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Rafael Pass Cornell University Limits of Provable Security From Standard Assumptions

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1.Precisely define security goal (e.g., secure encryption) 2.Precisely stipulate computational intractability assumption (e.g., hardness of factoring) 1.Security Reduction: prove that any attacker A that break security of scheme π can be used to violate the intractability assumption. Modern Cryptography

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A Celebrated Example: Commitments from OWFs [Naor,HILL] Task: Commitment Scheme –Binding + Hiding –Non-interactive Intractability Assumption: existence of OWF f –f is easy to compute but hard to invert Security reduction [Naor,HILL]: Com f, PPT R s. t. for every algorithm A that breaks hiding of Com f, R A inverts f –Reduction R only uses attacker A as a black-box; i.e., R is a Turing Reduction.

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r CRARA Security reduction: R A breaks C whenever A breaks Hiding f(r) Reduction R may rewind and restart A. Turing Reductions

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Provable Security In the last three decades, lots of amazing tasks have been securely realized under well-studied intractability assumptions –Key Exchange, Public-Key Encryption, Secure Computation, Zero- Knowledge, PIR, Secure Voting, Identity based encryption, Fully homomorphic Encryption, Leakage-resilient Encryption… But: several tasks/schemes have resisted security reductions under well-studied intractability assumptions.

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Schnorr’s Identification Scheme [Sch’89] One of the most famous and widely employed identification schemes (e.g., Blackberry router protocol) Secure under a passive “eaves-dropper” attack based on the discrete logarithm assumption What about active attacks? –[BP’02] proven it secure under a new type of “one-more” inversion assumption –Can we based security on more standard assumptions?

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Commitment Schemes under Selective Opening [DNRS’99] A commits to n values v 1, …, v n B adaptively asks A to open up, say, half of them. Security: Unopened commitments remain hidden –Problem originated in the distributed computing literature over 25 years ago Can we base selective opening security of non- interactive commitments on any standard assumption?

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One-More Inversion Assumptions [BNPS’02] You get n target points y 1,…, y n in group G with generator g. Can you find the discrete logarithm to all n of them if you may make n- 1 queries to a discrete logarithm oracle (for G and g) One-more DLOG assumption states that no PPT algorithm can succeed with non-negligible probability –[BNPS] and follow-up work: Very useful for proving security of practical schemes Can the one-more DLOG assumption be based on more standard assumptions? –What about if we weaken the assumption and only give the attacker n^eps queries?

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Unique Non-interactive Blind Signatures [Chaum’82] Signature Scheme where a user U may ask the signer S to sign a message m, while keeping m hidden from S. –Futhermore, there only exists a single valid signature per message –Chaum provided a first implementation in 1982; very useful in e.g., E-cash –[BNPS] give a proof of security in the Random Oracle Model based on a one-more RSA assumption. Can we base security of Chaum’s scheme, or any other unique blind signature scheme, on any standard assumption?

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Sequential Witness Hiding of O(1)-round public-coin protocols Take any of the classic O(1)-round public-coin ZK protocols (e.g., GMW, Blum) Repeat them in parallel to get negligible soundness error. Do they suddenly leak the witness to the statement proved? [Feige’90] –Sequential WH: No verifier can recover the witness after sequentially participating in polynomially many proofs. Can sequential WH of those protocols be based on any standard assumption?

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Main Result For a general class of intractability assumptions, there do NOT exists Turing security reductions for demonstrating security of any those schemes/tasks/assumptions Any security reduction R itself must constitutes an attack on assumption

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Intractability Assumptions Following [Naor’03], we model an intractability assumption as a interaction between a Challenger C and an attacker A. –The goal of A is to make C accept –C may be computationally unbounded (different from [Naor’03], [GW’11]) –The only restriction is that the number of communication rounds is an a-priori bounded polynomial. r CA f(r) Intractability assumption (C,t) : “no PPT can make C output 1 w.p. significantly above t” E.g., 2-round: f is a OWF, Factoring, G is a PRG, DDH, Factoring, … O(1)-round: Enc is semantically secure (FHE), (P,V) is WH, O(1)-round with unbounded C: (P,V) is sound

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Main Theorem Let (C,t) be a k(.)-round intractability assumption where k is a polynomial. If there exists a PPT reduction R for basing security of any of previously mentioned schemes, on the hardness of (C,t), then there exists a PPT attacker B that breaks (C,t) Note: restriction on C being bounded-round is necessary; otherwise we include the assumptions that the schemes are secure!

