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International Journal of Quantum Information

Quantum Key Recycling with 8-state encoding (The Quantum One-Time Pad is more interesting than we thought)

Boris Škorić, Manon de Vries

Published Apr 1, 2017• 11 citations
Publisher

Abstract

Perfect encryption of quantum states using the Quantum One-Time Pad (QOTP) requires two 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 [Formula: see text] suffices to encrypt [Formula: see text] qubits in such a way that the cipherstate’s [Formula: see text]-distance from uniformity is upperbounded by [Formula: see text]. For a stricter security definition involving the [Formula: see text]-norm, we prove sufficient key length [Formula: see text], where [Formula: see text] is a small probability of failure. Our proof uses Pauli operators, whereas previous results on the [Formula: see text]-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 [Formula: see text] 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 (QKR) and Unclonable Encryption (UE). 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 nine qubits.

DOI
10.1142/s0219749917500162

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Perfect encryption of quantum states using the Quantum One-Time Pad (QOTP) requires two 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 [Formula: see text] suffices to encrypt [Formula: see text] qubits in such a way that the cipherstate’s [Formula: see text]-distance from uniformity is upperbounded by [Formula: see text]. For a stricter security definition involving the [Formula: see text]-norm, we prove sufficient key length [Formula: see text], where [Formula: see text] is a small probability of failure. Our proof uses Pauli operators, whereas previous results on the [Formula: see text]-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 [Formula: see text] 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 (QKR) and Unclonable Encryption (UE). 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 nine qubits.
Perfect encryption of a qubit state using the Quantum One-Time Pad (QOTP) requires 2 classical key bits. More generally, perfect QOTP encryption of a 2n2^n2n-dimensional state requires 2n2n2n classical bits. However, almost-perfect encryption, with information-theoretic security, can be achieved with only little more than 111 key bit per qubit. In this paper we slightly improve the key length. We show that key length n+23log⁡1εn+\frac23\log\frac1\varepsilonn+32​logε1​ suffices to encrypt nnn qubits in such a way that the cipherstate has trace distance ≤ε\leq\varepsilon≤ε from the fully mixed state. The previous best result was n+2log⁡1εn+2\log\frac1\varepsilonn+2logε1​. Similar to the previous literature, we expand the key to a 2n2n2n-bit pseudorandom string which is then used as a QOTP key. In this expansion we make use of 2n2n2n bits of public randomness which are included as a classical part of the cipherstate. Our key expansion is slightly faster than in previous works.
Quantum Key Recycling aims to re-use the keys employed in quantum encryption and quantum authentication schemes. QKR protocols can achieve better round complexity than Quantum Key Distribution. We consider a QKR protocol that works with qubits, as opposed to high-dimensional qudits. A security proof was given by Fehr and Salvail in the case where there is practically no noise. A high-rate scheme for the noisy case was proposed by \v{S}kori\'{c} and de Vries, based on eight-state encoding. However, a security proof was not given. In this paper we introduce a protocol modification and provide a security proof. The modified protocol has high rate not only for 8-state encoding, but also 6-state and BB84 encoding. Our proof is based on a bound on the trace distance between the real quantum state of the system and a state in which the keys are completely secure. It turns out that the rate is higher than suggested by previous results. Asymptotically the rate equals the rate of Quantum Key Distribution with one-way postprocessing.
Quantum Key Recycling aims to re-use the keys employed in quantum encryption and quantum authentication schemes. QKR protocols can achieve better round complexity than Quantum Key Distribution. We consider a QKR protocol that works with qubits, as opposed to high-dimensional qudits. A security proof was given by Fehr and Salvail [1] in the case where there is practically no noise. A high-rate scheme for the noisy case was proposed by Škorić and de Vries [2], based on eight-state encoding. However, a security proof was not given. In this paper we introduce a protocol modification to [2] and provide a security proof. The modified protocol has high rate not only for 8- state encoding, but also 6-state and BB84 encoding. Our proof is based on a bound on the trace distance between the real quantum state of the system and a state in which the keys are completely secure. It turns out that the rate is higher than suggested by previous results. Asymptotically the rate equals the rate of Quantum Key Distribution with one-way postprocessing.
Quantum Key Recycling (QKR) is a quantum cryptographic primitive that allows one to reuse keys in an unconditionally secure way. By removing the need to repeatedly generate new keys, it improves communication efficiency. Škorić and de Vries recently proposed a QKR scheme based on 8-state encoding (four bases). It does not require quantum computers for encryption/decryption but only single-qubit operations. We provide a missing ingredient in the security analysis of this scheme in the case of noisy channels: accurate upper bounds on the required amount of privacy amplification. We determine optimal attacks against the message and against the key, for 8-state encoding as well as 4-state and 6-state conjugate coding. We provide results in terms of min-entropy loss as well as accessible (Shannon) information. We show that the Shannon entropy analysis for 8-state encoding reduces to the analysis of quantum key distribution, whereas 4-state and 6-state suffer from additional leaks that make them less effective. From the optimal attacks we compute the required amount of privacy amplification and hence the achievable communication rate (useful information per qubit) of qubit-based QKR. Overall, 8-state encoding yields the highest communication rates.
Quantum Key Recycling (QKR) is a quantum-cryptographic primitive that allows one to re-use keys in an unconditionally secure way. By removing the need to repeatedly generate new keys it improves communication efficiency. Skoric and de Vries recently proposed a QKR scheme based on 8-state encoding (four bases). It does not require quantum computers for encryption/decryption but only single-qubit operations. We provide a missing ingredient in the security analysis of this scheme in the case of noisy channels: accurate bounds on the privacy amplification. We determine optimal attacks against the message and against the key, for 8-state encoding as well as 4-state and 6-state conjugate coding. We show that the Shannon entropy analysis for 8-state encoding reduces to the analysis of Quantum Key Distribution, whereas 4-state and 6-state suffer from additional leaks that make them less effective. We also provide results in terms of the min-entropy. Overall, 8-state encoding yields the highest capacity.
Quantum computing is an interdisciplinary field comprising of physics, mathematics and computer science that leverages quantum mechanics to complete complicated tasks more quickly than classical computers.It includes both the advancement of hardware research and application development.In the paper on quantum computing, quantum cryptography/Quantum Key Distribution, the revolutionary paradigm shift in computation is introduced, exploring the enigmatic world where quantum bits, or qubits, replace classical bits.Computational efficiency is increased through the manipulation of probabilities made easier by quantum interference.Furthermore, we examined the applications of quantum cryptography, which makes use of the ideas of quantum physics to guarantee secure communication.As a cornerstone, quantum key distribution appears, providing cryptographic keys encoded in quantum states with unbreakable encryption.The proposed work serves as a gateway to understand the profound implications and limitless potential of quantum computing and cryptography.The main contribution of the proposed work is to enhance the performance of the Quantum One-Time Pad (QOTP) algorithm in order to reduce the amount of classical key information needed while preserving the same level of security.The proposed approach combines entanglement with the QOTP, allowing secure communication without needing new key bits for each qubit encrypted.