Statistical Zero-Knowledge:

A survey of recent

developments

Amit SahaiMIT

Zero-knowledge Proofs [GMR85]

• One party (“the prover”) convinces another party (“the verifier”) that some assertion is true,

• The verifier learns nothing except that the assertionis true!

• Statistical zero-knowledge: variant in which “learns nothing” is interpreted in a very strong information-theoretic sense.

Organization

• Motivation

• What is statistical zero-knowledge?

• Complete Problems

• Honest verifier vs. any verifier

• Noninteractive statistical zero-knowledge

Will not address works on power of the prover [BP92] or knowledge complexity [GMR85,GP91,GOP94,ABV95,PT96]

Motivation from Cryptography

• Statistical ZK proofs: strongest security guarantee

• Identification schemes [GMR85,FFS87]

• “Cleanest” model of ZK:

– allows for unconditional results (eg., [Oka96, GSV98])

– most suitable for initial study, later generalize techniques to other

types of ZK (eg., [Ost91,OW93,GSV98]).

• Zero-knowledge cryptographic protocols [GMW87]

• But statistical ZK proofs not as expressive as computational ZK or ZK arguments [GMW86,BCC87,F87,AH87]

Still study of statistical ZK useful:

Motivation from Complexity

• Contains “hard” problems:

– QUADRATIC (NON)RESIDUOSITY [GMR85],

– GRAPH (NON)ISOMORPHISM [GMW86]

– DISCRETE LOG [GK88],

– APPROX SHORTEST AND CLOSEST VECTOR [GG97]

• Yet SZK AM coAM [F87,AH87], so unlikely to contain

NP-hard problems [BHZ87,Sch88]

• Has natural complete problems.

What is Statistical Zero-Knowledge?

YES NO YES NO

0,1 *0,1 *

Language Promise Problem

Example: UNIQUE SAT [VV86]

bleunsatisfia is

assignment satisfying 1exactly has

:US

:USY

N

excluded inputs

Promise Problems [ESY84]

Statistical Zero-Knowledge Proof [GMR85]for a promise problem

v1

p1

v2

pk

accept/reject

Prover Verifier

Interactive protocol in which computationally unbounded Prover tries to convince probabilistic poly-time Verifier that a string x is a YES instance.

• When x is a YES instance, Verifier accepts w.h.p.• When x is a NO instance, Verifier rejects w.h.p. no matter what strategy Prover uses.

Statistical Zero-Knowledge Proof (cont.)

v1

p1

v2

pk

accept/reject

When x is a YES instance, Verifier can simulate her view of the interaction on her own.

Formally, there is probabilistic poly-time simulator such that, when x is a YES instance, its output distribution is statistically close to Verifier’s view of interaction with Prover.

Note: ZK for “honest verifier” only.

HVSZK = {promise problems possessing such proofs}

Example: GRAPH ISOMORPHISM [GMW86]

10 ,GG Graphs :Input

.0G

H

ofcopy isomorphic random Let

.1,0R

coin Flip

.HGcoin ifAccept

H

1.

2.

4.

Prover Verifier

Claim: Protocol is an (honest ver) SZK proof.

10 GG :YES

10 GG :NO

coin

3.

.HGcoin and between misomorphis

(random) a be Let

Correctness of GRAPH ISO. SZK Proof

Completeness: accepts Verifier HGGG coin10

Soundness:

21 y probabilit withrejects Verifier

sends) prover whatmatter (no2

1 y probabilit with

H

HGGG coin10

What about zero-knowledgeness?

Zero-knowledgeness of GRAPH ISO. Proof

Simulator on input (G0,G1):

,,

).(

.

.1,0

coinH

GH

S

coin

coin

nR

R

Output 4.

Let 3.

npermutatio random a Choose 2.

Flip 1.

Analysis: If G0 G1, then, in both simulator & protocol,

• H is a random isomorphic copy of G0 (equivalently, G1).

• coin is random & independent of H.

• is a random isomorphism between Gcoin and H.

distributions are identical.

Other types of zero-knowledge proofs

• Different quality of simulation:HVPZK — “Perfect” : distributions identical

HVSZK — “Statistical”: statistically close (negligible deviation)

HVCZK — “Computational”: computationally indistinguishable.

