On Fri, 18 Feb 2005, John Richard Moser wrote:

> The authentication works as below:
>
> - N users may authenticate to access the data
> - A magnitude M of authenticated users is needed to access the data
> - N >= 3 > M >= 2
>
> Are there any known ways to do this?

Google for secret sharing or secret splitting. In particular, look for
Shamir's scheme, which seems to be the simplest. And there's always
Wikipedia: http://en.wikipedia.org/wiki/Secret_sharing

A brief overview of Shamir's scheme (it's so cool I can't resist):
Consider the M-th order polynomial:
N = c_{M-1} x^{M-1} + ... + c_1 x^1 + c_0 x^0
This polynomial is defined by c_0 .. c_{M-1}. So, M unknowns should
require M unknowns, right? Now let's say I tell you that I'm using M=2
(so N = c_1 x + c_0) and also tell you that:
N(1) = -1
N(2) = 1
and ask you for the password: c_0, c_1. You have two equations and two
unknowns, so you can solve it. What if person 2 was hit by a bus, and we
had to call in person 3 to access the data?
N(1) = -1
N(3) = 3
Either way, you can recover the coefficients (assuming you know
high-school math). And yet each individual person has zero knowledge.

> <EXAMPLE>
> N=3
> M=2
> Users X, Y, Z
> Key: [xxxx][yyyy][zzzz]
> X provides a key which decrypts xxxx
> Y provides a key which decrypts yyyy
> Z provides a key which decrypts zzzz

Very bad idea: each person knows enough to reduce the brute-force search
space dramatically.

As a side note, you mentioned that malicious attackers might have access
to the hardware. This is fine if they can only steal it and run their own
attacks on it. But an intelligent attacker would simply install a
keystroke logger, and grab a few keys. Guarding against this is left as
an exercise to the reader, but might involve splitting the secret amongst
multiple machines running different OSes in different locations adminned
by different people, possibly even running the secret-sharing software
written by different people. ;)

Damian Menscher
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