SEPARATING SUCCINCT NON-INTERACTIVE ARGUMENTS

FROM ALL FALSIFIABLE ASSUMPTIONS

Daniel WichsCraig GentryIBM NYU

MIT Seminar (Dec’ 10).

Non-Interactive Argument

Succinct?

Prove Language Membership

Language L µ {0,1}*. Want to show x 2 L.

NP = Non-Interactive Proofs with Efficient Verifier.

Question: How succinct can proofs for NP be?

If L has witness-size t(n) then L 2 DTIME( 2t(n)poly(n)). Sub-linear proofs for all NP ) NP 2

DTIME( 2o(n)). Generalizes to interactive proofs [GH98, GVW02].

Succinct Arguments for NP

Arguments = Comp Sound Proofs. [Kilian92, Micali 94] Cannot prove false statements x efficiently. Can prove true statements x efficiently given witness w. Succinct: size is poly(n)polylog(|x| + |w|).

n = security parameter.

What we know: Interactive (4 rounds): Assuming CRHFs [Kilian 92]. Non-interactive: Random Oracle model [Micali 94].

* Ignore: better efficiency for prover/verifier, languages outside of NP.

Succinct Non-Interactive Arguments

Question: Can we get Succinct Non-Interactive Arguments (SNARGs) in the standard model?

Problem: 9 small adversary with hard-coded false statement x and verifying proof ¼. Same reason why un-keyed CRHFs don’t exist.

Rest of talk: SNARGs initialized with a common reference string (CRS).

Do SNARGS exist?

Positive Evidence: Take [Micali 94] construction, replace RO with “complicated hash function” H (set CRS = H). Don’t know how to break it. Can conjecture security.

Can we prove any SNARG construction secure under OWFs, DDH, RSA, LWE,… ? “q-decisional-augmented-bilinear-Diffie-Hellman-exponent-

assumption” ?

This work: NO*. * Restrictions apply.

Main Result

No Black-Box-Reduction proof of security for any SNARG construction under any Falsifiable Assumption.

DDH, RSA, LWE,…q-ABDHE,…

Defining SNARGs

Completeness: Correctly generated proofs verify with overwhelming probability.

CRS Ã Gen(1n)

¼ Ã Prove(CRS, x, w) Verify(CRS, x, ¼) x, ¼

Defining SNARGs

Public Verifiability: any party can verify proofs.

CRS Ã Gen(1n)

¼ Ã Prove(CRS, x, w) Verify(CRS, x, ¼) x, ¼

Defining SNARGs

Public Verifiability: any party can verify proofs. Designated Verifier: only verifier that knows SK can

verify. All our results hold for Designated Verifier SNARGs.

Syntactically same as two-round interactive arguments. Challenge = CRS, Response = ¼.

(CRS, SK) Ã Gen(1n)

¼ Ã Prove(CRS, x, w) Verify(CRS, SK, x, ¼) x, ¼

Security of SNARGs

(x, ¼) Ã Adv (CRS)

(Adaptive) Soundness: For efficient Adv if (x, ¼)Ã Adv(CRS) Pr[ Verify(CRS, SK, x, ¼) = accept and x 2 L ] = negligible(n)

Natural for SNARGs. For 2-round arguments traditionally consider static

soundness.

(CRS, SK) Ã Gen(1n)

Verify(CRS, SK, x, ¼) x, ¼

Succinct Arguments: What we know?

4 round

3 round2 round

Publically Verifiable SNARG (CRS)

SNARG without CRS

Designated Verifier SNARG (CRS)

Doesn’t Exist

May exist (RO Heuristic)but cannot prove securevia BB reduction from falsifiable assumption.

??

Exist assuming CRHFs

(adaptive soundness)

(static soundness)

Main Result

No Black-Box-Reduction proof of security for any SNARG construction under any Falsifiable Assumption.

Falsifiable Assumptions

Falsifiable Assumption (in spirit of [Naor 03]): Interactive game between an efficient challenger and

adversary; challenger decides if adversary wins. For PPT Adv Pr[Adv wins] · negl(n).

Examples: DDH, RSA, LWE, QR,…, q-ABDHE,… “RSA Signatures (Full-Domain-Hash) with SHA-1 are secure”.

Not Falsifiable: “This Proof System is ZK”. (Not a game - requires Simulator) “This SNARG construction is secure”. (Inefficient Challenger) “Knowledge-of-Exponent” (KoE) Assumptions. [Dam91, HT98]

Main Result

No Black-Box-Reduction proof of security for any SNARG construction under any Falsifiable Assumption.

SNARG Attack

Assumption Attack

Black-Box Reductions

SNARG Security

Assumption

SNARG Attack

Assumption Attack

Black-Box Reductions

Black-Box Reduction: Constructive Proof. Efficient Reduction Algorithm. Given Black-Box access

to any SNARG-Attacker becomes an Assumption-Attacker.

