# Complexity of universal access structures

## TL;DR

This paper investigates the complexity of universal access structures in secret sharing schemes, establishing asymptotic bounds based on the number of minimal qualified sets.

## Contribution

It provides the first asymptotic bounds on the complexity of universal access structures relative to the number of minimal qualified sets.

## Key findings

- Complexity of universal structures is between n/log n and n/2.7182 asymptotically.
- Universal access structures are the richest possible given the number of minimal qualified sets.
- Every access structure is a substructure of the universal structure with the same minimal sets.

## Abstract

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.