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Exact and approximate encryption of quantum information

Jan Bouda

Published Jan 1, 2005• 0 citations
OpenAlex

Abstract

In this paper, we address properties of the private quantum channel (PQC), namely a security weakness occurring when at least two copies of the ciphertext are available. We discuss possible definitions of the known-plaintext, known-ciphertext and chosen-plaintext attack. We examine the ability of a PQC to randomize correlations and, finally, we show that specific PQCs encrypting a given set of plaintexts can achieve the approximate encryption of a larger set of plaintexts.

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We study in detail the problem of encryption of quantum information. We present an attack on a private quantum channel (PQC) which applies when partial classical description of a ciphertext is known (the so-called known-ciphertext attack) and we show how this situation can be avoided. The quantum analogue of the known plaintext attack is also discussed. We determine how correlations between quantum systems can be encrypted and we conclude that two PQCs on the subsystems form a PQC on the whole composite system. Finally, some applications of the PQC are suggested and a security of a noisy channel is discussed.

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The lack of perfect randomness can cause significant problems in securing communication between two parties. McInnes and Pinkas \cite{McInnesPinkas-ImpossibilityofPrivate-1991} proved that unconditionally secure encryption is impossible when the key is sampled from a weak random source. The adversary can always gain some information about the plaintext, regardless of the cryptosystem design. Most notably, the adversary can obtain full information about the plaintext if he has access to just two bits of information about the source (irrespective on length of the key). In this paper we show that for every weak random source there is a cryptosystem with a classical plaintext, a classical key, and a quantum ciphertext that bounds the adversary's probability $p$ to guess correctly the plaintext strictly under the McInnes-Pinkas bound, except for a single case, where it coincides with the bound. In addition, regardless of the source of randomness, the adversary's probability $p$ is strictly smaller than $1$ as long as there is some uncertainty in the key (Shannon/min-entropy is non-zero). These results are another demonstration that quantum information processing can solve cryptographic tasks with strictly higher security than classical information processing.

Quantum Information and Computation•May 1, 2012•0 citations
In this paper, we address properties of the private quantum
channel (PQC), namely a security weakness occurring when at
least two copies of the ciphertext are available. We discuss
possible definitions of the known-plaintext, known-ciphertext
and chosen-plaintext attack. We examine the ability of a PQC to
randomize correlations and, finally, we show that specific PQCs
encrypting a given set of plaintexts can achieve the
approximate encryption of a larger set of plaintexts.
We study in detail the problem of encryption of quantum information. We present an attack on a private quantum channel (PQC) which applies when partial classical description of a ciphertext is known (the so-called known-ciphertext attack) and we show how this situation can be avoided. The quantum analogue of the known plaintext attack is also discussed. We determine how correlations between quantum systems can be encrypted and we conclude that two PQCs on the subsystems form a PQC on the whole composite system. Finally, some applications of the PQC are suggested and a security of a noisy channel is discussed.
We addressed the question of optimality of private quantum channels. We have shown that the Shannon entropy of the classical key necessary to securely transfer the quantum information is lower bounded by the entropy exchange of the private quantum channel E\cal EE and von Neumann entropy of the ciphertext state ϱ(0)\varrho^{(0)}ϱ(0). Based on these bounds we have shown that decomposition of private quantum channels into orthogonal unitaries (if exists) is optimizing the entropy. For non-ancillary single qubit PQC we have derived the optimal entropy for arbitrary set of plaintexts. In particular, we have shown that except when the (closure of the) set of plaintexts contains all states, one bit key is sufficient. We characterized and analyzed all the possible single qubit private quantum channels for arbitrary set of plaintexts. For the set of plaintexts consisting of all qubit states we have characterized all possible approximate private quantum channels and we have derived the relation between the security parameter and the corresponding minimal entropy.
We review the notion of a classical random cipher and its advantages. We sharpen the usual description of random ciphers to a particular mathematical characterization suggested by the salient feature responsible for their increased security. We describe a concrete system known as $\ensuremath{\alpha}\ensuremath{\eta}$ and show that it is equivalent to a random cipher in which the required randomization is affected by coherent-state quantum noise. We describe the currently known security features of $\ensuremath{\alpha}\ensuremath{\eta}$ and similar systems, including lower bounds on the unicity distances against ciphertext-only and known-plaintext attacks. We show how $\ensuremath{\alpha}\ensuremath{\eta}$ used in conjunction with any standard stream cipher such as the Advanced Encryption Standard provides an additional, qualitatively different layer of security from physical encryption against known-plaintext attacks on the key. We refute some claims in the literature that $\ensuremath{\alpha}\ensuremath{\eta}$ is equivalent to a nonrandom stream cipher.
We study private quantum channels on a single qubit, which encrypt given set of plaintext states PPP. Specifically, we determine all achievable states ρ(0)ρ^{(0)}ρ(0) (average output of encryption) and for each particular set PPP we determine the entropy of the key necessary and sufficient to encrypt this set. It turns out that single bit of key is sufficient when the set PPP is two dimensional. However, the necessary and sufficient entropy of the key in case of three dimensional PPP varies continuously between 1 and 2 bits depending on the state ρ(0)ρ^{(0)}ρ(0). Finally, we derive private quantum channels achieving these bounds. We show that the impossibility of universal NOT operation on qubit can be derived from the fact that one bit of key is not sufficient to encrypt qubit.
The lack of perfect randomness can cause significant problems in securing communication between two parties. McInnes and Pinkas \cite{McInnesPinkas-ImpossibilityofPrivate-1991} proved that unconditionally secure encryption is impossible when the key is sampled from a weak random source. The adversary can always gain some information about the plaintext, regardless of the cryptosystem design. Most notably, the adversary can obtain full information about the plaintext if he has access to just two bits of information about the source (irrespective on length of the key). In this paper we show that for every weak random source there is a cryptosystem with a classical plaintext, a classical key, and a quantum ciphertext that bounds the adversary's probability ppp to guess correctly the plaintext strictly under the McInnes-Pinkas bound, except for a single case, where it coincides with the bound. In addition, regardless of the source of randomness, the adversary's probability ppp is strictly smaller than 111 as long as there is some uncertainty in the key (Shannon/min-entropy is non-zero). These results are another demonstration that quantum information processing can solve cryptographic tasks with strictly higher security than classical information processing.