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Related Work Several earlier lower bounds: –One-more inversion assumptions [BMV’08] –Selective opening [BHY’09] –Witness Hiding [P’06,HRS’09,PTV’10] –Blind Signatures [FS’10] But they only consider restricted types of reductions (a la [FF’93,BT’04]), or (restricted types of) black-box constructions (a la [IR’88]) –Only exceptions [P’06,PTV’10] provide conditional lower-bounds on constructions of certain types of WH proofs based on OWF Our result applies to ANY Turing security reduction and also non-black-box constructions.

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Proof Outline 1.Sequential Witness Hiding is “complete” –A positive answer to any of the questions implies the existence of a “special” O(1)-round sequential WH proof/argument for a language with unique witnesses. 2.Sequential WH of “special” O(1)-round proofs/arguments for languages with unique witnesses cannot be based on poly-round intractability assumptions using a Turing reduction.

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Special-sound proofs [CDS,Bl] X is true a a b0b0 c0c0 b1b1 c1c1 Can extract a witness w b 0, b 1 R {0,1} n Relaxations: multiple rounds computationally sound protocols (a.k.a. arguments) need p(n) transcripts (instead of just 2) to extract w Generalized special-sound

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Main Lemma Let (C,t) be a k(.)-round intractability assumption where k is a polynomial. Let (P,V) be a O(1)-round generalized special-sound proof of a language L with unique witnesses. If there exists a PPT reduction R for basing sequential WH of (P,V) on the hardness of (C,t), then there exists a PPT attacker B that breaks (C,t)

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Proof Idea r CRARA Assume R A breaks C whenever A completely recovers witness of any statement x it hears sufficiently many sequential proofs of. f(r) Goal: Emulate in PPT a successful A’ for R thus break C in PPT (the idea goes back to [BV’99] “meta-reduction”, and even earlier [Bra’79])

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Proof Idea r CR Assume R A breaks C whenever A breaks seq WH of some special-sound proof for language with unique witness f(r) Assume reduction R is “nice” [BMV’08,HRS’09,FS’10] Only asks a single query to its oracle (or asks queries sequentially) Then, simply “rewind” R feeding it a new “challenge” and extract witness x w Unique witness requirement crucial to ensure we emulate a good oracle A’

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General Reductions: Problem I R x1x1 Problem: R might nest its oracle calls. “naïve extraction” requires exponential time (c.f., Concurrent ZK [DNS’99]) Solution: If we require R to provide many sequential proofs, then we can find (recursively) find one proof where nesting depth is “small” Use Techniques reminiscent of Concurrent ZK a la [RK’99], [CPS’10] x2x2 x3x3 rewinding here: redo work of nested sessions w2w2 w3w3 w1w1

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General Reductions: Problem II Problem: R might not only nest its oracle calls, but may also rewind its oracle Special-soundness might no longer hold under such rewindings. Solution: Pick messages from oracle using hashfunction. Use Techniques reminiscent of Black-box ZK lower-bound of [GK’90],[P’06] O(1)-round restriction on (P,V) is here crucial

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General Reductions: Problem III CR x w Problem: Oracle calls may be intertwined with interaction with C Solution: If we require R to provide many sequential proofs, then at least one proof is guaranteed not to intertwine

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1.Security of several “classic” cryptographic tasks/schemes---which are believed to be secure--- cannot be proven secure (using Turing reduction) based on “standard” intractability assumptions. 1.Establish a connection between lower-bounds for security reductions and Concurrent Security In Sum

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The GOOD: Provably secure under standard assumptions The BAD: broken The ANNOYING : not broken, not provably secure* …but very efficient

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Ways Around It? Super Polynomial Security Reductions: Basing security on “super-poly” intractability assumptions Possible to overcome some, but not all, lower-bounds Full characterization in the paper. Non-black-box security reductions: Allow R to look at the code of A Our lower-bound do NOT apply Possible to overcome the Main Lemma [B’01,PR’06] PPT Turing security reductions provide stronger security guarantees: any attacker---even if I don’t know the description of his brain--with reproducible behavior can be be efficiently used to break challenge New types of assumptions? Instead of intractability, tractability [W’10]? “knowledge”-assumptions? Hard to “falsify” [Naor’03]

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