Paper 2016/1122

Quantum Key Recycling with eight-state encoding (The Quantum One Time Pad is more interesting than we thought)

B. Skoric and M. de Vries

Abstract

Perfect encryption of quantum states using the Quantum One-Time Pad (QOTP) requires 2 classical key bits per qubit. Almost-perfect encryption, with information-theoretic security, requires only slightly more than 1. We slightly improve lower bounds on the key length. We show that key length $n+2\log \frac{1}{\epsilon }$ suffices to encrypt $n$ qubits in such a way that the cipherstate's $L_1$-distance from uniformity is upperbounded by $\epsilon$. For a stricter security definition involving the $\mathrm{\infty }$-norm, we prove sufficient key length $n+\log n+2\log \frac{1}{\epsilon }+1+\frac{1}{n}\log \frac{1}{\delta }+\log \frac{\ln 2}{1-\epsilon }$, where $\delta$ is a small probability of failure. Our proof uses Pauli operators, whereas previous results on the $\mathrm{\infty }$-norm needed Haar measure sampling. We show how to QOTP-encrypt classical plaintext in a nontrivial way: we encode a plaintext bit as the vector $$ on the Bloch sphere. Applying the Pauli encryption operators results in eight possible cipherstates which are equally spread out on the Bloch sphere. This encoding, especially when combined with the half-keylength option of QOTP, has advantages over 4-state and 6-state encoding in applications such as Quantum Key Recycling and Unclonable Encryption. We propose a key recycling scheme that is more efficient and can tolerate more noise than a recent scheme by Fehr and Salvail. For 8-state QOTP encryption with pseudorandom keys we do a statistical analysis of the cipherstate eigenvalues. We present numerics up to 9 qubits.

Note: Improved the Key Recycling scheme. Extended the results on key length.