 3-round protocol for semi-honest players • Background tools • Some of our new techniques • Our 4-round protocol (if time permits) • Proof of security (if time permits) • Modifications needed for Dynamic Adv. • Conclusions. 30/48

  31. Recall: 1-2-OT [EGL] • Sender has (v0, v1); • Receiver has b, 1-2-OT: • Receiver gets vb • Sender gets nothing 31/48

  32. Semi-honest 1-2-OT [EGL,GMW] • S: generate td perm. (f, f-1); send f • R: yb = f(zb), y1-b rand; send (y0, y1) • S: send ui = h(f-1(yi))vi, for i=0,1 • R computes vb = h(zb)ub Note: extends easily for strings in semi-honest setting 32/48

  33. Yao’s “garbled circuit” • Algorithms (Y1, Y2) s.t.: • Y1(y) outputs “circuit” C, input-wire labels {Zi,b}, • [C “represents” F(.,y)] • Y2(C, Z1,x1, …, Zk,xk) outputs v Correctness: v = F(x, y) 33/48

  34. 3-round semi-honest 2PC • Player 2 sends Yao’s C, f for OT • Player 1 sends OT pairs {(yi,0, yi,1)} • Player 2 sends {(ui,0, ui,1)} to Player 1. Player 1 recovers v. 34/48

  35. Malicious 2PC? • Standard method [GMW87] increases round-complexity: • Coin tossing into the well to fix random tapes of players; • Players commit to their inputs; • ZK arguments of correctness after every round; High round complexity of compilation 35/48

  36. Malicious 2PC in 4 rounds • Our goal: do everything in 4 rounds, (player 1 gets the output) forcing “good” behavior from both sides! • Intuition: do everything “as early as possible” but …things “don’t fit” – we need new tricks to cram it all.. • Surprise: we must “delay” proofs to make it work. 36/48

  37. Reminder:3-Round WI proofs [FS] P claims that graph G has a HC • PV: commit n cycle graphs C1..Cn • VP: random n-bit string Q • PV: for each bit of Q, either • open entire matrix Ci OR • show perm of G onto Ci open non-edges of G in Ci. 37/48

  38. OBSERVATION • Graph G can be determined in the last round. • IF G is determined in the 1st round  this is WI proof of knowledge • IF G is determined in the 3rd round  this is only a WIproof, but it is still sound! 38/48

  39. NEW PROPERTIES FOR 3-ROUND WI-PROOF 39/48

  40. Next: [FS] 4-round ZK • Q can we get similar result for [FS] 4-round ZK argument? 40/48

  41. [FS] 4-round ZK argument: 2 interleaved WI-proofs 41/48

  42. [FS] 4-round ZK-argument 2 interleaved WI proofs: • PV: gives y1,y2 s.t. f(a1)=y1,f(a2)=y2 and WI proof of this fact (3 rounds) • PV: WI proof of witness w that x is in L or w is one of the a’s (starting on the 2nd round). Total of 4 rounds. Proof of knowledge; also ZK. 42/48

  43. New FS properties needed: • Observation: In FS, prover needs to determine the statement in the second round. • Goal: to defer parts of statement to last (4th) round. Previous ideas are not sufficient… 43/48

  44. Technical lemma - we extend [FS] to FS’ so that: • FS’ is a 4-round Zero-knowledge argument where statements can be “Postponed”. • FS’ define conjunctive parts of statement in the second round (with knowledge extraction) and part of statement in the 4th round (without extraction but still sound!) • It is of independent interest (requires equivocal commitment, some other tools) 44/48

  45. [FS] 4-round ZK argument: 2 interleaved WI-proofs 45/48

  46. OUR PROTOCOL PROOF-FLOWS 46/48

  47. Simulation on both sides? we need more tools… • Malicious player 2 gains nothing by using non-random tape in Yao. • Player 1 cannot freely choose his random tape, but full-blown coin-tossing is not necessary (i.e., we don’t need simulatability on both sides) • Player 2 has to commit Yao’s garbled circuit in round 2, but the simulator need to open it arbitrary, so use equivocal comm. 47/48

  48. Equivocal commitments • (Informal): in real execution, sender committed to a single value; in simulation, can open arbitrarily • Construction: Equiv(b) = Com(b0), Com’(b1)ZK argument that b0 = b1Open by opening either b0 or b1 • Can “fold” ZK argument into larger statement already used in 4th round of FS’ 48/48

  49. And now… the 4-round protocol… (only 4 slides, 1 msg per slide) 49/48

  50. Round 1: P1(x)P2(y) • P1 commits {(ri,0, ri,1)}; (random) • starts 3-round WI PoK of either ri,0 or ri,1; • Starts FS’1 (statement TBA by P2 partly in round 2, partly in round 4) 50/48

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