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On ideal and weakly-ideal access structures

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  • For more than two decades, proving or refuting the following statement has remained a challenging open problem in the theory of secret sharing schemes (SSSs): every ideal access structure admits an ideal perfect multi-linear SSS. The class of group-characterizable (GC) SSSs include the multi-linear ones. Hence, if the above statement is true, then so is the following weaker statement: every ideal access structure admits an ideal perfect GC SSS. One contribution of this paper is to show that ideal SSSs are not necessarily GC. Our second contribution is to study the above two statements with respect to several variations of weakly-ideal access structures. Recently, Mejia and Montoya studied ideal access structures that admit ideal multi-linear schemes and provided a classification-like theorem for them. We additionally present some tools that are useful to extend their result.

    Mathematics Subject Classification: Primary: 94A62, 05B35; Secondary: 05B15.


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  • Table 1.  Summary of answers to the problem of whether every ideal (resp. weakly-ideal) access structure is realizable by an ideal scheme (resp. weakly-ideal family of schemes) from different classes

    Ideal Nearly-ideal Stat.-ideal Almost-ideal Quasi-ideal
    Multi-linear $ \text{Unsolved} $ $ \text{No} $ $ \text{No} $ $ \text{No} $ $ \text{No} $
    GC with normal secret group $ \text{Unsolved} $ $ \text{Unsolved} $ $ \text{Unsolved} $
    GC $ \text{Unsolved} $ $ \text{Unsolved} $ $ \text{Unsolved} $ $ \text{Unsolved} $ $ \text{Yes} $
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