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arXiv:quant-ph/0108133

On Classical and Quantum Cryptography

I. V. Volovich, Ya.I. Volovich

Published Aug 29, 2001• 12 citations
arXivHTMLPDF

Abstract

Lectures on classical and quantum cryptography. Contents: Private key cryptosystems. Elements of number theory. Public key cryptography and RSA cryptosystem. Shannon`s entropy and mutual information. Entropic uncertainty relations. The no cloning theorem. The BB84 quantum cryptographic protocol. Security proofs. Bell`s theorem. The EPRBE quantum cryptographic protocol.

DOI
10.48550/arxiv.quant-ph/0108133

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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).
This final chapter concerns cryptography , the principle of securing information against access or tampering by third parties. Classical cryptography refers to the manipulation of classical bits for this purpose, while quantum cryptography can be viewed as doing the same with qubits . I describe these two approaches in the same chapter, as in my view the field of cryptography should be understood as a whole and appreciated within such a broader framework, as opposed to focusing on the specific applications offered by the quantum approach. I, thus, begin by introducing the notions of message encryption , message decryption , and code breaking , the action of retrieving the message information contents without knowledge of the code's secret algorithm or secret key . I then consider the basic algorithms to achieve encryption and decryption with binary numbers, which leads to the early IBM concept of the Lucifer cryptosystem , which is the ancestor of the first data encryption standard (DES). The principle of double-key encryption , which alleviates the problem of key exchange , is first considered as an elegant solution but it is unsafe against code-breaking. Then the revolutionary principles of cryptography without key exchange and public-key cryptography (PKC) are considered, the latter also being known as RSA . The PKC–RSA cryptosystem is based on the extreme difficulty of factorizing large numbers . This is the reason for the description made earlier in Chapter 20 concerning Shor's factorization algorithm .
It is well known that Shor's quantum algorithm for integer factorization can break down the RSA public-key cryptosystem, which is widely used in many cryptographic applications. Thus, public-key cryptosystems in the quantum computational setting are longed for cryptology. In order to define the security notions of public-key cryptosystems, we have to model the power of the sender, receiver, adversary and channel. While we may consider a setting where quantum computers are available only to adversaries, we generally discuss what are the right security notions for (quantum) public-key cryptosystems in the quantum computational setting. Moreover, we consider the security of quantum public-key cryptosystems known so far.
Quantum cryptography converts structural restrictions on quantum information into operational secrecy. This article develops the chain connecting nonorthogonal-state discrimination, trace distance and fidelity, the Helstrom bound, no-cloning, no-broadcasting, information-disturbance relations, conditional entropy, and privacy amplification. A linearity argument proves that no universal device can copy arbitrary pure states, while the mixed-state no-broadcasting theorem identifies pairwise commutativity as the exact boundary of exact broadcastability. These results explain why an eavesdropper cannot retain a perfect replica of every signal and wait for later basis disclosure. They are not, however, complete security proofs. Modern quantum key distribution must cover coherent attacks, quantum memory, public reconciliation leakage, imperfect sources and detectors, finite samples, authentication, and composition with later applications. The review connects state discrimination to adversarial guessing probability through conditional min-entropy and explains how entropic uncertainty and entanglement-based reductions transform observed statistics into a lower bound on extractable key. Approximate cloning, unambiguous discrimination, channel loss, side channels, and device-independent certification are examined as qualifications to elementary intuition. The central result is a hierarchy: nonorthogonality restricts distinguishability; data processing forbids perfect information amplification; parameter estimation bounds an adversary’s conditional states; and quantum-proof hashing yields a key close in trace distance to an ideal independent key. Cryptographic security therefore depends on quantum limitations embedded within a fully specified protocol and adversarial model.