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The Complexity of Zero Knowledge Salil Vadhan Harvard University

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A Successful Marriage Complexity Theory: Which problems are “computationally hard” to solve? Cryptography: Design protocols that are “computationally hard” to break. hard problems, techniques revisit notions, adversarial view

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Two Areas of Interaction Pseudorandomness: generating objects that “look random” despite being constructed with little or no randomness. –Cryptography: many unpredictable bits from short key –Complexity: power of randomized algs (RP vs. P, RL vs. L) Zero-knowledge proofs: interactive proofs that reveal nothing other than validity of assertion being proven –Cryptography: central in study of crypto protocols –Complexity: augments NP $ “efficiently verifiable proofs”

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Cryptography Zero Knowledge Complexity Protocols [B82,...] Def of ZK, IP [GMR85] IP=PSPACE [LFKN90,S90] NP µ ZK [GMW86 ] NP-completeness [C71,L73,K72] Secure Computation [Yao86,GMW87, BGW88,CCD88] Multiprover ZK [BGKW88] MIP=NEXP PCP Theorem [BFL91...ALMSS92] Polylog-eff ZK Args [K92,M94] Random Oracle Model [FS86,BR93,CGH98] Concurrency [F90,DNS98] Diagonalization [T36] Non-BB Simulation [B01] ?

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This Talk Complexity-theoretic study of zero-knowledge proofs: Characterize the expressiveness of ZK. Prove general theorems about ZK. Minimize or eliminate complexity assumptions.

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ZK Complexity Classes SZKP SZKA CZKP CZKA Zero Knowledge statisticalcomputational statistical (“proofs”) computational (“arguments”) Soundness Verifier learns nothing Prover cannot convince Verifier of false statements [GMR85] [BCC86]

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Conditional Results on ZK SZKP SZKA CZKP CZKA Zero Knowledge statisticalcomputational statistical (“proofs”) computational (“arguments”) Soundness Complexity assumptions ) understand CZKP, SZKA, CZKA very well

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes. 2. Pick random edge. (1,4) Accept if colors different. 3. Send keys for endpoints. Com( )…Com( ) (,K 1 ),(,K 4 )

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Commitment Schemes Bit-commitment: Hiding: Com( ) & Com( ) indistinguishable. ( ) zero knowledge) Binding: W.h.p. z can be opened to only one value 2 {0,1}. ) soundness Sender Receiver commit stage: reveal stage: ( ,K) K z accept/ reject

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Assuming one-way functions exist... Conditional Results on CZKP 9 comp. hiding, stat. binding commitments [HILL90,N91] NP µ CZKP [GMW86] CZKP=IP=PSPACE [IY87,BGG+88,LFKN90,S90] CZKP = CZKP w/ public coins, perfect completeness [GS86,FGMSZ87] CZKP = honest-verifier CZKP CZKP closed under union, complement... CZKP Å NP has ZK pfs w/ poly-time prover (given witness) and O(1) rounds Thms:

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Conditional Results on SZKA 9 stat. hiding, comp. 1-out-of-2-binding commitments […,NOV06] NP µ SZKA [GMW86,BCC86] SZKA=MA (randomized NP) SZKA=SZKA w/ public coins, perfect completeness [GS86,FGMSZ87] SZKA=honest-verifier SZKA SZKA closed under union,… where SZKA=statistical ZK arguments w/poly-time prover Thms: Q: What can we prove about ZK unconditionally? Assuming one-way functions exist...

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Unconditional Results on ZK SZKP SZKA CZKP CZKA Zero Knowledge statisticalcomputational statistical (“proofs”) computational (“arguments”) Soundness Complexity assumptions don’t seem useful for SZKP (stat hiding, stat binding commitments impossible)

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Unconditional Results on SZKP SZKP contains Q UADRATIC R ESIDUOSITY [GMR85], G RAPH I SOMORPHISM [GMW86],... SZKP=SZKP w/public coins, perfect completeness [O96] SZKP closed under complement, union [O96] Complete Problems [SV97,GV99] SZKP=honest-verifier SZKP [DGW94,DOY97,GSV98] SZKP Å NP has SZKP pfs w/poly-time prover [NV06] And more [DDPY98,DSY00...] But more constrained: SZKP µ AM Å coAM [F86,AH87] ) unlikely to contain NP. Thms:

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Unconditional Results on CZKP New characterizations of CZKP CZKP = CZKP with public coins, perfect completeness CZKP = honest-verifier CZKP CZKP closed under union CZKP \ NP has CZKP proofs w/poly-time prover... Thm [V04,NV06]: Assuming one-way functions exist...

