Cryptography Reference
In-Depth Information
(x,y)
Figure 4.2: A secret,
x
, is split into
n
parts by finding
n
random lines
that intersect at
(
x, y
)
.(
y
is chosen at random.) Any pair is enough to
recover the secret.
can be used if the geometry is extended into
=3
,
then planes are used instead of lines. Three planes will intersect only
at the point. Two planes will form a line when they intersect. The
point
(
k
dimensions. If
k
)
will be somewhere along the line, but it is impossible
to determine where it is.
x, y, z
Stephan Brands uses
this technique in his
digital cash scheme
[Bra93].
It is also possible to flip this process on its head. Instead of hiding
the secret as the intersection point of several lines, you can make the
line the secret and distribute points along it. The place where the line
meets the
axismightbethesecret.Oritcouldbetheslopeofthe
line. In either case, knowing two points along the line will reveal the
secret. Figure 4.3 shows this approach.
Each of these systems offers a pretty basic way to split up a secret
keyorafilesothatsomesubsetofpeoplemustbepresent.Itshould
be easy to see that the geometric systems that hide the secret as the
intersection point are as secure as a one-time pad. If you only have
one line, then it is impossible to guess where the intersection lies
along this line.
y
= 41243
. In fact, owning
one part gives you no more insight into the secret than owning no
part. In either case, all you know is that it is some value of
x
=23
is just as likely as
x
x
.Thisis
often called a
perfect
secret-sharing system.
Some people might be tempted to cut corners and hide infor-
mation in both the
x
and the
y
coordinate of the intersection point.
