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The problem of minimizing the amount of secret information (secret bits) required for certain key management schemes related to data access control techniques is addressed. Particularly note that the importance of the secret storage minimization originates from the fact that this storage should be read-proof and tamper-proof one. This paper points out to a novel approach for minimization of the secret storage dimension and compare this approach with a straightforward one based on establishing a virtual secret storage employing data encryption. The novel approach yields: (i) provable security; (ii) it is not based on storing the encrypted data into a public storage, and (iii) it requires lower processing complexity. The proposed approach yields the secret storage minimization via exchange of a secret storage to a public one based on the efficient one-way mapping of the secret bits yielding significant additional flexibility and reduction of the secret storage overhead at the user's side as an appropriate trade-off with the required public storage and processing complexity. The overheads of the proposed technique are compared with the related previously reported ones, and advantages of the novel approach are pointed out.