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Unconditional Results on CZKA New characterizations of CZKA CZKA = CZKA with public coins, perfect completeness CZKA = honest-verifier CZKA CZKA closed under union CZKA Å coMA closed under complement... Thm [OV06]: Assuming one-way functions exist...

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Unconditional Results on SZKA New characterizations of SZKA SZKA = SZKA with public coins, perfect completeness SZKA = honest-verifier SZKA SZKA closed under union SZKA = coCZKP Å MA... Thm [OV06]: Assuming one-way functions exist...

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How to get unconditional results on ZK? Thm [OW93]: If CZKA BPP, then a “weak form” of one- way functions exist. Idea: Case analysis. –Case I: CZKA=BPP. Everything trivial. –Case II: CZKA BPP. Use above OWF in conditional results. Problem: “Weak form” of OWF not enough (cf. [DOY97]) Our approach: –replace BPP by SZKP –case analysis on input-by-input basis –combine OWF-based results w/unconditional results on SZKP

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YESNOYESNO LanguagePromise Problem Example: U NIQUE S AT [VV86] excluded inputs Promise Problems [ESY84] Generalize all definitions (eg IP,CZKA) in natural way.

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SZKP/OWF T RIPLETS Def: ( J) with I µ Y, J µ N, is an SZKP/OWF T RIPLET if 9 poly-time { f x (y)} x 2 {0,1} * s.t. 1.Ignoring I and J, is in SZKP. 2.When x 2 I [ J, f x is hard to invert. 8 (nonuniform) poly-time A, x 2 I [ J Pr[A inverts f x (U poly(|x|) )] · negl(|x|) Y N I in SZKP instances yield OWF Note: 9 OWF ) every problem satisfies above. J Y N

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CZKP Characterization Theorem Thm [V04]: 2 CZKP m 2 IP and 9 I s.t. ( , I, ; ) is a SZKP/OWF T RIPLET Y N I in SZKP instances yield OWF J Y N

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CZKA Characterization Theorem Thm [OV06]: 2 CZKA m 2 MA and 9 I, J s.t. ( , I, J) is a SZKP/OWF T RIPLET Y N I in SZKP instances yield OWF J Y N

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SZKA Characterization Theorem Thm [OV06]: 2 SZKA m 2 MA and 9 J s.t. ( , ;, J) is a SZKP/OWF T RIPLET Y N in SZKP instances yield OWF J Y N

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SZKP/OWF Triplets: Summary SZKP SZKA CZKP CZKA Zero Knowledge statisticalcomputational statistical (“proofs”) computational (“arguments”) Soundness I= ;, J= ; I= ; J= ; Y N I in SZKP instances yield OWF J Y N “Zero Knowledge & Soundness are Symmetric”

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CZKA Characterization Theorem Thm [OV06]: 2 CZKA m 2 MA and 9 I, J s.t. ( , I, J) is a SZKP/OWF T RIPLET Y N I in SZKP instances yield OWF J Y N

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Proof of the Characterization Thms 2 honest-verifier CZKA even w/inefficient prover 9 I, J s.t. ( , I, J) is SZKP/OWF T RIPLET. 2 CZKA w/public coins, perfect completeness, poly-time prover proof system J= ; statistical ZK I= ; + 2 MA

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From ZK to SZKP/OWF T RIPLETS Lemma: If has an honest-verifier CZKA system (even w/inefficient prover), then 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET. Proof: Let (P,V) = honest-verifier CZKA system S = simulator Know: –x 2 Y ) S(x) comp. indistinguishable from (P,V)(x) –x 2 N ) no poly-time P * makes V accept w/nonnegl. prob. –WLOG S always outputs accepting transcripts.

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Analyzing the Simulator [F87,AH88,O91,PT96,SV97,GV99,…] S(x) (inefficient) strategies P S (x) and V S (x) Respond m i+1 to history (m 1,…,m i ) w.p. Pr[S(x) i+1 =m i+1 | S(x) 1…i =(m 1,…,m i )] Measure (statistical) “similarity” between V S (x) and V(x).

