Security of a Grover's Algorithm-based secret sharing protocol, generalized for an arbitrary number of participants, against interception attacks

TL;DR

This paper analyzes the security of a Grover's algorithm-based secret sharing protocol for any number of participants, focusing on interception attacks and deriving probabilities of breach depending on initial states and Householder angles.

Contribution

It generalizes the secret sharing protocol based on Grover's algorithm to any number of participants and derives analytical formulas for security probabilities.

Findings

Exact solutions for 2 and 3 participants

Numerical simulations for 4 to 7 participants

Extrapolated formulas for arbitrary participants

Abstract

In this work, we study interception attacks against a secret sharing protocol based on Grovers search algorithm. Unlike previous works that only give the algorithm for two and three participants, we have generalized the algorithm for any number of participants. Both reflections used in the algorithm are constructed using a generalized Householder reflection. Our main goal is to obtain the probability for an eavesdropper to break the secret depending on the true initial state and the one assumed by the eavesdropper and on the Householder angle. In cases where there are two and three participants, we give an exact analytical solution. These formulas are consistent with the numerical results. We use simulations for the case of between 4 and 7 participants to extrapolate the analytical formula for any number of participants.