  In single-cloud storage, ciphertext-policy attribute-based encryption
(CP-ABE) allows one to encrypt any data under an access structure to a cloud
server, specifying what attributes are required to decrypt. In multi-cloud
storage, a secret sharing scheme (SSS) allows one to split any data into
multiple shares, one to a single server, and specify which subset of the
servers are able to recover the data. It is an interesting problem to remove
some attributes/servers but still enable the remaining attributes/servers in
every authorized set to recover the data. The problem is related to the
contraction problem of access structures for SSSs. In this paper, we propose a
method that can efficiently transform a given SSS for an access structure to
SSSs for contractions of the access structure. We show its applications in
solving the attribute removal problem in the CP-ABE based single-cloud storage
and the data relocating problem in multi-cloud storage. Our method results in
solutions that require either less server storage or even no additional server
storage.