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Constructing the Triplet I = {x 2 Y : V S (x) not “similar” to V(x)} J = {x 2 N : V S (x) not “far” from V(x)} ( Y n I, N n J ) 2 SZKP: Distinguishing whether two samplable distributions are statistically “similar” vs. “far” is complete for SZKP [SV97,GV99] Y N I in SZKP instances yield OWF J Y N

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Constructing the Triplet I = {x 2 Y : V S (x) not “similar” to V(x)} J = {x 2 N : V S (x) not “far” from V(x)} OWF on I : S and (P,V)(x) computationally indistinguishable but statistically far ) OWF [HILL90,G90] Difficulty: (P,V)(x) not sampable given x Y N I in SZKP instances yield OWF J Y N

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Constructing the Triplet I = {x 2 Y : V S (x) not “similar” to V(x)} J = {x 2 N : V S (x) not “far” from V(x)} OWF on J: P S makes V S accept w.p. 1 ) P S makes V accept w.p..01 ) P S hard to approximate ) Simulator hard to invert [O91] Y N I in SZKP instances yield OWF J Y N

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Analyzing the Simulator [F87,AH88,O91,PT96,SV97,GV99,…] S(x) (inefficient) strategies P S (x) and V S (x) Respond m i+1 to history (m 1,…,m i ) w.p. Pr[S(x) i+1 =m i+1 | S(x) 1…i =(m 1,…,m i )] Measure (statistical) “similarity” between V S (x) and V(x). D(x) = entropy of V’s msgs – entropy of V S ’s msgs = #coins(V) - i H( S(x) 2i | S(x) 1…2i-1 ) (WLOG V sends even-numbered msgs, reveals coins at end.)

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Proof of the Characterization Thms 2 honest-verifier CZKA even w/inefficient prover 9 I, J s.t. ( , I, J) is SZKP/OWF T RIPLET. 2 CZKA w/public coins, perfect completeness, poly-time prover proof system J= ; statistical ZK I= ; + 2 MA

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From SZKP/OWF to ZK Idea: Use SZKP proof when x I [ J, use NP proof system when x 2 I [ J (with f x as OWF) Problem: cannot efficiently decide whether x 2 I [ J. Lemma: If 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET and 2 NP, then has a CZKA system with public coins, perfect completeness, and a poly-time prover. Y N I J SZKP OWF

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Sol’n: Instance-dependent Commitments Def [IOS94,MV03]: In an I.D. commitment scheme for , sender & receiver receive auxiliary input x s.t. –x 2 Y ) hiding –x 2 N ) binding Example [BMO90]: G RAPH I SOMORPHISM –aux. input = (G 0,G 1 ) –commitment to = random isomorphic copy of G –perfectly hiding and perfectly binding! H B

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Usefulness of I.D. Commitments –x 2 Y ) hiding –x 2 N ) binding Many ZK pfs only use hiding on YES instances (for ZK), binding on NO instances (for soundness). Example: Convoluted ZK proof for G RAPH I SOMORPHISM –Reduce (G 0,G 1 ) to instance G of 3-C OLORING. –Run [GMW86] protocol on G. –Using (G 0,G 1 ) to do the commitments. H B

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I.D. Commitments from SZKP/OWF H B H B SZKP has stat. hiding, stat. 1-out-of-2-binding i.d. commitments [NV06] OWF ) comp. hiding, stat. binding commitments [HILL90,N91] OWF ) stat. hiding, comp. 1-out-of-2-binding commitments [NOV06] Com SZKP Com I Com J SZKP/OWF Triplet ) comp. hiding comp. 1-out-of-2-binding i.d. commitments Com SZKP (b © r), Com I (r), Com J (b) H B B H

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Putting it Together Lemma: If 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET and 2 NP, then has a CZKA system with public coins, perfect completeness, and a poly-time prover. Proof: 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET ) has instance-dependent commitment Run generic NP protocol for with instance- dependent commitment.

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poly-time Verifier Prover 1. Randomly permute coloring & send in locked boxes. 2. Pick random edge. (1,4) Accept if colors different. 3. Send keys for endpoints. Com x ( )…Com x ( ) x (,K 1 ),(,K 4 ) Putting it Together

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Conclusions ZK continues to be an lively interface between cryptography and complexity theory. SZKP/OWF Characterizations of ZK ) unconditional results Variations on commitments –Instance-dependent commitments –1-out-of-2-binding commitments (next talk!)

