Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 11
Due date: Wednesday 26/01 at 13:45
Exercise 1. Prove:
(1) Lemma 7.1.3.(4): A Carmichael number has at least 3prime factors.
(2) n is aCarmichael number ⇐⇒ n is composite and1 (∀p∈P: p|n =⇒ p−1|n−1).
(3) If {6k+ 1,12k+ 1,18k+ 1} ⊂Pfor a k ∈Nthen the product (6k+ 1)(12k+ 1)(18k+ 1)
of the entries is aCarmichaelnumber. Find the two smallest k’s with this property.
Exercise 2.
(1) Let n = 3α for an α ∈ N. Denote by U0 the subgroup of the Fermat nonwitnesses of n (as in the proof of Theorem 7.1.10).
(a) Determine U0 and its index (Z/nZ)∗/U0 for the case α= 2.
(b) Show that 9|ord(Z/nZ)∗(4) for all α >2.
(2) Let G be an Abelian group. Denote by Gm the kernel of the group endo- morphism G→G, x7→xm. Show that for all a∈G:
(ordG(a), m) = 1 =⇒ G:Gm ≥ordG(a).
(Compare with Lemma 7.1.9).
Exercise 3. Characterize the set of alln∈N for which (1) there exists x, y ∈Zsuch that x2−y2 =n.
(2) there exists x, y, k∈Z such that x2 −y2 =kn.
Exercise 4. Solve Exercise 8.1.2:
Describe Pollard’s p −1 factorization algorithm. Use this algorithm to factor 1633797455657959.
1This is the converse of Lemma 7.1.3.(3).
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