Difference between hierarchical and multi level secret sharing

Question

In this paper, the definitions are simply as follows.

Multi-Level secret sharing: Higher level participants can help lower participants to get the secret.

Compartmented secret sharing: Each level participants has to get its own secret. (No help from higher level participants)

According to this paper

I found that Disjunctive Multi-Level secret sharing is identical to Multi-Level secret sharing and Conjunctive Multi-Level secret sharing is identical to Compartmented secret sharing.

Is above statement true?

My doubt is what is the difference between hierarchical secret sharing and multi-level secret sharing?

Is hierarchical secret sharing same as Disjunctive Multi-Level secret sharing?

If I add a restriction to the compartmented secret sharing that 'participants at level $i$ can recover its secret only if participants at level $i-1$ recovers their secret', then what we can call this type of secret sharing?

Answer 1

In a multilevel secret sharing scheme there is only one secret $S$ that all levels seek to find. In a compartemnted scheme we can have multiple secrets that correspond to each level that must be found before $S$ can be recovered. In a compartmented scheme a set of compartments must work together to recover the secret, where each compartment contains a number of participants. In a hierarchical scheme participants from different levels can work together to discover the secret, but they don't have to.

As far as I know hierarchical is interchangeable with multilevel. A disjunctive scheme is indeed simply a hierarchial scheme where participants from different levels can work together but can also find the secret by cooperating with participants on their own level. However a conjunctive scheme is what Tassa terms a hierarchial threshold scheme (see: http://www.openu.ac.il/personal_sites/tamirtassa/download/journals/htss.pdf).

To give an example of this type imagine we have a scenario where there is a safe at a bank that must be opened by employees, however they cannot do this without being in the presence of two senior employees or a manager. This type of scheme is not the same as a compartmental scheme as not all levels must cooperate together to find the secret.

Additionally I think Nojoumian and Stinson have done something similar to what you propose: http://isyou.info/jisis/vol5/no2/jisis-2015-vol5-no2-02.pdf

Hope this helps.