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Outline Review of Zero Knowledge “An Unconditional Study of Computational Zero Knowledge” (V., FOCS `04) “Zero Knowledge with Efficient Provers” (Nguyen-V., STOC `06) “Statistical Zero-Knowledge Arguments for NP from Any One-Way Function” (Nguyen-Ong-V. `06)

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Efficient Provers Thm [Nguyen-V06]: Every 2 ZK Å NP has a zero- knowledge proof where prover poly-time w/NP witness. –SZK Å NP ! statistical zero-knowledge w/poly-time prover –Improves BPP NP in [BP92,V04] Proof idea: Construct instance-dependent 1-out-of-2-binding commitments for all of SZK & ZK. Show these suffice to construct ZK proofs for NP.

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1-out-of-2-Binding Commitments Sender Receiver commit 1 : reveal 1 : ( ,K 1 ) K1K1 z1z1 commit 2 : reveal 2 : ( ,K 2 ) K2K2 z1z1 Hiding: Both phases hiding ) ZK Binding: Sender can change value at most once ) Soundness

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1-out-of-2-binding Commitments ) ZK for NP Prover Verifier Commit 1 (coloring) Hiding: Both phases hiding ) ZK Binding: Sender can change value at most once ) Soundness Edge Reveal 1 Commit 2 (coloring) Edge Reveal 2 Intuitive idea: Run 3-coloring protocol twice

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1-out-of-2 Binding Commitments b2b2 b1b1

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b2b2 b1b1 Case 1: Change value of b 1 ) b 2 binding _

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1-out-of-2 Binding Commitments b2b2 b1b1 Case 1: Change value of b 1 ) b 2 binding Case 2: Keep value of b 1 ) b 2 not nec. binding _

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Efficient Provers Thm [Nguyen-V06]: Every 2 ZK Å NP has a zero- knowledge proof where prover poly-time w/NP witness. –SZK Å NP ! statistical zero-knowledge w/poly-time prover –Improves BPP NP in [BP92,V04] Proof idea: Construct instance-dependent 1-out-of-2-binding commitments for all of SZK & ZK. Show these suffice to construct ZK proofs for NP. (Std. I.D.-commitments still of interest, e.g. imply ZK=concurrent-ZK [MOPS05])

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Outline Review of Zero Knowledge “An Unconditional Study of Computational Zero Knowledge” (V., FOCS `04) “Zero Knowledge with Efficient Provers” (Nguyen-V., STOC `06) “Statistical Zero-Knowledge Arguments for NP from Any One-Way Function” (Nguyen-Ong-V. `06)

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commitments impossible! SZK Statistical ZK Arguments Soundness (Binding) ZK (Hiding) statistical computational statistical computational 9 commitments iff 9 one-way functions [HILL89,Nao89] 9 under various complexity assumptions (“proofs”) (“arguments”) ZK

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Complexity of SZK arguments for NP number-theoretic assumptions claw-free perm SZK arguments stat. hiding comp. binding commitments [BCC] [GMR,BKK] [NY] collision-resistant hash functions [GMR, Damgard] [GK]

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Complexity of SZK arguments for NP number-theoretic assumptions claw-free perm one-way perm regular OWF SZK arguments stat. hiding comp. binding commitments [HHK + 05] [NOVY 92] [BCC] [GMR,BKK] [NY] collision-resistant hash functions [GK]

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Complexity of SZK arguments for NP [Nguyen-Ong-V06] number-theoretic assumptions claw-free perm one-way perm regular OWF one-way function SZK arguments stat. hiding comp.1-out-of-2-binding commitments stat. hiding comp. binding commitments [HHK + 05] [NOVY 92] [BCC] [NY] collision-resistant hash functions [GMR,BKK] [GK]

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Conclusion ZK continues to be an exciting interface between cryptography and complexity theory. Future impacts on complexity theory? –Non-black-box reductions –SZK-completeness

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Zero-Knowledge Proofs [GMR85] Interactive proofs that reveal nothing other than the validity of assertion being proven. Central tool in study of cryptographic protocols Major source of interaction between cryptography & complexity theory

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Outline Zero Knowledge & the Complexity-Crypto Interface Non-Black-Box Zero Knowledge Unconditional Results on Zero Knowledge

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes. 2. Pick random edge. (1,4) Accept if colors different. 3. Send keys for endpoints. (Perfect) Completeness: graph 3-colorable ) V accepts w.p. 1

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes. 2. Pick random edge. (1,4) Soundness: graph not 3-colorable ) 8 P * V rejects w.p. ¸ 1/(#edges) 4. Accept if colors different. 3. Send keys for endpoints.

