Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 5
Due date: Wednesday 01/12 at 13:45
Exercise 1. Prove (1) and the equivalence in Lemma 2.4.15:
SettingH0(K) :=H(K |P C). Then (1) H(P, K) =H(P, C) +H0(K).
(2) H(K) = H(C |P) +H0(K).
(3) H(K |C) =H(P |C) +H0(K).
Further:
Kis free ⇐⇒H0(K) = 0⇐⇒I(K, P C) =H(K).
In particular: The key equivocation and the plaintext equivocation coincide in free cryptosystems.
Exercise 2. Prove 2.2.5.(10):
There exists a (row) vectort ∈K1×ℓ with perhc, ti= perc.
Exercise 3. Prove exercise 3.2.6:
For06=v ∈V let Uϕ,v :=hϕi(v)|i∈N0i ≤V. Then (1) mϕ,v =mϕ|
Uϕ,v.
(2) dimKUϕ,v = min{d ∈ N | (v, ϕ(v), . . . , ϕd(v)) are K-linearly dependent} ≥ 1.
(3) degmϕ,v = dimKUϕ,v.
(4) mϕ = lcm{mϕ,v | 0 6= v ∈ V}. This gives an algorithm to compute the minimal polynomial of ϕ as the lcm of at most n minimal polynomials mϕ,v1, . . . , mϕ,vℓ, where ℓ= dimKV.
(5) α ∈EndK(V) is an automorphism if and only if mα(0) 6= 0∈K. This gives an algorithm to compute the inverse of α.
Exercise 4. Prove exercise 3.2.14:
Classify all irreducible4-bit LFSRs. How many of them are transitive?
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