Complexity of universal access structures
This work addresses a fundamental question in cryptography for researchers and practitioners, providing precise bounds on the complexity of access structures.
The paper tackles the problem of determining the complexity of universal access structures in secret sharing schemes, showing that the complexity lies asymptotically between n/log₂ n and n/2.7182 for n minimal qualified sets.
An important parameter in a secret sharing scheme is the number of minimal qualified sets. Given this number, the universal access structure is the richest possible structure, namely the one in which there are one or more participants in every possible Boolean combination of the minimal qualified sets. Every access structure is a substructure of the universal structure for the same number of minimal qualified subsets, thus universal access structures have the highest complexity given the number of minimal qualified sets. We show that the complexity of the universal structure with $n$ minimal qualified sets is between $n/\log_2 n$ and $n/2.7182$ asymptotically.