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NP µ ZK [GMW86] ZK pf for G RAPH 3-C OLORING poly-time Verifier unbounded Prover 1. Randomly permute coloring & send in locked boxes. 2. Pick random edge. (1,4) Zero Knowledge: graph 3-colorable ) can simulate interaction w/o prover 4. Accept if colors different. 3. Send keys for endpoints.

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Flavors of Commitments & ZK Binding ( ) Soundness) Hiding ( ) ZK) statistical computational statistical computational iff 9 one-way functions [HILL89,Nao89] (“proofs”) (“arguments”) ZK poly-time f : {0,1} * ! {0,1} * is a one-way function if 8 nonuniform poly-time A 8 n Pr[A inverts f(U n )] · negl(n)

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Flavors of Zero-Knowledge Proofs Quality of ZK/Simulation: –Perfect (PZK) –Statistical (SZK) –Computational (ZK) Verifier strategies considered: –Honest-verifier zero knowledge (HVZK) –General zero knowledge (ZK) Prover strategies considered in Soundness: –Proof systems: unbounded provers –Arguments: poly-time provers

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Complexity Issues Soundness error –Can be reduced by sequential repetitions –ZK not preserved under parallel repetition [FS90,GK90] Round complexity –Constant rounds with negligible error? [FS89,GK88] Communication complexity –Can be reduced to polylog for arguments [K92,M94], using PCP Theorem Computational complexity –Prover polynomial time given NP witness Minimizing assumptions Public coins (aka Arthur-Merlin [B85]) vs. private coins (cf. [GS86])

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Complexity-theoretic interest in ZK NP: What can be proven to an efficient verifier? IP: Do randomness & interaction add power? ZK: What can be proven with secrecy?

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Outline Zero Knowledge & the Complexity-Crypto Interface Non-Black-Box Zero Knowledge Unconditional Results on Zero Knowledge

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Cryptography Zero Knowledge Complexity Protocols [B82,...] Def of ZK, IP [GMR85] IP=PSPACE [LFKN90,S90] NP µ ZK [GMW86 ] NP-completeness [C71,L73,K72] Secure Computation [Yao86,GMW87, BGW88,CCD88] Multiprover ZK [BGKW88] MIP=NEXP PCP Theorem [BFL91...ALMSS92] Polylog-eff ZK Args [K92,M94] Random Oracle Model [FS86,BR93,CGH98] Concurrency [F90,DNS98] Diagonalization [T36] Non-BB Simulation [B01] ?

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Non-black-box Simulation Thm [Barak01]: Assuming 9 collision-resistant hash functions, NP has ZK arguments with 1.O(1) rounds 2.Negligible soundness error 3.Public coins 4.“Bounded-concurrent” ZK Impossible w/simulators that use (malicious) verifier as a “black-box” [GK90] Tool: Witness-Indistinguishable Proofs [FS90] – 8 w 1,w 2,V * (P(w 1 ),V * ) ´ (P(w 2 ),V * ) –Preserved under parallel & concurrent composition

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Barak’s Protocol Verifier Prover Completeness: prover uses (A) w/real NP witness x B) “I know V’s program & coin tosses” z=Com( ) r Ã {0,1} n B) 9 s.t. z=Com( ), (z)=r WI Proof that A) x 2 L OR Soundness: (B) ) z “predicts” r or commitment broken ) negligible prob. Zero Knowledge: simulate malicious V* using ( ¢ ) = V * ( ¢ ;r) V* non-bb! V*(z;r)

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Barak’s Protocol Verifier Prover x z=Com( ) r Ã {0,1} n B) 9 s.t. z=Com( ), (z)=r WI Proof that A) x 2 L OR V* V*(z;r) Problem: running time of V* not bounded by a fixed poly. Solution: Use WI arguments for NTIME(t) with running time poly(log(t),n). constructed in [K92,M94] using PCP Theorem.

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Back to Complexity Theory Standard def: P reduces to Q if can solve P in poly- time given a black-box for Q. Have non-BB “reductions” in complexity: –SAT 2 P ) PH=P What else can we do with them? –Derandomization [IW98] –Worst-case/avg-case connections for NP [GST05] –Non-relativizing separations?

