ОЦЕНКИ МОЩНОСТИ КЛАССОВ ОТОБРАЖЕНИЙ, ПРИМЕНЯЕМЫХ В ПРОТОКОЛАХ КВАНТОВОГО РАСПРЕДЕЛЕНИЯ КЛЮЧЕЙ

Abstract

Introduction: In the near future, quantum cryptography will play an important role in maintaining a sufficient level of information security of modern telecommunication networks in the conditions of a quantum challenge, which refers to the emergence of quantum computers that will be able to effectively solve the mathematical problems on which, for example, modern key distribution systems are based. Now quantum cryptography is actively used by commercial and government agencies around the world and, in particular, in the Russian Federation. At the same time, a large amount of research is being carried out in the field of development and implementation of quantum key distribution systems, as the main part of quantum cryptography. In this regard, the task of developing new and refining existing protocols for quantum key distribution, as well as studying various mathematical and physical objects that are associated with these protocols, is an urgent task. In particular, one of the stages of the classical BB84 protocol implement ed in a noisy quantum channel is associated with the problem of studying correlation immune and stable mappings, part of which is the problem of estimating their number, which has not been completely solved. Purpose: to find mathematical expressions for exact and asymptotic estimates of the cardinalities of classes of (n,m,k) stable and correlation immune of order k boolean mappings. Results: The best currently asymptotic upper and lower bounds for the number of such classes of mappings with the number of outputs greater than or equal to five are obtained. Recurrent relations were also proved, which allow one to find the exact distribution of the cardinalities of classes of similar mappings for the case of small numbers n and m. Practical relevance: the results obtained allow us to estimate the probability that with a random choice of mapping to enhance secrecy at the stage of secondary processing of the BB84 protocol, the situation will be neutralized when the adversary has access to k photons sent over a communication chan nel of his choice.