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arXiv:quant-ph/0102016

A Talk on Quantum Cryptography, or How Alice Outwits Eve

Samuel J. Lomonaco

Published Feb 2, 2001• 0 citations
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Abstract

Alice and Bob wish to communicate without the archvillainess Eve eavesdropping on their conversation. Alice, decides to take two college courses, one in cryptography, the other in quantum mechanics. During the courses, she discovers she can use what she has just learned to devise a cryptographic communication system that automatically detects whether or not Eve is up to her villainous eavesdropping. Some of the topics discussed are Heisenberg's Uncertainty Principle, the Vernam cipher, the BB84 and B92 cryptographic protocols. The talk ends with a discussion of some of Eve's possible eavesdropping strategies, opaque eavesdropping, translucent eavesdropping, and translucent eavesdropping with entanglement.

DOI
10.48550/arxiv.quant-ph/0102016

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Alice and Bob wish to communicate without the archvillainess Eve eavesdropping on their conversation. Alice, decides to take two college courses, one in cryptography, the other in quantum mechanics. During the courses, she discovers she can use what she has just learned to devise a cryptographic communication system that automatically detects whether or not Eve is up to her villainous eavesdropping. Some of the topics discussed are Heisenberg's Uncertainty Principle, the Vernam cipher, the BB84 and B92 cryptographic protocols. The talk ends with a discussion of some of Eve's possible eavesdropping strategies, opaque eavesdropping, translucent eavesdropping, and translucent eavesdropping with entanglement.
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).
The recent application of the principles of quantum mechanics to cryptography has led to a remarkable new dimension in secret communication. As a result of these new developments, it is now possible to construct cryptographic communication systems which detect unauthorized eavesdropping should it occur, and which give a guarantee of no eavesdropping should it not occur. CONTENTS P3. Cryptographic systems before quantum cryptography P7. Preamble to quantum cryptography P10. The BB84 quantum cryptographic protocol without noise P16. The BB84 quantum cryptographic protocol with noise P19..The B92 quantum cryptographic protocol P21. EPR quantum cryptographic protocols P25. Other protocols P25. Eavesdropping stategies and counter measures P26. Conclusion P29. Appendix A. The no cloning theorem P30. Appendix B. Proof that an undetectable eavesdropper can obtain no information from the B92 protocol P31. Appendix C. Part of a Rosetta stone for quantum mechanics P44. References
Alice communicates with words drawn uniformly amongst {∣j⟩}j=1..n\{\ket{j}\}_{j=1..n}{∣j⟩}j=1..n​, the canonical orthonormal basis. Sometimes however Alice interleaves quantum decoys {∣j⟩+i∣k⟩2}\{\frac{\ket{j}+i\ket{k}}{\sqrt{2}}\}{2​∣j⟩+i∣k⟩​} between her messages. Such pairwise superpositions of possible words cannot be distinguished from the message words. Thus as malevolent Eve observes the quantum channel, she runs the risk of damaging the superpositions (by causing a collapse). At the receiving end honest Bob, whom we assume is warned of the quantum decoys' distribution, checks upon their integrity with a measurement. The present work establishes, in the case of individual attacks, the tradeoff between Eve's information gain (her chances, if a message word was sent, of guessing which) and the disturbance she induces (Bob's chances, if a quantum decoy was sent, to detect tampering). Besides secure channel protocols, quantum decoys seem a powerful primitive for constructing n-dimensional quantum cryptographic applications. Moreover the methods employed in this article should be of strong interest to anyone concerned with the old but fundamental problem of how much information may be gained about a system, versus how much this will disturb the system, in quantum mechanics. Keywords: d-level systems cryptography
Alice communicates with words drawn uniformly amongst {|j>}j=1…n, the canonical orthonormal basis. Sometimes however, Alice interleaves quantum decoys [Formula: see text] between her messages. Such pairwise superpositions of possible words cannot be distinguished from the message words. Thus as malevolent Eve observes the quantum channel, she runs the risk of damaging the superpositions (by causing a collapse). At the receiving end honest Bob, whom we assume is warned of the quantum decoys' distribution, checks upon their integrity with a measurement. The present work establishes, in the case of individual attacks, the tradeoff between Eve's information gain (her chances, if a message word was sent, of guessing which) and the disturbance she induces (Bob's chances, if a quantum decoy was sent, to detect tampering). Whilst not directly applicable to secure channel protocols, quantum decoys seem a powerful primitive for constructing other n-dimensional quantum cryptographic applications. Moreover, the methods employed in this article should be of strong interest to anyone concerned with the old but fundamental problem of how much information may be gained about a system, versus how much this will disturb the system, in quantum mechanics.
In quantum key distribution (QKD), Alice selects a sequence of characters from a finite alphabet and transmits the corresponding signals, each described by a quantum state, to Bob. Between Alice and Bob, Eve---with her own receiver and transmitter---can eavesdrop. Eve is assumed to know beforehand all the possible states from among which Alice chooses. The central question is: Can Eve gain significant information about the key without influencing what Bob receives in ways that Alice and Bob can detect? This formulation implies that many options are available to Eve. It is the purpose here to discuss one of these options, where extensive use is made of the Schrodinger equation with spatial variables. This procedure of Eve---which consists of letting the quantum state of Alice scatter from a quantum memory and then, using the information thus obtained, sending a suitably chosen quantum state to Bob---is discussed in detail. Furthermore, there are ways for Alice to defeat this procedure by an unusual choice of her quantum states. This counter-measure is also presented and discussed.