Shannon Perfect Secrecy in a Discrete Hilbert Space

TL;DR

This paper extends Shannon's perfect secrecy from classical one-time-pad to a discrete Hilbert space framework using permutation matrices over finite fields, maintaining perfect secrecy and reusability.

Contribution

It introduces a novel quantum-inspired encryption scheme in a discrete Hilbert space that preserves perfect secrecy and allows reusable pads.

Findings

The scheme achieves perfect secrecy with permutation matrices.

Reusability of the quantum permutation pad is demonstrated.

The encryption maintains 1-to-1 mapping decoupled in discrete Hilbert space.

Abstract

The One-time-pad (OTP) was mathematically proven to be perfectly secure by Shannon in 1949. We propose to extend the classical OTP from an n-bit finite field to the entire symmetric group over the finite field. Within this context the symmetric group can be represented by a discrete Hilbert sphere (DHS) over an n-bit computational basis. Unlike the continuous Hilbert space defined over a complex field in quantum computing, a DHS is defined over the finite field GF(2). Within this DHS, the entire symmetric group can be completely described by the complete set of n-bit binary permutation matrices. Encoding of a plaintext can be done by randomly selecting a permutation matrix from the symmetric group to multiply with the computational basis vector associated with the state corresponding to the data to be encoded. Then, the resulting vector is converted to an output state as the ciphertext. The decoding is the same procedure but with the transpose of the pre-shared permutation matrix. We demonstrate that under this extension, the 1-to-1 mapping in the classical OTP is equally likely decoupled in Discrete Hilbert Space. The uncertainty relationship between permutation matrices protects the selected pad, consisting of M permutation matrices (also called Quantum permutation pad, or QPP). QPP not only maintains the perfect secrecy feature of the classical formulation but is also reusable without invalidating the perfect secrecy property. The extended Shannon perfect secrecy is then stated such that the ciphertext C gives absolutely no information about the plaintext P and the pad.