Docsity
Docsity
Prepare for your exams
Prepare for your exams

Study with the several resources on Docsity

Earn points to download
Earn points to download

Earn points by helping other students or get them with a premium plan

Guidelines and tips
Guidelines and tips
Sell on Docsity
Sell on Docsity
Docsity AI
Docsity AI
Log inSign up

Prepare for your exams

Study with the several resources on Docsity

Find documents

Prepare for your exams with the study notes shared by other students like you on Docsity

Search for your university

Find the specific documents for your university's exams

Docsity AINEW

Summarize your documents, ask them questions, convert them into quizzes and concept maps

Explore questions

Clear up your doubts by reading the answers to questions asked by your fellow students

Earn points to download

Earn points by helping other students or get them with a premium plan

Share documents
download point icon20 Points

For each uploaded document

Answer questions
download point icon5 Points

For each given answer (max 1 per day)

All the ways to get free points
Get points immediately

Choose a premium plan with all the points you need

Study Opportunities

Choose your next study program

Get in touch with the best universities in the world. Search through thousands of universities and official partners

Community

Ask the community

Ask the community for help and clear up your study doubts

Free resources

Our save-the-student-ebooks!

Download our free guides on studying techniques, anxiety management strategies, and thesis advice from Docsity tutors

Crytography Zero Knowledge Proof Application, Lecture Notes - Computer Science, Study notes of Cryptography and System Security

Harvard UniversityCryptography and System Security

Prof. Salil Vadhan, Prof. Alon Rosen, Computer Science, Cryptography, Zero-Knowledge Proofs, Compiling Protocols to Handle Malicious Adversaries, Harvard, Lecture Notes

Typology: Study notes

2010/2011

Uploaded on 11/02/2011

thecoral
thecoral 🇺🇸

4.5

(30)

395 documents

1 / 3

Toggle sidebar

This page cannot be seen from the preview

Don't miss anything!

bg1
CS 120/ E-177: Introduction to Cryptography
Salil Vadhan and Alon Rosen Dec. 14, 2006
Lecture Notes 21:
Zero-Knowledge Proofs II
Recommended Reading.
•Vadhan, Interactive & Zero-Knowledge Proofs, from IAS/PCMI Summer School on Compu-
tational Complexity, Secs 1.1, 1.2, 2.1, 2.2.
•Goldreich, Chapter 4 (up to 4.4)
1 Zero Knowledge for NP
An NP-complete problem: Graph 3-Coloring.
•An (undirected) graph G= (W, E) is 3-colorable if there is a function C:W→ {R, Y, B}
such that for all (u, v)∈E,C(u)6=C(v).
•3COL ={G:Gis 3-colorable}.
•For every L∈NP, there is a poly-time fsuch that x∈L⇔f(x)∈3COL.
•Moreover, given any NP proof system for L, we can choose fsuch that valid NP proofs for
x∈Lcan be mapped in poly-time to valid 3-colorings of f(x).
Cut and Choose:
•G∈3COL ⇔ ∃C³V(u,v)∈EC(u)6=C(v)´.
•If we randomly permute the 3 colors, each pair (C(u), C(v)) for u6=vreveals no information.
•Have prover ‘commit’ to randomized coloring C, verifier pick a random edge.
Physical Zero-Knowledge Proof: See video.
1
pf3

Partial preview of the text

Download Crytography Zero Knowledge Proof Application, Lecture Notes - Computer Science and more Study notes Cryptography and System Security in PDF only on Docsity!

CS 120/ E-177: Introduction to Cryptography Salil Vadhan and Alon Rosen Dec. 14, 2006

Lecture Notes 21:

Zero-Knowledge Proofs II

Recommended Reading.

  • Vadhan, Interactive & Zero-Knowledge Proofs, from IAS/PCMI Summer School on Compu- tational Complexity, Secs 1.1, 1.2, 2.1, 2.2.
  • Goldreich, Chapter 4 (up to 4.4)

1 Zero Knowledge for NP

An NP-complete problem: Graph 3-Coloring.

  • An (undirected) graph G = (W, E) is 3-colorable if there is a function C : W → {R, Y, B} such that for all (u, v) ∈ E, C(u) 6 = C(v).
  • 3COL = {G : G is 3-colorable}.
  • For every L ∈ NP, there is a poly-time f such that x ∈ L ⇔ f (x) ∈ 3COL.
  • Moreover, given any NP proof system for L, we can choose f such that valid NP proofs for x ∈ L can be mapped in poly-time to valid 3-colorings of f (x).

Cut and Choose:

  • G ∈ 3COL ⇔ ∃C

(u,v)∈E C(u)^6 =^ C(v)

  • If we randomly permute the 3 colors, each pair (C(u), C(v)) for u 6 = v reveals no information.
  • Have prover ‘commit’ to randomized coloring C, verifier pick a random edge.

Physical Zero-Knowledge Proof: See video.

Definition 1 A commitment scheme over message space P = ⋃ n Pn is a polynomial-time com- putable function Com(m, k) satisfying:

  • (Hiding) For every m, m′^ ∈ Pn such that ‖m‖ = ‖m′‖, Com(m, K) ≡c Com(m′, K), when K ← {R 0 , 1 }n.
  • (Binding) There do not exist m 6 = m′^ and k, k′^ such that Com(m, k) = Com(m, k′).

Zero-Knowledge Proof for Graph 3-Coloring

Common input: A graph G = (W, E) on n vertices. Prover’s input: A valid 3-coloring C : W → {R, Y, B} (in case G ∈ 3COL)

  1. P : Choose a permutation π : {R, Y, B} → {R, Y, B} uniformly at random, and set C′^ = π ◦C. For every vertex w ∈ W , choose kw ← {R 0 , 1 }n^ and send zw = Com(C′(w), kw) to V.
  2. V : Choose an edge (u, v) ←R E, and send (u, v) to P.
  3. P : Check that (u, v) ∈ E, and if so send C′(u), C′(v), ku, kv to V.
  4. V : Accept if C′(u) 6 = C′(v), zu = Com(C′(u), ku) and zv = Com(C′(v), kv).

Theorem 2 Above is a zero-knowledge proof for Graph 3-Coloring.

Proof:

  • Perfect completeness.
  • Soundness error 1 − 1 /|E|. Reduce by repetition.

Simulator SV^ ∗ , on input G = (W, E):

  1. Select (u, v) ←R E.
  2. Define a coloring C′^ by setting (C′(u), C′(v)) to be two random distinct colors in {R, Y, B}, and setting C′(w) = R for all other vertices w.
  3. For every w ∈ W , choose kw ← {R 0 , 1 }n, and set zw = Com(C′(w), kw).
  4. Select random coin tosses r for V ∗, and let (u∗, v∗) = V ∗(G, {zw}w∈W ; r).
  5. If (u∗, v∗) 6 = (u, v), output fail. Otherwise, output ({zw}w∈W , (u, v), (ku, kv, C′(u), C′(v)); r).

Claim 3 For every PPT V ∗^ and G ∈ 3COL, we have

  1. SV^ ∗ (G) succeeds with probability at least 1 /|E| − neg(n), and