Fachbereich Mathematik Mohamed Barakat
Wintersemester 2010/11 Simon Hampe
Cryptography
Homework assignment 6
Due date: Wednesday 08/12 at 13:45
Exercise 1. Solve Exercise 3.3.6:
Prove that the F2-algebra L:= F2[x]/hx4 +x2+ 1i is not a field. Find all nonzero noninvertible elements in L. Show that x is invertible inL. Determine the minimal polynomialmF2,x and show that it is reducible. Find all invertible elements different from1and invert them either by using their minimal polynomial (cf. Exercise 3.2.6) or by using theextended Euclidian algorithm.
Exercise 2. Solve Exercise 3.3.11:
LetL:=F2[x]/hfiand
(1) f :=x3+x+ 1. Prove thatf is a primitive polynomial, or equivalently, that
¯
x∈L is a primitive element, i.e. L∗ =h¯xi.
(2) f := x4 +x3 +x2 +x+ 1. First prove that L is a field. Prove that f is an imprimitive polynomial, or equivalently, that x¯ ∈ L is an imprimitive1 element, i.e. L∗ 6=h¯xi
Exercise 3. Solve Exercise 3.3.22:
LetK be a field and f ∈K[x] with f(0)6= 0. Then ordf |k ⇐⇒ f |xk−1.
Exercise 4.
(1) Determine the order of the polynomial f :=x7+x4 +x3+x2+ 1∈F2[x].
(2) Draw the subfield lattices of Fp45.
(3) Show that the converse of Proposition 3.2.9.(3) is false.
(4) Which elements of F16 :=F2[x]/hx4+x+ 1iare contained in F4? (Compare with Example 3.3.12)
(5) Determine the orbits of the Frobenius automorphism x7→x2 onF16. Hint: (1) First show thatf is irreducible.
1Although¯xis a primitive element of the field extensionF2≤F2[¯x] =F2[x]/hx4+x3+x2+x+1i.
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