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arXiv:1210.8251

Cryptography based on operator theory (I): quantum no-key protocols

Yang Li, Min Liang

Published Oct 31, 2012• 1 citation
arXivHTMLPDF

Abstract

We study cryptography based on operator theory, and propose quantum no-key (QNK) protocols from the perspective of operator theory, then present a framework of QNK protocols. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Then we defined the information-theoretical security of QNK protocols and the security of identification keys. Two kinds of QNK protocols are also proposed. The first scheme is constructed based on unitary transformation, and the other is constructed based on two multiplicative commutative sets.

DOI
10.48550/arxiv.1210.8251

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We study cryptography based on operator theory, and propose quantum no-key (QNK) protocols from the perspective of operator theory, then present a framework of QNK protocols. The framework is expressed in two forms: trace-preserving quantum operators and natural presentations. Then we defined the information-theoretical security of QNK protocols and the security of identification keys. Two kinds of QNK protocols are also proposed. The first scheme is constructed based on unitary transformation, and the other is constructed based on two multiplicative commutative sets.
Quantum cryptography promises a paradigm shift in secure communications by leveraging the principles of quantum mechanics. This article explores mathematical frameworks central to quantum cryptographic protocols, such as linear algebra, Hilbert spaces, quantum probability, and number theory. It focuses on the theoretical underpinnings of key distribution, quantum security proofs, and error correction. The paper also highlights how mathematical tools like entropic uncertainty relations, complexity theory, and operator algebras underpin advancements in quantum cryptographic systems. These mathematical approaches ensure not only the security but also the efficiency and scalability of next-generation quantum communication networks.
The emergence of quantum computer will threaten the security of existing public-key cryptosystems, including the Diffie Hellman key exchange protocol, encryption scheme and etc, and it makes the study of resistant quantum cryptography very urgent. This motivate us to design a new key exchange protocol and encryption scheme in this paper. Firstly, some acknowledged mathematical problems was introduced, such as ergodic matrix problem and tensor decomposition problem, the two problems have been proved to NPC hard. From the computational complexity prospective, NPC problems have been considered that there is no polynomial-time quantum algorithm to solve them. From the algebraic structures prospective, non-commutative cryptography has been considered to resist quantum. The matrix and tensor operator we adopted also satisfied with this non-commutative algebraic structures, so they can be used as candidate problems for resisting quantum from perspective of computational complexity theory and algebraic structures. Secondly, a new problem was constructed based on the introduced problems in this paper, then a key exchange protocol and a public key encryption scheme were proposed based on it. Finally the security analysis, efficiency, recommended parameters, performance evaluation and etc. were also been given. The two schemes has the following characteristics, provable security, security bits can be scalable, to achieve high efficiency, quantum resistance, and etc.
We present a bidirectional quantum key distribution protocol with minimal encoding operations derived from the use of only two “nonorthogonal” unitary transformations selected from two mutually unbiased unitary bases. Differently from many bidirectional protocols, these transformations are indistinguishable in principle for a single use. Along with its decoding procedure, it is unique compared to its “orthogonal encoding” predecessors. Given the nature of such protocols where key rates are usually dependent on two different types of error rates, we define a more relevant notion of security threshold for such protocols to allow for proper comparisons to be made. The current protocol outperforms its predecessor in terms of security as the amount of information an eavesdropper can glean is limited by the indistinguishability of the transformations. We further propose adaptations for a practical scenario and report on a proof of concept experimental scheme based on polarised photons from an attenuated pulsed laser for qubits, demonstrating the feasibility of such a protocol.
Methods of quantum mechanics promise information-theoretic security for various protocols in cryptography. However, impossibility of some cryptographic applications such as standard bit commitment, oblivious transfer, multiparty secure computations and ideal coin tossing in quantum regime leaves an obvious question on the completeness of quantum cryptography. Instead of using wide range of rules and techniques for a variety of cryptographic applications, we demonstrate here a unified structure for quantum cryptography based on quantum non-local correlations. The unified framework achieves same goals in information-theoretic way as classical cryptography does with computational hardness. To cover the broad range of cryptographic applications, we show that the framework (i) assures secrecy by providing encryption completely unintelligible to eavesdroppers, (ii) guarantees that input from distant parties is concealed unless they are willing to reveal, (iii) assures binding, (iv) allows splitting information between several parties securely and more generally, (v) evades both quantum and classical attacks from internal as well as external eavesdropping.