On Feb 17, 2009, at 6:03 PM, R.A. Hettinga wrote:
Begin forwarded message:
From: Sarad AV <jtrjtrjtr2...@yahoo.com>
Date: February 17, 2009 9:51:09 AM EST
To: cypherpu...@al-qaeda.net
Subject: Shamir secret sharing and information theoretic security
hi,
I was going through the wikipedia example of shamir secret sharing
which says it is information theoretically secure.
http://en.wikipedia.org/wiki/Shamir%27s_Secret_Sharing
In the example in that url, they have a polynomial
f(x) = 1234 + 166.x + 94.x^2
they construct 6 points from the polynomial
(1,1494);(2,1942);(3,2578);(4,3402);(5,4414);(6,5615)
the secret here is S=1234. The threshold k=3 and the number of
participants n=6.
If say, first two users collude then
1494 = S + c1 .1 + c2.1
1942 = S + c1 .2 + c2.2
clearly, one can start making inferences about the sizes of the
unknown co-efficients c1 and c2 and S.
However, it is said in the URL above that Shamir secret is
information theoretically secure
It is. Knowing some of the coefficients, or some constraints on some
of the coefficients, is just dual to knowing some of the points.
Neither affects the security of the system, because the coefficients
*aren't secrets* any more than the values of f() at particular points
are. They are *shares* of secrets, and the security claim is that
without enough shares, you have no information about the remaining
shares.
The argument for information-theoretic security is straightforward:
An n'th degree polynomial is uniquely specified if you know its value
at n+1 points - or, dually, if you know n+1 coefficients. On the
other hand, *any* set of n+1 points (equivalently, any set of n+1
coefficients) corresponds to a polynomial. Taking a simple approach
where the secret is the value of the polynomial at 0, given v_1,
v_2, ..., v_n and *any* value v, there is a (unique) polynomial of
degree at most n with p(0) = v and p(i) = v_i for i from 1 to n.
Dually, the value p(0) is exactly the constant term in the
polynomial. Given any fixed set of values c_1, c_2, ..., c_n, and any
other value c there is obviously a polynomial p(x) = Sum_{0 to n}(c_i
x^i), where c_0 = c, and indeed p(0) = c.
Or ... in terms of your problem: Even if I give you, not just a pair
of linear equations in c1, c2, and S, but the actual values c1 and c2
- the constant term (the secret) can still be anything at all.
The description above is nominally for polynomials over the reals. It
works equally for polynomials over any field - like the integers mod
some prime, for example. For a finite field, there is an obvious
interpretation of probability (the uniform probability distribution),
and given that, "no information" can be interpreted in terms of the
difference between your a priori and a posterio estimates of the
probability that p(0) takes on any particular value, the values of
p(1), ..., p(n) (and that differences is exactly 0). Because there
can be no uniform probability distribution over all the reals, you
can't state things in quite the same way, and "information theoretic
security" is a bit of a vague notion. Then again, no one does
computations over the reals. FP values - say, IEEE single precision -
aren't a field and there are undoubtedly big biases if you try to use
Shamir's technique there. (It should work over infinite-precision
rationals.)
-- Jerry
in the url below they say
http://en.wikipedia.org/wiki/Information_theoretic_security
"Secret sharing schemes such as Shamir's are information
theoretically secure (and in fact perfectly secure) in that less
than the requisite number of shares of the secret provide no
information about the secret."
how can that be true? we already are able to make inferences.
Moreover say that, we have 3 planes intersecting at a single point
in euclidean space, where each plane is a secret share(Blakely's
scheme). With 2 plane equations, we cannot find the point of
intersection but we can certainly narrow down to the line where the
planes intersect. There is information loss about the secret.
from this it appears that Shamir's secret sharing scheme leaks
information from its shares but why is it then considered
information theoretically secure?
They do appear to leak information as similar to k-threshold schemes
using chinese remainder theorem.
what am i missing?
Thanks,
Sarad.
---------------------------------------------------------------------
The Cryptography Mailing List
Unsubscribe by sending "unsubscribe cryptography" to majord...@metzdowd.com
---------------------------------------------------------------------
The Cryptography Mailing List
Unsubscribe by sending "unsubscribe cryptography" to majord...@metzdowd.com
