gfshare - explanation of Shamir Secret Sharing in gf(2**8)
In simple terms, this package provides a library for implementing the
sharing of secrets and two tools for simple use-cases of the algorithm.
The library implements what is known as Shamir’s method for secret
sharing in the Galois Field 2**8. In slightly simpler words, this is
N-of-M secret-sharing byte-by-byte. Essentially this allows us to
split a secret S into any M shares S(1) to S(M) such that any N of
those shares can be used to reconstruct S but any less than N shares
yields no information whatsoever.
EXAMPLE USE CASE
Alice has a GPG secret key on a usb keyring. If she loses that keyring,
she will have to revoke the key. This sucks because she go to
conferences lots and is scared that she will, eventually, lose the key
somewhere. So, if, instead she needed both her laptop and the usb
keyring in order to have her secret key, losing one or the other does
not compromise her gpg key. Now, if she splits the key into a 3-of-5
share, put one share on her desktop, one on the laptop, one on her
server at home, and two on the keyring, then the keyring-plus-any-
machine will yield the secret gpg key, but if she loses the keyring,
She can reconstruct the gpg key (and thus make a new share, rendering
the shares on the lost usb keyring worthless) with her three machines
THE PRINCIPLES BEHIND SHAMIR’S METHOD
What Shamir’s method relies on is the creation of a random polynomial,
the sampling of various coordinates along the curve of the polynomial
and then the interpolation of those points in order to re-calculate the
y-intercept of the polynomial in order to reconstruct the secret.
Consider the formula for a straight line: Y = Mx + C. This formula
(given values for M and C) uniquely defines a line in two dimensions
and such a formula is a polynomial of order 1. Any line in two
dimensions can also be uniquely defined by stating any two points along
the line. The number of points required to uniquely define a polynomial
is thus one higher than the order of the polynomial. So a line needs
two points where a quadratic curve needs three, a cubic curve four,
When we create a N-of-M share, we encode the secret as the y-intercept
of a polynomial of order N-1 since such a polynomial needs N points to
uniquely define it. Let us consider the situation where N is 2: We need
a polynomial of order 1 (a straight line). Let us also consider the
secret to be 9, giving the formula for our polynomial as: Y = Ax + 9.
We now pick a random coefficient for the graph, we’ll use 3 in this
example. This yields the final polynomial: Sx = 3x + 9. Thus the share
of the secret at point x is easily calculated. We want some number of
shares to give out to our secret-keepers; let’s choose three as this
number. We now need to select three points on the graph for our shares.
For simplicity’s sake, let us choose 1, 2 and 3. This makes our shares
have the values 12, 15 and 18. No single share gives away any
information whatsoever about the value of the coefficient A and thus no
single share can be used to reconstruct the secret.
Now, consider the shares as coordinates (1, 12) (2, 15) and (3, 18) -
again, no single share is of any use, but any two of the shares
uniquely define a line in two-dimensional space. Let us consider the
use of the second and third shares. They give us the pair of
simulaneous equations: 15 = 2M + S and 18 = 3M + S. We can trivially
solve these equations for A and S and thus recover our secret of 9.
Solving simultaneous equations isn’t ideal for our use due to its
complexity, so we use something called a ’Lagrange Interpolating
Polynomial’. Such a polynomial is defined as being the polynomial P(x)
of degree n-1 which passes through the n points (x1, y1 = f(x1)) ...
(xn, yn = f(xn)). There is a long and complex formula which can then be
used to interpolate the y-intercept of P(x) given the n sets of
coordinates. There is a good explanation of this at
OKAY, SO WHAT IS A GALOIS FIELD THEN?
A Galois Field is essentially a finite set of values. In particular,
the field we are using in this library is gf(2**8) or gf(256) which is
the values 0 to 255. This is, cunningly enough, exactly the field of a
byte and is thus rather convenient for use in manipulating arbitrary
amounts of data. In particular, performing the share in gf(256) has the
property of yielding shares of exactly the same size as the secret.
Mathematics within this field has various properties which we can use
to great effect. In particular, addition in any Galois Field of the
form gf(2**n) is directly equivalent to bitwise exclusive-or (an
operation computers are quite fast at indeed). Also, given that (X op
Y) mod F == ((X mod F) op (Y mod F)) mod F we can perform maths on
values inside the field and keep them within the field trivially by
truncating them to the relevant number of bits (eight).
OKAY, SO WHY IS THERE NO MULTIPLICATION IN THIS IMPLEMENTATION?
For speed reasons, this implementation uses log and exp as lookup
tables to perform multiplication in the field. Since exp( log(X) +
log(Y) ) == X * Y and since table lookups are much faster than
multiplication and then truncation to fit in a byte, this is a faster
but still 100% correct way to do the maths.
Written by Daniel Silverstone.
Report bugs against the libgfshare product on www.launchpad.net.
libgfshare is copyright © 2006 Daniel Silverstone.
This is free software. You may redistribute copies of it under the
terms of the MIT licence (the COPYRIGHT file in the source
distribution). There is NO WARRANTY, to the extent permitted by law.
gfsplit(1), gfcombine(1), libgfshare(3)