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Outline Zero Knowedge & the Complexity-Crypto Interface Non-Black-Box Zero Knowledge Unconditional Results on Zero Knowledge

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commitments impossible! SZK Statistical Zero Knowledge Soundness (Binding) ZK (Hiding) statistical computational statistical computational 9 commitments iff 9 one-way functions [HILL89,Nao89] (“proofs”) (“arguments”) ZK

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Outline Review of Zero Knowledge “An Unconditional Study of Computational Zero Knowledge” (V., FOCS `04) “Zero Knowledge with Efficient Provers” (Nguyen-V., STOC `06) “Statistical Zero-Knowledge Arguments for NP from Any One-Way Function” (Nguyen-Ong-V. `06)

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Outline Review of Zero Knowledge “An Unconditional Study of Computational Zero Knowledge” (V., FOCS `04) “Zero Knowledge with Efficient Provers” (Nguyen-V., STOC `06) “Statistical Zero-Knowledge Arguments for NP from Any One-Way Function” (Nguyen-Ong-V. `06)

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The SZK/OWF C HARACTERIZATION Def: satisfies the SZK/OWF C ONDITION if 9 I µ Y, poly-time { f x (y)} x 2 {0,1} *... Main Thm: 2 ZK if and only if 2 IP and satisfies the SZK/OWF C ONDITION. Y N I in SZK Y N OWF instances yield OWF

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OWF vs all poly-time Comparison w/[OW93] Corollary: ZK SZK ) 9 poly-time { f x (y)} x 2 {0,1} * 9 infinite set I 8 PPT A, x 2 I Pr[A inverts f x (U poly(|x|) )] · negl(|x|) Theorem [OW93]: ZK BPP ) 9 poly-time { f x (y)} x 2 {0,1} * 8 PPT A 9 1 ’ly many x Pr[A inverts f x (U poly(|x|) )] · negl(|x|) OWF vs Time( n ) OWF vs Time( n 2 ) OWF vs Time( n 3 ) OWF vs Time( n 4 ) OWF vs Time( n 5 )

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CZKA Characterization Theorem Thm [OV06]: For 2 MA, the following are equivalent: 2 CZKA 2 honest-verifier CZKA (even w/inefficient prover) 3. 9 I,J s.t. ( , I, J) is a SZKP/OWF T RIPLET has a CZKA protocol w/public coins, perfect completeness, and poly-time prover (if 2 MA) Y N I in SZKP instances yield OWF J Y N

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SZKA Characterization Theorem Thm [OV06]: For 2 MA, the following are equivalent: 2 SZKA 2 honest-verifier SZKA (even w/inefficient prover) 3. 9 J s.t. ( , ;, J) is a SZKP/OWF T RIPLET has an SZKA protocol w/public coins, perfect completeness, and poly-time prover (if 2 MA) Y N I in SZKP instances yield OWF J Y N

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I.D. commitments for SZK Thm: Every problem in SZK has an instance-dependent commitment scheme. +Public coins +Statistically hiding & statistically binding –Most technical part of paper, uses [SV97,GV99,O96] –Sender not poly-time, but BPP NP

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I.D. commitments for ZK Thm: Every problem satisfying SZK/OWF C ONDITION has an instance-dependent commitment scheme. Public coins, BPP NP sender, computationally hiding Pf Sketch: To commit to , –Randomly decompose as = 1 © 2. –Commit to 1 w/ SZK commitment –Commit to 2 w/ OWF-commitment from f x. H B 1© 21© 2 11 H B H 22 B

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Putting it Together Thm [V04]: Every 2 ZK has a ZK proof with public coins perfect completeness BPP NP prover, if 2 NP Proof: 2 ZK ) satisfies SZK/OWF C ONDITION ) has instance-dependent commitment Use general NP/IP-to-ZK construction ([GMW86,IY87,BGG+88]), but with instance-dependent commitment.

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Proof of Characterization Thms Lemma: If has an honest-verifier CZKA system (even w/inefficient prover), then 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET. Moreover, proof system ) J= ;, statistical ZK ) I= ;. Lemma: If 9 I, J s.t. ( , I, J) is an SZKP/OWF T RIPLET and 2 MA, then has a CZKA system with public coins, perfect completeness, and a poly-time prover. Moreover, J= ; ) proof system, I= ; ) statistical ZK.

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