Elliptic curve cryptography (ECC) is widely acknowledged as a method for implementing public key cryptography on devices with limited resources thanks to its use of small keys. A crucial and complex operation in ECC calculations is scalar point multiplication. To improve its execution time and computational complexity in low-power devices, such as embedded systems, several algorithms have been suggested for scalar point multiplication, with each featuring different techniques and mathematical formulas. In this research, we focused on combining some techniques to produce a scalar point multiplication algorithm for elliptic curves over finite fields. The employed methodology involved mathematical analysis to investigate commonly used point multiplication methods. The aim was to propose an efficient algorithm that combined the best computational techniques, resulting in lower computational requirements. The findings show that the proposed method can overcome certain implementation issues found in other multiplication algorithms. In certain scenarios, the proposed method offers a more efficient approach by reducing the number of point doubling and point addition operations on elliptic curves using the inverse of the targeted point.

This research highlights the importance of error-correcting codes in ensuring secure and efficient data transmission over noisy channels. This paper aims to address the issue of limited information regarding the factors that contribute to the effectiveness of the implementation of Convolutional Codes in GPS systems. The research problem revolves around the insufficiency of scholarly sources elucidating the rationale behind the utilization of convolutional codes, specifically in GPS systems, rather than others. Through an in-depth analysis of these factors, this study strives to achieve a comprehensive understanding of the application of Convolutional Codes in GPS. To tackle this research problem, a novel methodology involving comparative analysis is employed. The coding techniques commonly used in satellite communication systems (such as BCH, LDPC, and turbo codes) are carefully examined and compared to the advantages and suitability of Convolutional codes and the Viterbi algorithm for GPS systems. Each coding technique is evaluated based on factors such as error detection and correction capabilities, bandwidth efficiency, computational complexity, and resilience to noise. The key findings of this study shed light on the unique advantages offered by Convolutional codes and the Viterbi algorithm for GPS systems. The analysis reveals that these coding techniques exhibit superior error detection and correction capabilities, efficient bandwidth utilization, and the ability to withstand noise in the GPS communication channel. The results also highlight the computational complexity associated with these techniques, providing valuable insights for the implementation of convolutional codes in GPS systems. Overall, this article contributes to the existing knowledge by providing a comprehensive understanding of the reasons behind the suitability of convolutional codes for GPS systems. The findings of this study serve as a resource for researchers, engineers, and practitioners in the field of satellite communication, aiding in the comprehensive understanding, advancement, and optimization of GPS system designs.

ПОВЫШЕНИЕ ПРОИЗВОДИТЕЛЬНОСТИ И СНИЖЕНИЕ ПОТРЕБЛЕНИЯ РЕСУРСОВ ДЛЯ ЭЛЕКТРОННОЙ ПОДПИСИ НА 8-БИТНОМ МИКРОКОНТРОЛЛЕРЕ

Implementing elliptic curve point multiplication (ECPM), the core operation in elliptic curve cryptography (ECC), on microcontrollers is challenging due to limited processing power and memory, restricting its application in environments like embedded systems and autonomous vehicles, where security and performance are essential. This study presents an optimized ECPM approach for the Arduino Atmega 2560, leveraging low-level C and assembly programming for enhanced hardware resource control. By incorporating the cyclic group property of elliptic curve points, the additive inverse property in group theory, and the elimination of a curve-shaping parameter (specifically the parameter a, which influences point arithmetic in twisted Edwards curves) during doubling operations in the Extended Twisted Edwards Coordinates, along with the windowed method on Edwards25519, the approach achieves 14% to 71% reductions in execution time and cycle count, a 15% decrease in program storage space (flash memory) usage, and 77% to 80% lower dynamic memory (SRAM) consumption compared to previous studies. These results highlight the method’s efficiency, providing an optimal balance of speed, memory utilization, and security for constrained microcontroller applications.