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Muthuramakrishnan Venkitasubramaniam WORKSHOP: THEORY AND PRACTICE OF SECURE MULTIPARTY COMPUTATION Adaptive UC from New Notions of Non-Malleability Adaptive UC from New Notions of Non-Malleability 15 years of UC-Security [Canetti00] 25 years of Adaptive Security [Beaver89] dynamic Joint with Dana Dachman-Soled, Maryana Raykova, Tal Malkin

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How can we achieve semi-honest 2-party computation? YAO O(1)-rnd ^

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IDEALREAL ARAR AIAI x 2 y 2 x2y2x2y2 Security by Comparison x 1 y 1 “as correct & private as” Correctness: The output of every player is the same in real and ideal Mesgs Privacy: Mesgs can be generated from the simulator’s input & output Simulator

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IDEALREAL Concurrent Security many executions of different protocols many executions with independent trusted parties

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Arbitrary network REAL WORLDIDEAL WORLD Universal Composability [C] ARAR AIAI Simulate messages without honest input Independence of executions

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Theorem [CF, CKL, L]: It is impossible to achieve UC-security for all “non-trivial functionalities” What can we implement with UC- Security? SOLUTION: Get some “limited” help from a trusted party OR Relax definition of security

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… … Static Corruption Adaptive Corruption corrupt in the beginning corrupt adaptively during execution

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Stronger definition of security Static security does not imply adaptive security Implies leakage resilience* [BCH12,NVZ13] Relevant to cloud security [RTSS09] Adaptively co-locate VMs Side channel attacks Why Adaptive Security?

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— Common Reference String [CLOS02,DN02,DG03,CPS07] — Public Key Registration [BCNP04] Trusted Setups General Results in Adaptive UC-Security? Relaxed Security — Super-Poly Time Simulation (SPS) [BS05] What about Static UC-Security?

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— Common Reference String [CLOS02,DN02,DG03,CPS07,DNO10] — Public Key Registration [BCNP04,DNO10] — Tamper-Proof Hardware [Kat07,CGS08,GISVW10] — Timing Model [DNS98,KLP05] Trusted Setups What about Static UC-Security? Relaxed Security — Super-Poly Time Simulation (SPS) [Pas03, BS05, GGJS12] — Angel-based Security Model [PS04, MMY06,CLP10] — Bounded (Player) Concurrent [Barak] — Non-Uniform Simulation [LPV09]

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— A unified framework to achieve security in any setup under minimal trusted infrastructure [LPV09] — Can achieve security assuming only SA-OT [DNO10,LPV12] Static Security : State of the Art Adaptive Security : — Construction only in a few trusted setups — Constructions based on specific assumptions such as dense cryptosystems, trapdoor simulatable PKE — Require independent setups for every pair of parties, e.g sunspots [CPS07]

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UC-puzzle Simulation Trusted Setup Stand-Alone Non-malleabilty Stand-Alone Non-malleabilty One-Way Functions Non-malleability UC-Security Achieving UC-Security - Static Case [LPV09] Puzzle

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Static Security : Static OT Puzzle Static UC Static UC This work: When, and at what cost, can Adaptive UC security be acheived? Achieving UC-Security - Static Case [LPV09,LPV12] NMC

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Static Security : Ideally… Adaptive Security : Static OT Puzzle Static UC Static UC Adap. OT Adap. Puzzle Adap. Puzzle Adap. UC Adap. UC NMC ?

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Static Security : Our Work Adaptive Security : Static OT Puzzle Static UC Static UC Adap. OT Adap. Puzzle Adap. Puzzle Adap. UC Adap. UC NMC ?

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Static Security : Adaptive Security : Static OT Puzzle Static UC Static UC Adap. OT Adap. Puzzle Adap. Puzzle Adap. UC Adap. UC Simul. PKE Our Work NMC ?

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Static Security : Adaptive Security : Static OT Puzzle Static UC Static UC Adap. OT Adap. Puzzle Adap. Puzzle Adap. UC Adap. UC Simul. PKE Our Work NMC NM*

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Adaptive Security : Adap. OT Adap. Puzzle Adap. Puzzle Adap. UC Adap. UC Simul. PKE Our Work Simulatable Public Key Encryption [DN00] Oblivious Sampling of Public Keys/Ciphertexts Invertable randomness for oblivious algs. => Non-commiting Encryption [CFGN96,DN00] NM*

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Assuming existence of simulatable PKE, Adaptive UC-security is achievable in any setup that admits an Adaptive Puzzle Main Theorem Previous results - simple corollaries Improved complexity assumptions New models – non-uniform, bounded conc.

