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arXiv:quant-ph/0604207 · Physical Review A

Quantum mutual information and the one-time pad

Benjamin W. Schumacher, Michael D. Westmoreland

Published Oct 9, 2006• 132 citations
arXivHTMLPDF

Abstract

Alice and Bob share a correlated composite quantum system $AB$. If $AB$ is used as the key for a one-time pad cryptographic system, we show that the maximum amount of information that Alice can send securely to Bob is the quantum mutual information of $AB$.

DOI
10.1103/physreva.74.042305

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Alice and Bob share a correlated composite quantum system ABABAB. If ABABAB is used as the key for a one-time pad cryptographic system, we show that the maximum amount of information that Alice can send securely to Bob is the quantum mutual information of ABABAB.
Suppose that Alice and Bob are located in distant laboratories, which are connected by an ideal quantum channel. Suppose further that they share many copies of a quantum state ρ_{ABE}, such that Alice possesses the A systems and Bob the BE systems. In our model, there is an identifiable part of Bob's laboratory that is insecure: a third party named Eve has infiltrated Bob's laboratory and gained control of the E systems. Alice, knowing this, would like use their shared state and the ideal quantum channel to communicate a message in such a way that Bob, who has access to the whole of his laboratory (BE systems), can decode it, while Eve, who has access only to a sector of Bob's laboratory (E systems) and the ideal quantum channel connecting Alice to Bob, cannot learn anything about Alice's transmitted message. We call this task the conditional one-time pad, and in this Letter, we prove that the optimal rate of secret communication for this task is equal to the conditional quantum mutual information I(A;B|E) of their shared state. We thus give the conditional quantum mutual information an operational meaning that is different from those given in prior works, via state redistribution, conditional erasure, or state deconstruction. We also generalize the model and method in several ways, one of which is a secret-sharing task, i.e., the case in which Alice's message should be secure from someone possessing only the AB or AE systems, but should be decodable by someone possessing all systems A, B, and E.
ABSTRACT We propose a protocol to transfer a one‐time pad (in a probabilistic manner) from Alice to Bob, over a public channel. The proposed protocol is unique because Bob merely acts as a receiver of the pad (secret key); that is, Bob does not need to send any message back to Alice unless he detects eavesdropping. Such a secure transfer of one‐time pad, over public channel, is not possible in classical cryptography, and in quantum cryptography, all previous protocols require Bob to send almost as many messages back to Alice as she does to Bob to establish a key. Copyright © 2013 John Wiley & Sons, Ltd.
Cryptographic protocols can be classified by the type of security against eavesdropping which they provide. There exist mathematically secure schemes whose security relies on mathematical proofs (like the Vernam cipher discussed below) or conjectures (like public key RSA encryption) about the complexity of deciphering the message without possessing the correct key. The majority of current secure public Internet connections rely on such schemes. Alternatively, a cryptographic setup may provide a physically secure method for communicating. In these setups the security is provided by the physical laws governing the communication protocol. Here we first discuss a provably secure classical communication protocol and then quantum methods for distributing the necessary keys. The BB84 protocol relies on the impossibility of perfectly distinguishing non-orthogonal quantum states from one copy of the quantum system. The scheme was invented in 1984 and is the first of its kind. In contrast, the Ekert91 protocol makes use of Bell correlations between entangled pairs of photons. These correlations are destroyed when Eve attempts to make a measurement on one of the particles but also when imperfections, such as decoherence processes, affect the scheme. As long as no such “elements of reality” are introduced, and Bell correlations violating local realism can be generated by Alice and Bob, no eavesdropper can be present. One-time pads and the Vernam cipher The Vernam cipher is a cryptographic protocol which allows the encryption and decryption protocol to be publicly known. The security of the protocol relies entirely on the key which is private and not publicly known. Alice and Bob share identical n -bit secret key strings (one-time pad).
We introduce and analyze an information theoretical task that we call the quantum multiple-access one-time pad. Here, a number of senders initially share a correlated quantum state with a receiver and an eavesdropper. Each sender performs a local operation to encode a classical message and sends their system to the receiver, who subsequently performs a measurement to decode the messages. The receiver will be able to decode the messages almost perfectly, while the eavesdropper must not be able to extract information about the messages even if they have access to the quantum systems transmitted. We consider a “conditional” scenario in which a portion of the receiver’s side information is also accessible to the eavesdropper. We investigate the maximum amount of classical information that can be encoded by each of the senders. We derive a single-letter characterization for the achievable rate region in an asymptotic limit of infinitely many copies and vanishingly small error.
This study shows that secret information can be shared or passed from a sender to a receiver even if not encoded in a secret message. In the protocol designed in this study, no parts of the original secret information ever travel via communication channels between the source and the destination, no encoding/decoding key is ever used. The two communicating partners, Alice and Bob, are endowed with coherent qubits that can be read and set and keep their quantum values over time. Additionally, there exists a central authority that is capable of identifying Alice and Bob to share with each half of entangled qubit pairs. The central authority also performs entanglement swapping. Our protocol relies on the assumption that public information can be protected, an assumption present in all cryptographic protocols. Also any classical communication channel need not be authenticated. As each piece of secret information has a distinct public encoding, the protocol is equivalent to a one-time pad protocol.