Walter Unger WS 2013/2014
Sascha Geulen October 31, 2013
Exercise
Algorithmic Cryptography
Sheet 2
Exercise 2.1 (4 points)
Discuss the security of the following protocol. A wants to send a message m to B. For this, A and B generate n strings of length |m|. Then, they send the following messages:
A B
m, a1, . . . , an m⊕a1 b1, . . . , bn
m⊕a1⊕b1 m⊕a2⊕b1 m⊕a2⊕b2
...
m⊕an⊕bn−1
m⊕an⊕bn
m⊕bn
The protocol is an extension of the protocol without secure key-exchange presented in the lecture. Is this protocol for n≥2 secure?
Exercise 2.2 (4 points)
Construct a public-key system based on the following NP-complete problem:
SUBSET PRODUCT
Input: A= (a1, . . . , an)∈Nn and b∈N. Problem: Is there a subsetI ⊆ {1, . . . , n}with Q
i∈Iai =b?
Hint: Add to the plaintext, coded as 0-1-sequence, an appropriated padding in order to ensure a necessary condition on the number of ones in the sequence.
Exercise 2.3 (4 points)
Let p1, . . . , pn be distinct prime numbers, P = Qn
i=1pi, and A = (a1, . . . , an), where ai =P/pi.
Prove: The knapsack problem with input (A, α) can be solved efficiently for all α∈N.
— please turn over —
Exercise 2.4 (4 points) A numberα∈N is calledrepresentableby a knapsack vectorA if the knapsack problem with input (A, α) is solvable.
Prove:
(a) Each knapsack vectorB of length nhas at least as many representable numbers as the knapsack vectorAn= (1,2,3,4, . . . , n), for all n∈N.
(b) Each knapsack vectorB of length nhas at most as many representable numbers as the knapsack vectorA0n= (1,2,4,8, . . . ,2n−1), for all n∈N.
Note: In a knapsack vectorA = (a1, . . . , an) all numbers ai are distinct.
Deadline: Thursday, November 7, 2013, 10:15 a.m.,
in the lecture or in the letterbox in front of i1.
Walter Unger WS 2013/2014
Sascha Geulen October 31, 2013
Exercise
Algorithmic Cryptography
Sheet 2
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 2.1 E 2.2 E 2.3 E 2.4