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UC-puzzle Adap. Simulation Trusted Setup Adap. Non-malleability Adaptive UC-Security Achieving UC-Security - Adaptive Case Cannot decouple! stand alone adaptivity requires setup

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UC-puzzle Adap. Simulation Trusted Setup Adap. Non-malleability Adaptive UC-Security Achieving UC-Security - Adaptive Case Adap. UC-Puzzle [LPV09] TODAY

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Commitment Scheme The “digital analogue” of sealed envelopes. Com(v) Decommitment phase Sender/committer Receiver Hiding: The commitment hides the committed value Commitment phase d Binding: The commitment can only open to one value

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Com(u) MIM Attack on Commitments [DDN91] Receiver/Sender Sender Receiver Com(u+1) Man in the Middle MIM ”mauls” left commitment into another to a related value

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Non-Malleable w.r.t commitment [DDN91, PR05, LPV08] i j ≠ i IDEAL REAL Ci(u)Ci(u)Cj(v)Cj(v) Simulator MIM Output v’ = v C j ( v’ ) Can construct O(1) round concurrent NMC w.r.t commitment based on OWFs [LP12,Goy12]

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Non-Malleable w.r.t opening [CIO98,FF00,PR05] i j ≠ i IDEAL REAL Ci(u)Ci(u)Cj(v)Cj(v) MIM C j ( v’ ) u v u v' Simulator Can construct O(1) round stand-alone NMC w.r.t opening based on CRHs for sychronized adversaries [PR05]

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What we need? C i3 ( w ) C j1 ( v ) MIM w v C i1 ( u ) u C i2 ( t ) t C i4 ( x ) x C i5 ( y ) y C j2 ( v’ ) v' C j3 ( u’ ) u' Concurrent Non-Malleable Commitments w.r.t opening Adaptively Secure

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Concurrent Non-Malleable Commitments w.r.t opening Adaptively Secure MIM C i1 ( u ) u C i2 ( w ) w C j ( v’ ) v' … … Simulator u w C j ( v’ ) v' … … v’ Relaxation: Left commitments are i.i.d samples

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Main Lemma: Assuming OWFs and Puzzle, O(n)-round Adaptively-secure Conc. NMC w.r.t opening and i.i.d samples No additional trusted infrastructure to achieve non-malleability! A single CRS/URS/sunspot is sufficient same gains as static case Relaxation: Left commitments are i.i.d samples “What is a few rounds of communication between friends”

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i.e., Receiving Green does not help giving Orange and vice versa Non-Malleable Sub-protocols Ingredient I – Scheduling [DDN]

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Can rewind the right without rewinding the left! Id = 0Id = 1

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Simulation Soundness Challenger Solver No Malicious Solver can output trapdoor after interaction TRAPDOOR NP-statement Concurrent Adversary Challenger A, Simulator S that simulates all puzzles indistinguishably while extracting the trapdoor Puzzle NP-witness UC-

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Ingredient II – Instance Based Comm. [LZ09] W/O Trapdoor: Commitment is binding With Trapdoor: Reveal it to 0 and 1 UC-Puzzle NP-statement UC-Puzzle NP-statement Hamiltonian Circuit Scheme: Commit to adjacency matrix Commit 0 : Commit to true adjacency matrix Commit 1 : Commit to a simple cycle Equivocate : Commit to true adjacency matrix

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Application: Conc. NM Coin Tossing ANMCOM(r) r' r Coin toss output = r+r’ IDEA FOR UC-COM: Create two URS Sender to Receiver (URS1) – equivocate (using OWF) Receiver to Sender (URS2) – extract (using sim PKE)

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Main Theorem Assuming existence of sim. PKE and Adap.UC Puzzle, Adaptive UC-security is achievable Assuming existence of OWFs and Adap.UC Puzzle, O(n)-round Adaptively-secure Concurrent NMC w.r.t opening and i.i.d samples Main Lemma UC-Puzzle: Hard for Adversary to solve in real world Easy for Simulator to obtain trapdoor

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— Common Reference String [CLOS02,CPS07,CDPW07,DNO10] — Public Key Registration [BCNP04,DNO10] — Tamper-Proof Hardware [Kat07,CGS08,GISVW10] — Timing Model [DNS98,KLP05] Trusted Setups Corollaries Relaxed Security — Super-Poly Time Simulation (SPS) [Pas03, BS05, GGJS12] — Angel-based Security Model [PS04, MMY06,CLP10] — Bounded (Player) Concurrent [Barak, Goyal1, Goyal2] — Non-Uniform Simulation [LPV09] ✓ ✓ ✓ ✓ ✓ ✓ ✓

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Adaptive UC Security Sim. PKE and Puzzle O(nd)-rounds (d = depth(C)) Not Everything! [IKOS10] Static vs Adaptive Static UC Security AssumptionsSA-OT and Puzzle NECESS. And SUFF. RoundsO(1)-rounds What can we compute? Any PPT computation

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Conclusion Characterize when Adaptive UC is achievable Next… Reduce complexity assumptions – trapdoor simulatable PKE are suff. for NCE [CDMW09] – improve round complexity [Recent] UC-Adaptive Security in O(d)-rounds [V14] Angel Based UC-Security [PS04,CLP10,…] – reasonable model without any setup – implies SPS – linear-blowup in rounds with black-box tech. [GS12]

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How can we achieve semi-honest 2-party computation? O(1)-rnd adaptive ^ … still open

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THANKS

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