Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 2
Due date: Wednesday 10/11 at 13:45
Exercise 1. Decipher the following German ciphertext
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
Hint: The cryptosystem is monoalphabetic substitution of the latin alphabet to- gether with the space character. If you decide to use the crypto package you can initialize the key using
gap> key := CryptoKey( rec( ) );
and set the nonstandard alphabet for the key using
gap> key!.ALPHABET := "abcdefghijklmnopq0123456789";
To unbind a (possibly wrong) assignment of 9, for example, use gap> Unbind( key!.9 );
Exercise 2. Consider the following simple probabilistic algorithm with a perfect standard 6-face dice being the random source: In each step throw the dice an read the upper face name. The algorithms terminates as soon as 6 is thrown.
Define theestimated runtime of the algorithm as t :=X
n∈N
np(n),
wherep(n)is the probability that the algorithms terminates aftern steps. Compute the estimated runtime t.
1
Exercise 3. Prove exercise 1.2.2:
(1) LetF, F′ :M N be two multi-valued maps withF ⊂F′. ThenF′injective implies F injective.
(2) Let F :M N be surjective. Then (a) F−1 is surjective and (F−1)−1 =F.
(b) F is injective (and hence bijective) iffF−1 is a (surjective) map.
(3) Each bijective mutli-valued mapF :M N is the multi-valued inverse g−1 of a surjective map g :N →M (viewed as a multi-valued map).
Exercise 4. Prove exercise 1.4.2:
The multi-valued mapΦe is injective for all e∈K.
