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325
A The Special Conjugacy Problem in GL(2,R)
Let Inn(g):GL(2,R) → GL(2,R) be a public inner automorphism. We assume
that Inn(g) is given as a black box, i.e. an attacker is able to calculate images
under Inn(g) but does not know the used g ∈ GL(2,R). This approach assures
that our calculations are independent of the presentation of Inn(g). We now
show that the special conjugacy problem is efficiently solvable in GL(2,R).
Let B, C,X ∈ GL(2,R)andB,XBX
−1
=
ˆ
B =
ˆ
b
1
ˆ
b
2
ˆ
b
3
ˆ
b
4
and C, XCX
−1
=
ˆ
C =
ˆc
1
ˆc
2
ˆc
3
ˆc
4
be two simultaneous instances of the conjugacy problem in GL(2,R).
Let
ˆ
X ∈ GL(2,R) be a solution of these two instances. Then
ˆ
X = Z · X with
z
1
z
2
z
3
z
4
= Z ∈ Z(
ˆ
B) ∩ Z(
ˆ
C). By comparing the components of Z ·
ˆ
B,
ˆ
B · Z and
Z ·
ˆ
C,
ˆ
C · Z we get:
1
1
Since
ˆ
X could also be expressed as
ˆ
X = X ·
ˆ
Z for a
ˆ
Z ∈ Z(B) ∩ Z(C), the following
paragraph is also true if
ˆ
b
i
and ˆc
i
are replaced by b
i
and c
i
. In particular B ∈
Z(C) ⇔
ˆ
B ∈ Z(
ˆ
C).
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