Should work even if SNARG-Attacker is inefficient. (If SNARG-Attacker is stateless can ignore rewinding).

Reduction

Assumption

Challenger

Main Result

No Black-Box-Reduction proof of security for any SNARG construction under any Falsifiable Assumption.

• Assuming the falsifiable assumption isn’t

false. • Assuming sub-exponentially hard OWFs exist.

Main Result

If there is a Black-Box-Reduction proof for some SNARG construction under some Falsifiable Assumption then one of the following holds: The falsifiable assumption is false! There are no sub-exponentially hard OWFs.

Main Idea: Simulatable Attacker

Inefficient Attacker. Breaks soundness (outputs false statements,

“proofs”). Efficient Simulator.

Does not break soundness (outputs true statements, proofs).

No efficient distinguisher can tell them apart.

SNARG Attack

Simulator≈

Separation via Simulatable Attack

Existence of Simulatable Attack for any SNARG.

Simulatable Attack implies Black-Box Separation.

Simulatable Attack ) Separation

SNARG Attack

Assumption Attack

Reduction

Assumption

Challenger

Given access to the “Simulatable Attacker” reduction breaks assumption.

Attacker

WINS

Simulatable Attack ) Separation

SNARG Attack

Reduction

Assumption

Challenger

Given access to the “Simulatable Attacker” reduction breaks assumption.

Efficient

Attacker

WINS

Simulatable Attack ) Separation

Reduction

Assumption

Challenger

Given access to the “Simulatable Attacker” reduction breaks assumption.

Replace “Simulatable Attacker” with efficient Simulator.

Attacker

WINS

Simulator Efficient

Simulatable Attack ) Separation

Reduction

Assumption

Challenger

There is an efficient attack on the assumption. ) Assumption is false!

Attacker

WINS

Simulator

Efficient Attack

on Assumption

Separation via Simulatable Attack

Existence of Simulatable Attack for any SNARG.

Simulatable Attack implies Black-Box Separation. BB Reduction under Falsifiable Assumption

) Assumption false.

Existence of Simulatable Attack

If NP has poly-logarithmic witnesses, there may not be any attacks at all!

Assumption: Sub-exponentially-hard subset-membership problems in NP. An NP language L. Distributions: G µ L , B µ

{0,1}*\L. Can efficiently sample x à G along with a witness w. Cannot distinguish G from B in time 2n± with

probability 2-n±.

Implied by sub-exponentially secure PRGs, OWFs.

Existence of Simulatable Attack

Naïve Idea: try all ¼ until one verifies. Might not look at all like correct distribution!

Show: Way to sample “correct looking” ¼ for x à B.

SNARG Attack

Simulator≈

CRS (x, ¼)x à G witness w

x à B¼ à Prov(CRS, x, w)How to sample ¼ ?

x à G witness w

x à B¼ à Prov(x, w)¼ à Prov*(x)

≈

8 efficient Prov w/ short output 9 inefficient function Prov*:

(x, ¼) (x, ¼)

Existence of Simulatable Attack

If G, B are (s, ²)-indistinguishable thens* = s/poly(2|¼| ²), ²* = 2²

x à G¼ à Prov(x)

≈

8 inefficient Prov w/ short output 9 inefficient function Prov*:

(x, ¼) (x, ¼)

Indisitinguishability w/ Auxiliary Info

x à B¼ à Prov*(x)

Proof coming up soon.Assuming the Lemma…

(s*, ²*)

Existence of Simulatable Attack

Security of G,B exponential in size of proof. Proof-size nc polylog(|x| + |w|) = o(nc+1). Choose large enough statements to get security 2nc+1.

Distinguisher can ask many queries – hybrid argument.

SNARG Attack

Simulator≈

CRS (x, ¼)x à G witness wx à B

¼ Ã Prov(CRS, x, w)¼ Ã Prov*(CRS, x)

Simulator

Existence of Simulatable Attack

Problem: Who gets which security parameter? D can “lie” about security parameter to “oracle”.

Solution: Simulator gives false statements when m ¼ log(n). Annoying and messy! Simulator gets n and depends

on D.

SNARG Attack ≈

D(n)

CRS (x, ¼)x à G witness wx à B

¼ Ã Prov(CRS, x, w)¼ Ã Prov*(CRS, x)

Sec = m

Simulator

Existence of Simulatable Attack

Why is this a legitimate attack? Do proofs verify? Set D to be the verifier of the SNARG.

SNARG Attack ≈

D(n)

CRS (x, ¼)x à G witness wx à B

¼ Ã Prov(CRS, x, w)¼ Ã Prov*(CRS, x)

Sec = m

Separation via Simulatable Attack

Existence of Simulatable Attack for any SNARG. Any SNARG for a sub-exp hard membership

problem. Any SNARG for NP assuming sub-exp hard OWF.

