Rafael Pass Cornell University Constant-round Non-malleability From Any One-way Function Joint work with Huijia (Rachel) Lin

Commitment Scheme The “digital analogue” of sealed envelops. Commitment Reveal Sender Receiver One of the most basic cryptographic tasks. Part of essentially all more involved secure computations Can be constructed from any one way function. [N’89, HILL’ 99]

“Right” abstraction if: AliceBob

But life is:

Possible that v’ = v+1 Even though MIM does not know v! Receiver/Sender MIM C(v) C(v’) Sender Receiver

Non-Malleable Commitments [Dolev Dwork Naor’91] Non-malleability: Either MIM forwards : v = v’ Or v’ is “independent” of v ij Receiver/Sender MIM C(v’) Sender Receiver C(v)

Non-Malleable Commitments [Dolev Dwork Naor’91] Receiver/Sender Non-malleability: if then, v’ is “independent” of v MIM C(i,v) C(j, v’) i  j Sender Receiver ij

Man-in-the-middle execution: Simulation: j i  ji  j Non-Malleable Commitments [Dolev Dwork Naor’91] ij Non-malleability: For every MIM, there exists a “simulator”, such that value committed by MIM is indistinguishable from value committed by simulator

Non-Malleable Commitments [Dolev Dwork Naor’91] ij Important in practice “Test-bed” for other tasks Applications to MPC

Non-malleable Commitments Original Work by [DDN’91] –OWF –black-box techniques –But: O(log n) rounds Main question: how many rounds do we need? With set-up solved: 1-round, OWF: [DiCreczenzo-Ishai- Ostrovsky’99,DKO,CF,FF,…,DG] Without set-up: [Barak’02]: O(1)-round Subexp CRH + dense crypto: [P’04,P-Rosen’05]: O(1) rounds using CRH [Lin-P’09]: O(1)^log* n round using OWF [P-Wee’10]: O(1) using Subexp OWF [Wee’10]: O(log^* n) using OWF Non BB

Non-malleable Commitments Original Work by [DDN’91] –OWF –black-box techniques –But: O(log n) rounds Main question: how many rounds do we need? With set-up solved: 1-round, OWF: [DiCreczenzo-Ishai- Ostrovsky’99,DKO,CF,FF,…,DG] Without set-up: O(1)-round from CRH or Subexp OWF O(log^* n) from OWF Sd

Main Theorem Thm: Assume one-way functions. Then there exists a O(1)- round non-malleable commitment with a black-box proof of security. Note: Since commitment schemes imply OWF, we have that unconditionally that any commitments scheme can be turned into one that is O(1)-round and non-malleable. Note: As we shall see, this also weakens assumptions for O(1)- round secure multi-party computation.

DDN Protocol Idea Blue does not help Red and vice versa i = 01…1 j = C(i,v) C(j, v’)

The Idea: What if we could run the message scheduling in the head? Let us focus on non-aborting and synchronizing adversaries. (never send invalid mess in left exec)

c=C(v) Com(id,v): I know v s.t. c=C(v) Or I have “seen” sequence WI-POK id = 00101

Signature Chains Consider 2 “fixed-length” signature schemes Sig 0, Sig 1 (i.e., signatures are always of length n) with keys vk 0, vk 1. Def: (s,id) is a signature-chain if for all i, s i+1 is a signature of “(i,s 0 )” using scheme id i s 0 = r s 1 = Sig 0 (0,s 0 )id 1 = 0 s 2 = Sig 0 (1,s 1 )id 2 = 0 s 3 = Sig 1 (2,s 2 )id 3 = 1 s 4 = Sig 0 (3,s 3 )id 4 = 0

Signature Games You have given vk 0, vk 1 and you have access to signing oracles Sig 0, Sig 1. Let  denote the access pattern to the oracle; –that is  i = b if in the i’th iteraction you access oracle b. Claim: If you output a signature-chain (s,id) Then, w.h.p, id is a substring of the access pattern .

c=C(v) Com(id,v): I know v s.t. c=C(v) Or I have “seen” sequence WI-POK id = vk 0 r0r0 Sign 0 (r 0 ) vk 1 r1r1 Sign 1 (r 1 )

c=C(v) Com(id,v): WI-POK id = vk 0 r0r0 Sign 0 (r 0 ) vk 1 r1r1 Sign 1 (r 1 ) I know v s.t. c=C(v) Or I know a sig-chain (s,id) w.r.t id

c=C(v) WI-POK vk 0 r0r0 Sign 0 (r 0 ) vk 1 r1r1 Sign 1 (r 1 ) c=C(v) WI-POK vk 0 r0r0 Sign 0 (r 0 ) vk 1 r1r1 Sign 1 (r 1 ) w.r.t i i = j = w.r.t j Non-malleability through dance * In actual protocol need “many” seq WIPOK a la [LP‘09]

Dealing with Aborting Adversaries Problem 1: –MIM will notice that I ask him to sign a signature chain –Solution: Don’t. Ask him to sign commitments of sigs… (need to add a POK of commitment to prove sig game lemma) Problem 2: –I might have to “rewind” many times on left to get a single signature –So if I have id = 01011, access pattern on the right is 0*1*0*1*... –Solution: Use 3 keys (0,1,2); require chain w.r.t 2id 1 2id 2 2id 3 …

Main Theorem Main Technique Exploit rewinding pattern (instead of just location) Thm: Assume one-way functions. Then there exists a O(1)- round non-malleable commitment with a black-box proof of security. Some applications

Secure Multi-party Computation [Yao,GMW] A set of parties with private inputs. Wish to jointly compute a function of their inputs while preserving privacy of inputs (as much as possible) Security must be preserved even if some of the parties are malicious.

Original work of [Goldreich-Micali-Wigderson’87] –TDP, n rounds More Recent: “Stronger assumption, less rounds” –[Katz-Ostrovsky-Smith’03] TDP, dense cryptosystems, log n rounds TDP, CRH+dense crypto with SubExp sec, O(1)-rounds, non-BB –[P’04] TDP, CRH, O(1)-round, non-BB Secure Multi-party Computation [Yao,GMW]

NMC v.s. MPC Thm [ Lin-P-Venkitasubramaniam’09 ]: TPD + k-round “robust” NMC  O(k)-round MPC Holds both for stand-alone MPC and UC-MPC (in a number of set-up models) Corollary: TDP  O(1)-round MPC

NM ZK Thm [ Lin-P-Tseng-Venkitasubramaniam’10 ]: k-round “robust” NMC  O(k)-round NMZK Corollary: OWF  O(1)-round NMZK Can also get Conc NMZK if adding ω(log n) rounds

What’s Next – Adaptive Hardness Consider the Factoring problem: Given the product N of 2 random n-bit primes p,q, can you provide the factorization Adaptive Factoring Problem: Given the product N of 2 random n-bit primes p,q, can you provide the factorization, if you have access to an oracle that factors all other N’ that are products of equal-length primes Are these problems equivalent? Unknown!

Adaptively-hard Commitments [Canetti-Lin-P’10] Commitment scheme that remains hiding even if Adv has access to a decommitment oracle Implies Non-malleability (and more!) Thm [CLP’10] Existence of commitments implies O(n^  )- round Adaptively-hard commitments What’s Next – Adaptive Hardness

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