• Cheating-verifier versions: PZK,SZK,CZK

• Complexity:– CZK=IP=PSPACE NP if one-way functions exist

[GMW86,IY87,BGG+88,LFKN90,Sha90]– but SZK unlikely to contain NP-hard problems [F87,AH87,BHZ87,Sch88]

Other types of zero-knowledge proofs

• Different quality of simulation:HVPZK — “Perfect” : distributions identical

HVSZK — “Statistical”: statistically close (negligible deviation)

HVCZK — “Computational”: computationally indistinguishable.

• Cheating-verifier versions: PZK,SZK,CZK

• Private coins vs. Public coins:– Private coins: No restrictions on Verifier.– Public coins: Verifier only sends random bits.

Results

• Complete problem for HVSZK [SV97]– New characterization of statistical zero-knowledge.– Simplify study of entire class.

• Applications of complete problems [SV97]– Very efficient HVSZK proofs.– Strong closure properties of HVSZK.– Simpler proofs of most previously known results.– Manipulating statistical properties of efficiently sampleable

distributions.– Knowledge complexity.

[Mostly joint work with Oded Goldreich and Salil Vadhan]

Results (cont.)

• Private coins vs. public coins [GV99]– Transform any HVSZK proof system into a “public coin” one

(i.e., verifier’s messages are just random coins flips)– Originally proved by Okamoto [Oka96]; new proof much simpler

• Honest verifiers vs. cheating verifiers [GSV98]– Transform public-coin honest-verifier ZK proofs to cheating-

verifier ZK proofs.– Combining w/previous result, HVSZK=SZK.– Honest-verifier ZK results translate to cheating-verifier ZK.

• “Noninteractive” SZK [GSV99]– Complete problems related to those for SZK– Use these to compare the two classes.

Complete Problems for HVSZK

The Complexity of SZK

• SZK contains “hard” problems [GMR85,GMW86,GK93,GG98]

• Fortnow’s Methodology [F87]:

1. Find properties of simulator’s output that distinguishbetween YES and NO instances.

2. Show that these properties can be decided in lowcomplexity.

• Using this: SZK AM coAM. [F87,AH87]

• Obtain upper-bound on complexity of SZK, butdoes not give a characterization of SZK.

Refinement of Fortnow Methodology [SV97]

is a complete problem for SZK, i.e– every problem in SZK reduces to (via 1,2). SZK (by 3).

1. Find properties of simulator’s output that distinguish between YES and NO instances.

2. Show that these properties can be decided in low complexity.

2. Embed these properties in a natural computational problem .

3. Exhibit a statistical zero-knowledge proof for .

A Complete Problem

Def: STATISTICAL DIFFERENCE (SD) is the following promise problem:

SDY C0 ,C1 : C0 C1 23

SDN C0 ,C1 : C0 C1 13

C0 ,C1 are

circuits

Thm [SV97]: SD is complete for SZK.

X Y X Y Area

2

Statistical Difference between distributions

How circuits define distributions

circuit

n1,0 on dist uniform m0,1 ondist output

Meaning of Completeness Thm

• “The assertions that can be proven in statistical zero knowledge are exactly those that can be cast as comparing the statistical difference between two sampleable distributions.”

• Characterizes HVSZK with no reference to interaction or zero knowledge.

• Tool for proving general theorems about HVSZK.

• Results about HVSZK Techniques for manipulating sampleable distributions

Refinement of Fortnow Methodology [SV97]

is a complete problem for SZK, i.e– every problem in SZK reduces to (via 1,2). SZK (by 3).

1. Find properties of simulator’s output that distinguish between YES and NO instances.

2. Show that these properties can be decided in low complexity.

2. Embed these properties in a natural computational problem .

3. Exhibit a statistical zero-knowledge proof for .

Proof Ideas: Analyzing the simulator

• We know: For a YES instance,1. Simulator outputs accepting conversations w.h.p., and2. Simulated verifier “behaves like” real verifier.

• Claim: For a NO instance, cannot have both conditions.

• “Pf:” If both hold, contradict soundness of proof system byprover strategy which mimics simulated prover.

• Easy to distinguish between simulator outputting accepting conversations with high probability vs. low probability.

• Main challenge: how to quantify “behaves like.”

Proof Ideas (cont.)• Thm I [Oka96]: SZK=public-coin SZK.

(i.e. can transform any SZK proof into one where verifier’s messages are just random coin flips)

• Now examine condition:2. Simulated verifier “behaves like” real verifier.

• In a public-coin proof, simulated verifier “behaves like”real verifier iff simulated verifier’s coins are • nearly uniform, and• nearly independent of conversation history.

• Key observation: Both properties can be captured by statistical difference between samplable distributions!