Simulatable Attack implies Black-Box Separation. BB reduction under falsifiable assumption

) Assumption false.

Returning to:

Indisitinguishability with

Auxiliary Information

x à G¼ à Aux(x)

≈

8 short inefficient Aux 9 inefficient Aux*:

(x, ¼) (x, ¼)

Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)

If G, B are (s, ²)-indistinguishable then s* = s/poly(2|¼| ²), ²* = 2²

(s*, ²*)

) L-bit leakage on seed of PRG reduces HILL entropy of output

by L bits. [DP08]

Proof related to Nisan’s proof of Impagliazzo Hardcore Lemma.

Pr[ D(x, ¼)=1] - Pr[D(x, ¼)=1] > ²* x à G

¼ Ã Aux(x)

9 short inefficient Aux

Proof: Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)

8 inefficient function Aux* 9 D of size s*

Distinguish G, B with s = s* poly(2|¼| ²) ² = ²* /2

Task:

Goal: switch quantifiers with Min-Max theorem.

Pr[ D(x, ¼)=1] - Pr[D(x, ¼)=1] > ²* x à G

¼ Ã Aux(x)

9 short inefficient Aux

Proof: Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)

min Aux* max D of size s*

Goal: switch quantifiers with Min-Max theorem.

Pr[ D(x, ¼)=1] - Pr[D(x, ¼)=1] > ²* x à G

¼ Ã Aux(x)

9 short inefficient Aux

Proof: Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)

min Aux* max Dist(over D of size s*)

D Ã Dist D Ã Dist

Goal: switch quantifiers with Min-Max theorem.

Pr[ D(x, ¼)=1] - Pr[D(x, ¼)=1] > ²* x à G

¼ Ã Aux(x)

9 short inefficient Aux

Proof: Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)D à Dist D à Dist

min Aux*max Dist(over D of size s*)

[von Neumann 28]

Pr[ D(x, ¼)=1] - Pr[D(x, ¼)=1] > ²* x à G

¼ Ã Aux(x)

9 short inefficient Aux,

Proof: Indisitinguishability w. Auxiliary Info

x à B¼ à Aux*(x)D à Dist D à Dist

min Aux*Dist(over D of size s*)

Val(x) := min¼ Pr[D(x, ¼) = 1]Goal: get rid of auxiliary information.

E[Val(x)] - E[Val(x)] > ²* x à B x à G

E[Val(x)] - E[Val(x)] > ²* x à B x à G

9 short inefficient Aux,

Proof: Indisitinguishability w. Auxiliary Info

Dist(over D of size s*)

Val(x) := min¼ Pr[D(x, ¼) = 1]

To distinguish if x comes from G, or B: Get estimate for Val(x).

Try all possible values of ¼. Run many D on each choice.

Output “B” with that probability.

size = poly(2|¼|²).

Main Result

If there is a Black-Box-Reduction proof for some SNARG construction under some Falsifiable Assumption then one of the following holds: The falsifiable assumption is false! There are no sub-exponentially hard OWFs.

Slightly succinct: sub-linear arguments.

No exponentially hard subset-membership problems.

Main Result

If there is a Black-Box-Reduction proof for some SNARG construction under some Falsifiable Assumption then one of the following holds: The falsifiable assumption is false! There are no sub-exponentially hard OWFs.

(sub)-exponential

(sub)-exponential version of

Comparison to other BB Separations

Notion A is not sufficient to realize B in a “black-box way”. [Impagliazzo Rudrich 89]: Separate KA from OWP. [Sim98]: Separate CRHFs from OWP. [GKM+00, GKTRV00, GMR01, RTV04, BPR+08 …]

Usually: Notion A is generic e.g. “existence of some OWP”. Construction of B using a generic instance of A as black-box.

(Reduction uses adversary as a black-box.)

Our result: Notion A can be a specific assumption e.g. “RSA is a OWP”. Reduction uses adversary as a black-box. Similar to: [DOP05, AF07,HH09].

BB Reductions for Succinct Arguments

[Rothblum-Vadhan 10] : Any interactive succinct argument with a black-box proof of security under a falsifiable assumption can be easily converted into a “PCP System”.

Not a separation since PCPs exist unconditionally.

Shows: heavy PCP machinery inherent in succinct args.

Summary & Open Problems

Black-box separation of SNARGs from Falsifiable Assumptions.

Non-black-box techniques? Only know [Bar01].

SNARGs under non-falsifiable assumptions (e.g. Knowledge of Exponent). Some results by [Gro10].

Succinct arguments with long CRS? Succinct in witness but not statement? Constructions of 2 or 3 round arguments? Or, do black-box separations extend?

THANK YOU!

QUESTIONS?