The basic branch of both discrete and continuous mathematics is Set theory, the study of which is a difficult task. Designed and developed in the Maple computer algebra system, the Maplets package–the set theory application–enables to provide assistance both in teaching (by demonstrating the performance of operations on sets, testing knowledge and skills in using operations), and in studying set theory and solving the simplest combinatorial tasks (can be used as a simulator for students). By using interactive worksheets and animated images in Maple, students are given the opportunity for numerous experiments that will contribute to the development of their math education as well as computational skills. The application will form a sustainable need for the use of computer systems by students in the study of various branches of mathematics.

One of the most important components of intelligent transportation systems (ITS) is the automotive self-organizing VANET network (vehicular ad hoc network). Its nodes are vehicles with specialized onboard units (OBU) installed on them. Such a network can be subject to various attacks. To reduce the effectiveness of a number of attacks on the VANET, it is advisable to use authentication protocols. Well-known authentication protocols support a security policy with full trust in roadside unit (RSU) base stations. The disadvantage of these authentication protocols is the ability of the RSU to track the route of the vehicle. This leads to a violation of the privacy and anonymity of the vehicle’s owner. To eliminate this drawback, the article proposes an adaptive authentication protocol. An advantage of this protocol is the provision of high imitation resistance without using symmetric and asymmetric ciphers. This result has been achieved by using a zero-knowledge authentication protocol. A scheme for adapting the protocol parameters depending on the intensity of the user’s traffic has been developed for the proposed protocol. The scientific novelty of this solution is to reduce time spent on authentication without changing the protocol execution algorithm by reducing the number of modular exponentiation operations when calculating true and “distorted” digests of the prover and verifying the correctness of responses, as well as by reducing the number of responses. Authentication, as before, takes place in one round without changing the bit depth of the modulus used in the protocol. To evaluate the effectiveness of the adaptive authentication protocol, the VANET model was implemented using NS-2. The obtained research results have shown that the adaptation of the authentication protocol in conditions of increased density of vehicles on the road makes it possible to increase the volume of data exchange between OBU and RSU by reducing the level of confidentiality. In addition, a mechanism for verifying the authority of the vehicle’s owner for provided services has been developed. As a result of the implementation of this mechanism, vehicle registration sites (VRS) calculate the public key of the vehicle without using encryption and provide necessary services to the owner.

The paper studies the problem of determining the lower boundary of number correction speed in computing systems operating in the basis of non-positional arithmetic of residual classes. The topicality of the problem is necessitated by the search of methods allowing reducing the time for digital processing of signals in non-positional neuroprocessors. The study considers several variants of error correction with a single and multiple control bases of the residue system. The calculations were performed that allowed determining the time necessary for zeroing operation. The method of paired zeroing of numbers in a system of residual classes was modified. It was demonstrated that the suggested solutions allow appreciably reducing the time consumption by digital processing of signals in neuroprocessors destined for operations of summation and multiplication.

Mathematical induction is one of the main methods of the proof that students study and use it in their studies in higher education. This important method, in addition to mathematics, is widely used in computer science and a number of other related disciplines. However, even if the principles of proof using the method of induction are taught, understood and mastered by students well, many students have great difficulty with algebraic transformations necessary for proving, for example, finding the inductive step. The use of the Maple computer algebra system allows students to overcome this obstacle and, more attention and effort to pay to understanding the concept of proofs using the method of mathematical induction. In addition, students can prove more complex algebraic statements, and their activities can be of a pronounced research nature.

The paper outlines ways in which the Maple computer algebra system can be used to master one of the branches of discrete mathematics – set theory. Computer algebra systems (CAS) expand the possibilities of creating, applying and using mathematical models on a daily basis by engineers, researchers, and foster the improved training of students in a wide range of disciplines. In addition, computer technology can help implement such methods as genetic approach and the use of various techniques to representing target objects in education. The most suitable for teaching discrete mathematics and mathematics at large are computer algebra systems such as Maple and Mathematica. The paper provides a Maple framework designed for the initial stages of mastering discrete mathematics within university curriculum, in particular, for working with the basic concepts and laws of set theory.