A Hierarchical Secret Sharing Scheme over Finite Fields of Characteristic 2

Abstract

Hierarchical secret sharing schemes are known for the way the secret is shared among a group of participants that is partitioned into levels. We examine these schemes in terms of how easily they delete a secret after it is distributed or namely for cases where the reliability of data deletion depends on deletion of the indispensable participants' share. In this paper, we consider Tassa's idea of using formal derivatives and Birkhoff interpolation so that his method will work well even over finite fields of characteristic 2, then we devise a method for derivatives. As a result, we propose a fast (k, n) hierarchical secret sharing scheme applicable to any level and report the software implementation evaluation. Moreover, taking practical use into consideration, we cover the optimization specialized for a ({1, 3}, n) hierarchical secret sharing scheme.

New Proof Techniques Using the Properties of Circulant Matrices for XOR-based (<i>k</i>, <i>n</i>) Threshold Secret Sharing Schemes

“…Shima et al reported in Ref. [8] that this computational cost can be one operation if we use a lookup table that has been precomputed for the multiplication operation over GF(2 L ); otherwise, there remains little choice but to practically choose L = 8 in terms of the amount of available memory. To extend the size of L, Ikarashi et al's technique [9] is suited for fast multiplication operations over GF (2 64 ).…”

Generalized threshold secret sharing and finite geometry

“…In the conjunctive case there are only few general solutions based on interpolation by Tassa [17], Tassa and Dyn [18], Shima and Doi [15] and on MDS codes by Tentu et al [19]. Furthermore, there are some constructions for special cases of two levels, like a (1, 3)-scheme by Fuji-Hara and Miao [8].…”

“…3.3. Apart from the general constructions [15,[17][18][19] on arbitrary levels , this is the first ideal conjunctive scheme on 3-levels. Note that neither of the above general methods yielding our geometry construction.…”

“…The proposed construction has no restrictions on the related finite field in contrast with the scheme of Tassa [17] working over fields of characteristic larger than 2, and the scheme of Shima and Doi [15] working only over fields of characteristic 2. Furthermore, the proposed constructions are unconditionally perfect in contrast with the solution of Tentu et al [19] which is probabilistic in the sense that a non-qualified subset can compute the secret with negligible probability.…”