# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021017
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## On ideal and weakly-ideal access structures

 Department of Mathematical Sciences, Sharif University of Technology, Tehran, Iran

* Corresponding author

Received  October 2020 Revised  April 2021 Early access June 2021

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.

Citation: Reza Kaboli, Shahram Khazaei, Maghsoud Parviz. On ideal and weakly-ideal access structures. Advances in Mathematics of Communications, doi: 10.3934/amc.2021017
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##### References:
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
 Weakly-ideal 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 $\text{Unsolved}$ $\text{Unsolved}$ $\text{Unsolved}$ $\text{Unsolved}$ $\text{Yes}$
 Weakly-ideal 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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