Public-coin proofs [Bab85]

random coinsanswer

random coins

answeraccept/reject

Prover Verifier

Proving that SD is complete for SZK (cont.)

• Have argued: Every problem in SZK reduces to SD.

• Still need: SD SZK.

A Polarization Lemma

Lemma: There exists a poly-time computable function such that

1010 ,1,, DDCC k

C0 C1 23 D0 D1 1 2 k

C0 C1 13 D0 D1 2 k

Not just Chernoff bounds!

Chernoff bounds only yield:

C0 C1 1 e m 2 mC0

m C1 m

where m X def

m independent copies of X

A Protocol for SD

C0 ,C1

).1,,Polarize(, 1010kCCDD compute Both

.

.1,0

coin

R

Dsample

coin

Sample

Flip

.

0

,]Pr[

]Pr[

1

0

1= let else

,= let

)

( If

guess

guess

sampleD

sampleD

.coinguess if Accept

sample

guess

1.

2.

3.

4.

Prover Verifier

Claim: Protocol is an (honest ver) SZK proof for SD.

Properties of D0 and D1

kN

kY

DDCC

DDCC

2,StatDiffSD,

21,StatDiffSD,

1010

1010

parametersecurity where k

Applications of Complete Problem Methodology

Efficient HVSZK proof systems

• Cor: Every problem in HVSZK has an honest-verifier statistical zero-knowledge proof system with:

– 2 messages

– 1 bit of prover-to-verifier communication.

– soundness error 1/2+2-k

– completeness error & simulator deviation 2-k

– deterministic prover

(where k is a “security parameter” independent of input length)

Other Benefits of Complete Problem [SV97]

• Simpler proofs of known results (e.g., [Ost91,Oka96-Thm II] )

• Closure properties:– Previous results focused on specific problems or subclasses of SZK [DDPY94,DC95].– Can apply techniques of [DDPY94] to STATISTICAL DIFFERENCE to obtain results about all of SZK.

Closure Properties of SZK

Thm [SV97]: LSZK (L) SZK, where

1)(,),(:,,,,)( 121 kLLk xxxxxL

= k-ary boolean formulaL= characteristic fn of L

Equivalently, SZK is closed under NC1-truth table reductions.

e.g. can prove “exactly k/2 of (x1, x2,..., xk) are in L” in SZK.

Simplifying Okamoto’s Thm I [GV98]

Use the “complete problem methodology”:

1)H()H(,ED

1)H()H(,ED

0110

1010

CCCC

CCCC

N

Y

: :

• Reduce every problem in SZK to ED. (Uses analysis of simulator from [AH87].)

• Show that ED has a public-coin SZK proof system.(Employs two subprotocols of [Oka96].)

Consider promise problem ENTROPY DIFFERENCE (ED):

Main steps in proof:

xXxXXx

PrlogPr)H( functionentropy

Simplifying Okamoto’s Thm I (cont.)

This gives:

• Simpler, modular proof that all of SZK haspublic-coins SZK proofs.

• ED is complete for SZK.

• (Yet another) proof that SZK is closed undercomplement.

• “weak-SZK” equals SZK.

Honest verifier vs. any verifier

Honest verifier vs. any verifier

• So far: zero-knowledge only vs. honest verifier, i.e. verifier that follows specified protocol.

• Cryptographic applications need zero-knowledge even vs. cheating verifiers.

• Main question: Does honest-verifier ZK=any-verifier ZK?

• Motivation?– honest verifier classes suitable for study

(e.g. complete problem, closure properties)– methodology: design honest-verifier proof and convert to any-verifier proof.

Any-verifier Statistical Zero-Knowledge

v1

p1

v2

pk

accept/reject

When x is a YES instance, Verifier can simulate her view of the interaction on her own.

Formally, for every poly-time verifier, there is probabilistic poly-time simulator such that, when x is a YES instance, its output distribution is statistically close to Verifier’s view of interaction with Prover.

Computational Zero-Knowledge (CZK): require simulator distribution to be computationally indistinguishable rather than statistically close.

