Walter Unger WS 2013/2014
Sascha Geulen November 21, 2013
Exercise
Algorithmic Cryptography
Sheet 5
Exercise 5.1 (4 points)
Consider the following protocol for Graph 3-Coloring:
Common Input: Graph G= (V, E).
Secret of the Prover: 3-Coloring col :V → {1,2,3}.
Prover: Choose permutation π over {1,2,3}. For each v ∈ V, send lockable box Bv with δ(Bv) =π(col(v)).
Verifier: Choose an edge {u, v} ∈E.
Prover: Show the permuted color of the vertices u and v, i.e., send δ(Bu)and δ(Bv).
Verifier: Check whether δ(Bu)6=δ(Bv).
(a) Prove:The protocol is a Zero-Knowledge-Proof, i.e., it satisfies the following three properties:
Completeness: If the prover knows the secret, he can convince the verifier.
Soundness: If the prover does not know the secret, he cannot convince the verifier, except with some small probability.
Zero-Knowledge: There exists a probabilistic simulator that computes an ac- cepting transcript in polynomial time that looks statistically (algorithmically) the same as a transcript between the pro- ver and a verifier.
(b) What is the cheating probability?
Exercise 5.2 (4 points)
Construct a Zero-Knowledge-Proof based on the following problem:
VERTEX COVER
Input: GraphG= (V, E)and k∈N.
Problem: Is there a vertex cover of size at most k for G, i.e., a subset C ⊆ V with |C| ≤k such that for each edge {u, v} ∈E at least one of u and v belongs toC?
— please turn over —
Exercise 5.3 (4 points) Construct a Zero-Knowledge-Proof based on the following problem:
THREE VERTEX-DISJOINT PATHS
Input: GraphG= (V, E)and s, t∈V.
Problem: Are there three vertex-disjoint paths froms to t?
Definition: Three paths are vertex-disjoint if the intersection of the vertex sets of the paths without the start and end vertex is empty.
Exercise 5.4 (4 points)
Construct a Zero-Knowledge-Proof based on the following problem:
NOT-ALL-EQUAL 3-SAT
Input: Set X of variables, collectionC of clauses over X such that each clause c∈C has |c|= 3.
Problem: Is there a truth assignment forX such that each clause in C has at least one true literal and at least one false literal?
Deadline: Thursday, November 28, 2013, 10:15 a.m.,
in the lecture or in the letterbox in front of i1.
Walter Unger WS 2013/2014
Sascha Geulen November 21, 2013
Exercise
Algorithmic Cryptography
Sheet 5
Please fill in your name and your student number and mark the exercises that you can present.
Then staple this page in front of your solution sheet.
Presentation of your exercise solution
I want to present the exercises marked in the following table in the tutorial. Each mark gives two points if you are present in the tutorial and your solution is good (at least two points).
Name Student Number E 5.1 E 5.2 E 5.3 E 5.4
