Client-Server Concurrent Zero Knowledge with Constant Rounds and Guaranteed Complexity Ran Canetti,...

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Client-Server Concurrent Zero Knowledgewith Constant Rounds

and Guaranteed Complexity

Ran Canetti, Abhishek Jain and Omer Paneth

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Zero-Knowledge Protocols

• Completeness• Soundness • Zero knowledge

𝑃 𝑉𝑥∈𝐿?

[Goldwasser-Micali-Rackoff 85]

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Completeness

𝑃 𝑉 Accept

𝑥∈𝐿𝑤

4

Soundness

𝑃∗ 𝑉 reject

𝑥∉𝐿

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Zero-knowledge

𝑃 𝑉 ∗ 𝑆≈𝑐𝑥∈𝐿

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Why do we care about zero-knowledge?

Used as a sub-protocol in larger cryptographic protocols and systems

Secure composition?

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Concurrent Composition

𝑃 𝑉

𝑃 𝑉

𝑃 𝑉

𝑥∈𝐿

𝑥∈𝐿

𝑥∈𝐿

𝑤

Session

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Concurrent Zero Knowledge

𝑉 ∗

[Dwork-Naor-Sahai 98]

𝑃

𝑃

𝑃

𝑥∈𝐿

𝑥∈𝐿

𝑥∈𝐿

𝑤 𝑆≈𝑐

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Rounds Assumption

Stand-alone zero knowledge

[Feige-Shamir 89][Bellare-Jakobson-Yung 97]

4 OWF

Concurrent zero knowledge

[Richardson-Kilian 99][Kilian-Petrank 01][Prabhakaran-Rosen-Sahai 02]

OWF

[Gupta-Sahai 12][Chung-Lin-Pass 13][Pandey-Prabhakaran-Sahai 13]

Strong assumption:interactive knowledge assumptions

statistically sound P-certificates differing input obfuscation

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Today

Constant-round protocols

from standard assumptions

Weaker notions of concurrent security

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Bounded Concurrent ZK[Barak 01]

sessions

Complexity of each sessionRounds

Communication

Assuming collision-resistant hash functions. For bound :

𝑃 𝑉

𝑃 𝑉

𝑃 𝑉

Barak

Barak

Barak

…

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Barak’s Protocol

Client

Server

Barak

[Persiano-Visconti 05]:set the bound only at protocol run time

This is too early

ClientBarak

ClientBarak

The bound on the number of concurrent sessions is set at protocol design time

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Standard Model for Concurrent ZK

𝑃 𝑉

𝑃 𝑉

𝑃 𝑉

𝑥∈𝐿

𝑥∈𝐿

𝑥∈𝐿

𝑤

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Client-Server Concurrent ZK

𝑉

𝑃 𝑉

𝑉

𝑥∈𝐿

𝑥∈𝐿

𝑥∈𝐿

𝑤

Server Clients

[Persiano-Visconti 05]

Increase the communicationas more session start

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The Persiano-Visconti Protocol

𝑃 𝑉Bonded concurrent

for sessions … active sessions

Finish session

Bonded concurrent for sessions … active sessions

Bonded concurrent for sessions … active sessions

A single session: Concurrent sessions:

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Protocol Complexity

Barak for sessions

Finish session

Barak for sessions

Barak for sessions Almost the same as

bounded concurrent ZK!

Complexity of each session(For concurrent sessions)

RoundsCommunication𝑃 𝑉

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The Persiano-Visconti Protocol

Client

Server

Persiano-ViscontiThis is

too lateClientPersiano-Visconti

ClientPersiano-Visconti

The communication complexity is changing at protocol run time

Client does not know what will be the communication complexity of the session!

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Example: Call Center

“All our lines are currently busy. please hold and your call will be answered shortly…”

“The estimated waiting time is 7 minutes.”

This work: the communication complexity is set at the beginning of every session

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Our Result

Assuming collision-resistant hash functions

there is a concurrent zero-knowledge protocol

in the client-server modelwith constant-rounds and guaranteed complexity.

Guaranteed complexity:The communication complexity of each session is determined in the beginning of the session

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for concurrent sessions

determined in the beginning of the session

not determined until the session terminates

This work [Persiano-Visconti]

Communication complexity

Round complexity

6

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The Protocol

𝑃Start session

Start session

Start session

First sessions to start run Barak’s protocol with bound .

Next sessions run Barak’s protocol with bound .

Next sessions run Barak’s protocol with bound .

Every session runs Barak’s protocol with some bound

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The Challenge

𝑃Start session

Barak’s protocol with bound

Start session

Start session

… new sessions 𝑉 ∗

Cannot rely directly on bounded concurrency

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Barak’s simulation

𝑆 sessions

Barak

… 𝑉 ∗Barak

Barak

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𝑆

𝑆

𝑆

Barak’s simulation

𝑆

sessions

Barak

… 𝑉 ∗Barak

Barak

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𝑆Barak’s simulation

Barak

Other protocol

Other protocol

… 𝑉 ∗𝑃

𝑃

sessions

Communication complexity Barak’s

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Proof

A session is of level- if it runs Barak’s protocol with bound .

Observation:If starts sessions,

sessions of level are easy to simulate.

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𝑉 ∗Level

Level

Level

Level

Level

Level

Level 𝑃…

…

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𝑆0𝑉∗

Level

Level

Level

Level

Level

Level

Level 𝑃

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𝑆1𝑆0𝑉∗

Level

Level

Level

Level

Level

Level

Level 𝑃

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𝑆2𝑆1𝑆0𝑉∗

Level

Level

Level

Level

Level

Level

Level

…

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Simulation Running Time

…

32[slide: Mira Belenkiy]

Thanks!