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the potential number of staff. The situation of staff
departing is somewhat more difficult. The shares of
departing staff must be revoked and we deal with
this in the next two sub-sections.
5.2 Revocation Lists
As staff leave, each data centre stores revoked
shares. To prevent a set of t shares, at least one of
which is revoked, from accessing a secret M
associated with a data centre, the polynomial (1) is
adjusted to the following:
s(x) = (M*Π(x – r
i
))/Π(x – r
i
) + s
1
x + … + s
t-1
x
t-1
(mod p) (2)
where the product is taken over the abscissa of each
revoked share. When a non-revoked value for x is
substituted, the coefficient of M is 1, and (2) reduces
to (1). However, when a revoked value for x is used,
the coefficient of M is not computable, M is not
revealed and access is denied.
Revocation can continue to the point where
precisely t associated shares are available to access a
data centre, but not beyond. The service provider
must ensure that enough shares are available at any
time in order to access the data centre. Thus, we
have the following lemma.
LEMMA 2: Any (t,w)-threshold secret sharing
scheme over GF(p) based on (2) with 2 ≤ t ≤ w ≤ p-
1, p a prime, permits revocation of up to w-t shares
while still operating as a (t,r)-threshold secret
sharing scheme over GF(p) based on (2) with 2 ≤ t
≤ r ≤ w ≤ p-1.
If a revoked share is used in attempting to gain
access to a data centre, the authentication centre can
determine which share it was and hence also to
whom it belongs. While equation (2) does not reveal
this, the following does. Once a set of t shares fails
to produce the secret, the authentication centre
compares each submitted share with each entry in
the list of revoked shares to find a match and thus
also identifies the owner.
5.3 Managing Revocation Lists
Since p-1 shares can be distributed, up to p-1-t
shares can be revoked with a usable secret sharing
scheme remaining. Once p-1-t+1 = p-t shares are
revoked, the scheme is no longer usable and a new
threshold secret sharing scheme must be deployed.
The current share holders may retain their
existing shares if a new polynomial is designed in
the following way: retain the t-1 or fewer existing
valid shares. Choose additional pairs randomly, but
excluding all revoked shares from the first scheme,
so that a total of t’ pairs is available where t’ is to be
the threshold of the new scheme. Use these t’ pairs
to determine uniquely a polynomial of the form (1)
over some (large) prime p’. This new polynomial
can now be used to formulate p’ shares, such that the
holders of old shares retain these.
6 CONCLUSIONS
AND FUTURE WORK
We have presented two approaches, which can be
combined, to protection of data inside a cloud
service centre. One, expressed in Figure 2, stores the
data in such a way as to separate data of similar
‘types’. The second deals with allocating shares to
cloud staff in such a way that access to more than
one data set of the same type is prevented. Thus, in
allocating shares, conditions (i) and (ii) must not be
violated; refer to Figure 2.
We showed that in such a setting revocation of
up to a certain number of shares distributed to staff
can be easily managed.
We leave for future work the following problem:
automate an efficient scheme for allocation of shares
in a dynamic environment (staff leaving and joining)
such that conditions (i) and (ii) are always valid.
Extend this scheme so that it includes the addition
and removal of data centres.
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