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Approximate quantum encryption with even shorter keys

Temel, Mehmet, Boris Škorić

Published Jan 1, 2022• 1 citation
OpenAlex

Abstract

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 $2^n$-dimensional state requires $2n$ classical bits. However, almost-perfect encryption, with information-theoretic security, can be achieved with only little more than $1$ key bit per qubit. In this paper we slightly improve the key length. We show that key length $n+\frac23\log\frac1\varepsilon$ suffices to encrypt $n$ qubits in such a way that the cipherstate has trace distance $\leq\varepsilon$ from the fully mixed state. The previous best result was $n+2\log\frac1\varepsilon$. Similar to the previous literature, we expand the key to a $2n$-bit pseudorandom string which is then used as a QOTP key. In this expansion we make use of $2n$ bits of public randomness which are included as a classical part of the cipherstate. Our key expansion is slightly faster than in previous works.

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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.

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We give an unconditional construction of information-theoretically secure one-time private-key unclonable encryption for one-bit messages, with efficient encryption and decryption and exponentially small unclonable-indistinguishability advantage. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf (arXiv:2603.08916) either only showed inverse-polynomial security loss or required inefficient encryption/decryption operations; we avoid both of these caveats. Our scheme's key is a uniformly random non-identity phase-free Pauli on $n$ qubits, and bit $a$ is encrypted as a random $(-1)^a$ eigenstate of that Pauli. We prove that the probability that both receivers recover the bit is at most $1/2 + O\left(2^{-n/2}\right).$ By a lower bound due to Broadbent, Culf, and Rochette (arXiv:2510.00903), this is the best probability bound achievable with $n$-qubit ciphertexts while ensuring perfect correctness (up to the constant hidden in the $O(\cdot)$).

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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.
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.
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.
Entropically secure encryption is a way to encrypt a large plaintext with a small key and still have information-theoretic security, thus in a certain sense circumventing Shannon's result that perfect encryption requires the key to be at least as long as the entropy of the plaintext. Entropically secure encryption is not perfect, and it works only if a lower bound is known on the entropy of the plaintext. The typical implementation is to expand the short key to the size of the plaintext, e.g. by multiplication with a public random string, and then use one-time pad encryption. This works in the classical as well as the quantum setting. In this paper, we introduce a new key expansion method that is faster than existing ones. We prove that it achieves the same security. The speed gain is most notable when the key length is a sizeable fraction of the message length. In particular, a factor of 2 is gained in the case of approximate randomization of quantum states.
It is well known that n bits of entropy are necessary and sufficient to perfectly encrypt n bits (one-time pad). Even if we allow the encryption to be approximate, the amount of entropy needed does not asymptotically change. However, this is not the case when we are encrypting quantum bits. For the perfect encryption of n quantum bits, 2n bits of entropy are necessary and sufficient (quantum one-time pad), but for approximate encryption one asymptotically needs only n bits of entropy. In this paper, we provide the optimal trade-off between the approximation measure ϵ and the amount of classical entropy used in the encryption of single quantum bits. Then, we consider n-qubit encryption schemes which are a composition of independent single-qubit ones and provide the optimal schemes both in the 2- and the ∞-norm. Moreover, we provide a counterexample to show that the encryption scheme of Ambainis-Smith [Proceedings of RANDOM ’04, pp. 249–260] based on small-bias sets does not work in the ∞-norm.
Abstract Entropically secure encryption is a way to encrypt a large plaintext with a small key and still have information-theoretic security, thus in a certain sense circumventing Shannon’s result that perfect encryption requires the key to be at least as long as the entropy of the plaintext. Entropically secure encryption is possible when a lower bound is known on the entropy of the plaintext from the adversary’s point of view. The typical implementation is to expand the short key to the size of the plaintext, e.g. by multiplication with a public random string, and then use one-time pad encryption. This works in the classical as well as the quantum setting. In this paper, we introduce a new key expansion method that is faster than existing ones. We prove that it achieves the same security. The speed gain is most notable when the key length is a sizeable fraction of the message length. In particular, a factor of 2 is gained in the case of approximate randomization of quantum states. In the classical case, we obtain a reduction of the ciphertext size.
We give an unconditional construction of information-theoretically secure one-time private-key unclonable encryption for one-bit messages, with efficient encryption and decryption and exponentially small unclonable-indistinguishability advantage. Previous work by Bhattacharyya and Culf (Nature Physics, 2026) and Bhattacharyya, Broadbent, and Culf (arXiv:2603.08916) either only showed inverse-polynomial security loss or required inefficient encryption/decryption operations; we avoid both of these caveats. Our scheme's key is a uniformly random non-identity phase-free Pauli on nnn qubits, and bit aaa is encrypted as a random (−1)a(-1)^a(−1)a eigenstate of that Pauli. We prove that the probability that both receivers recover the bit is at most 1/2+O(2−n/2).1/2 + O\left(2^{-n/2}\right).1/2+O(2−n/2). By a lower bound due to Broadbent, Culf, and Rochette (arXiv:2510.00903), this is the best probability bound achievable with nnn-qubit ciphertexts while ensuring perfect correctness (up to the constant hidden in the O(⋅)O(\cdot)O(⋅)).