Chapter 9 **Quantum Calculus with the Notion δ±-Periodicity and Its Applications 183**

Neslihan Nesliye Pelen, Ayşe Feza Güvenilir and Billur Kaymakçalan

Preface

One of the most effective ways to ensure confidentiality and data integrity during transmis‐ sion is cryptography. The purpose of the cryptographic system is to provide key distribu‐ tion, authentication, legitimate user authorization, and encryption. Today, key distribution is one of the most important problems of cryptography. This problem can be solved with the help of the following schemes: classical information-theoretic scheme, classical public-key cryptographic scheme, classical computationally secure symmetric-key cryptographic

QKD includes the number of protocols (such as BB84, SARG04, E91, B92, six-state protocol, Goldenberg-Vaidman protocol, and Koashi-Imoto protocol), and the main task of these is en‐ cryption key generation and distribution between the two users connecting through quantum and classical channels. The main advantages of all QKD protocols are as follows: (1) these protocols always allow eavesdropping to be detected because eavesdropper's connection brings much more error level (compared with natural error level) to the quantum channel. The laws of quantum mechanics allow eavesdropping to be detected, and the dependence between error level and intercepted information to be set. This allows to apply the privacy amplifica‐ tion procedure that decreases the quantity of information about the key, which can be inter‐ cepted by eavesdropper. Thus, QKD protocols have an unconditional (information-theoretic) security. (2) The information-theoretic security of QKD allows using an absolutely secret key for further encryption using the well-known classical symmetrical algorithms. Thus, the entire information security level increases. It is also possible to synthesize QKD protocols with the Vernam cipher (one-time pad), which, in complex with unconditionally secured authenticated

Besides, there are also disadvantages of QKD protocols that are as follows: (1) a system based only on QKD protocols cannot serve as a complete solution for key distribution in open networks because additional tools for authentication are needed. (2) The limitation of quantum channel length is caused by the fact that there is no possibility of amplification without quantum properties being lost. However, the technology of quantum repeaters could overcome this limitation in the near future. (3) There is a need for using the weak-co‐ herent pulses instead of single-photon pulses. This decreases the efficiency of protocol in practice. However, this technological limitation might be defeated in the near future. (4) The data-transfer rate decreases rapidly with an increase in the quantum channel length. (5) Pho‐ ton registration problem leads to a key rate of decrease in practice. (6) Photon depolarization in the quantum channel leads to errors during data transfer. Now, the typical error level equals a few percent, which is much greater than the error level in classical information and communication systems. (7) Difficulty in the practical realization of QKD protocols for *d*-lev‐ el (multilevel) quantum systems. (8) The high price of commercial QKD systems (more than

scheme, trusted courier key distribution, and quantum key distribution (QKD).

schemes, gives a totally secured system for transferring the information.

\$100,000 for two subscriber systems).