Conditional Results:

• honest-ver CZK=any-ver CZK=IP=PSPACE

[GMW86,IY87,BGG+88,Sha90]

If one-way functions exist,

• honest-ver SZK=any-ver SZK [BMO90,OVY93,Oka96]

Unconditional Results:

• For both computational and statistical zero-knowledge,honest-verifier=any-verifier for constant-round public-coin proofs [Dam93,DGW94]

Results on honest verifier vs. any verifier

• For both computational and statistical zero-knowledge,honest-verifier=any-verifier for constant-round public-coin proofs [Dam93,DGW94] [GSV98]

(+ [Oka96]) honest-ver SZK=any-ver SZK

Conditional Results:

• honest-ver CZK=any-ver CZK=IP=PSPACE

[GMW86,IY87,BGG+88,Sha90]

If one-way functions exist,

• honest-ver SZK=any-ver SZK [BMO90,OVY93,Oka96]

Unconditional Results:

• For both computational and statistical zero-knowledge,honest-verifier=any-verifier for constant-round public-coin proofs [Dam93,DGW94][GSV98]

(+ [Oka96]) honest-ver SZK=any-ver SZK

Results on honest verifier vs. any verifier

The Transformationrandom coins 1

answer 1random coins 2

answer kaccept/reject

answer 1

answer kaccept/reject

Random SelectionProtocol

1

Random SelectionProtocol

2

Honest-verifier Proof System

Any-verifier Proof System

Prover Verifier

Prover Verifier

Simulating the Transformed Pf System

answer 1

answer kaccept/reject

1

2

1. Use honest-verifier simulator to generate a transcript

1

12

kaccept/reject

2. “Fill in” transcripts ofRandom Selection

protocols

Desired Properties of Random Selection Protocol

• Dishonest prover:

Sdensity2Pr nS

S

Outcome

, messages verifier of setany For

(OK for soundness by parallel repetition of original proof system)

• Dishonest verifier:

– Outcome distributed almost uniformly.

– Simulability: For (almost) every , can simulate RS protocol transcripts yielding output .

• [GSV98] give a public-coin protocol with these properties(building on [DGW94]).

Noninteractive Statistical Zero-Knowledge

Noninteractive Statistical Zero-Knowledge [BFM88,BDMP91]

proof

accept/reject

Prover(unbounded)

Verifier(poly-time)

shared random string

On input x (instance of promise problem):

• When x is a YES instance, Verifier accepts w.h.p.• When x is a NO instance, Verifier rejects w.h.p. no matter what proof Prover sends.

Noninteractive Statistical ZK (cont.)

When x is a YES instance, Verifier can simulate her view on her own.

Formally, there is probabilistic poly-time simulator such that, when x is a YES instance, its output distribution is statistically close to Verifier’s view.

proofZK lstatistica tivenoninterac has : NISZK

proof

shared random string

Note: above is “one proof” version.

Study of Noninteractive ZK

• Motivation:– communication-efficient.– cryptography vs. active adversaries

[BFM88,BG89,NY90,DDN91]

• Examples of NISZK proofs and some initial study in [BDMP91,BR90,DDP94,DDP97].

• But most attention focused on NICZK, e.g. [FLS90,KP95].

• [DDPY98] apply “complete problem methodology” to show IMAGE DENSITY complete for NISZK.

Complete Problems for NISZK [GSV99]

STATISTICAL DIFFERENCE FROM UNIFORM (SDU):

1)H(,EA

1)H(,EA

kXkX

kXkX

N

Y

:

:

nUXX

nUXX

N

Y

11,StatDiffSDU

1,StatDiffSDU

:

:

on

dist. uniform nU

1,0

functionentropy H

Thm: The following problems are complete for NISZK:

ENTROPY APPROXIMATION (EA):

Relating SZK and NISZK

3

1,SD

32,SD

1010

1010

CCCC

CCCC

N

Y

:

: 1)H()H(,ED

1)H()H(,ED

0110

1010

CCCC

CCCC

N

Y

:

:

• Recall complete problems for SZK:

• NISZK’s complete problems are natural restrictions of these.

can use complete problems to relate SZK and NISZK.

• Thm [GSV98]: SZKBPP NISZKBPP.

• Thm [GSV98]: SZK=NISZK NISZK closed under complement.

Summary

• Recent work has refined our understanding of statisticalzero-knowledge.

• Main tools: – focus on public-coin proofs (via [Oka96])– complete problems [SV97]

• Questions addressed:– closure properties– honest verifier vs. any verifier– interactive vs. noninteractive

Open Problems

5. Does SZK=PZK (“Perfect” zero-knowledge)?

3. Does SZK=NISZK?

2. Combinatorial or number-theoretic complete problems?

1. Generalize more results/techniques to computational zero-knowledge or arguments.

4. Show that SZKBPP if one-way functions exist (“converse” to [Ost91]).