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sectors identified by the two radii connecting the centers of the areas with the
intersection points of original and obfuscated areas.
1
⎨
⎩
σ
2
r
2
sin
σ
+
γ
sin
γ
=
√
δπr · R
r
2
2
R
2
2
2
R
2
−
−
(5)
d
=
r
cos
2
+
R
cos
2
r
sin
2
=
R
sin
2
Solutions of this system can be obtained numerically.
Reducing the Radius
The third obfuscation technique consists in reducing the radius of a location
measurement from
r
to
r
, as showed in Fig. 1(c). The obfuscation effect is
produced by a correspondent reduction of the probability to find the real user
location within the returned area, whereas the joint pdf is fixed.
Let (
x
u
,y
u
) be the real user position coordinates, By assumption, the prob-
ability that the real user position falls in the location measurement of radius
r
is
P
((
x
u
,y
u
)
Area
(
r, x, y
)) = 1. When we obfuscate by reducing the radius,
an area of radius
r
∈
Area
(
r
,x,y
))
<
<r
is returned, where
P
((
x
u
,y
u
)
∈
P
((
x
u
,y
u
)
Area
(
r, x, y
)), since a circular ring having pdf greater than zero
has been excluded.
With regard to relevances
∈
R
Tech
and
R
Priv
, their relation can be defined
as:
P
((
x
u
,y
u
)
∈
Area
(
r
,x,y
))
P
((
x
u
,y
u
)
∈ Area
(
r, x, y
))
·R
Tech
=
r
2
r
2
with
r
<r
R
Priv
=
·R
Tech
,
(6)
0, the radius of the obfuscated area
r
is
calculated from (1) and (6) as follows:
Given a privacy preference
λ
≥
r
√
λ
+1
r
=
This relation permits to generate the obfuscated area by reducing radius
r
to radius
r
, which satisfies, according to our semantics, the user privacy
preference
λ
.
5 Integrating Obfuscation Techniques with LBAC
Systems
The definition of LBAC systems poses some architectural and functional is-
sues that w
ere never studied before in the context of traditional access control
1
The system of equation (5) is presented in the most general form of two areas
with different radii (i.e.,
r
and
R
).
