Cryptography Reference
In-Depth Information
the information that represents them. The Shamir scheme shown
here doesn't require more space. Each part is still just a point,
(
x, y
)
that lies on the polynomial. This is substantially more efficient.
4.3.1 Making Some More Equal
In each of the schemes described in this chapter, the secrets are split
into
parts has the same equal share of
control. Humans, being human, are never satisfied with anything as
fair as that—some people will want some parts to be more powerful
than others.
The most straightforward way to accomplish this is to give some
people more parts. For instance, imagine a scheme where you need
six parts to reconstruct the secret. That is, you might have a col-
lection of five-dimensional hyperplanes in a six-dimensional space.
Any set of six points is enough to uncover the secret, which for the
sake of example will be the launch codes for a nuclear missile. Let's
say that it takes two commanders, three sergeants, or six privates to
launch a missile. This can be accomplished by giving three parts to
the commanders, two parts to the sergeants, and one part to each
private.
One problem with this solution is that arbitrary combinations of
different ranks can join together. So, one commander, one sergeant,
and one private can work together to uncover the secret. This might
not be permitted in some cases. For example, the U.S. Congress re-
quires a majority of both the House and the Senate to pass a bill. But
the votes from one chamber can't be counted against the other. So
even though there are 100 Senators and 435 members of the House,
a Senator is not worth 4.35 House members. A bill won't pass just
because 99 Senators vote for it and only 10 House Representatives.
But this could be the situation if someone naively created a secret-
sharing scheme by parceling out parts to both sides of Congress from
the same shared secret.
A better solution to prevent this is to first split the secret into two
equal parts,
n
parts and each of the
n
X
S
, so that both are required to endorse a bill
with the digital signature of Congress. Then
X
H
and
H
R
would be split into
435 parts so that 218 are enough to recover it.
H
S
is split into 100
parts so that 51 are enough to recover it.
Numerous combinations canmake these schemes possible. Prac-
tically any scheme can be implemented using some combination
and layers of secrets. The only problem with very complicated sys-
tems is that they can require many different dimensions. For in-
stance, if you want a system that takes 17 privates, 13 sergeants, or
