Arithmetic operations’ complexity analysis in modular arithmetic within quadratic range

Abstract

The authors study the features and effectiveness of modular arithmetic within a quadratic range, including the implementation of basic arithmetic operations on commodity computers. The research aim is to analyze the structure and features of performing various arithmetic operations in the modular arithmetic within quadratic range and compare their time complexity with similar operations in traditional positional systems. To meet this objective, the authors investigated the structure and characteristics of modular arithmetic within a quadratic range. They implemented arithmetic operations in Python, performed experiments, and assessed the time complexity of operations. The research methods include a theoretical study of the basis of the modular arithmetic within quadratic range, the creation of algorithms for performing operations in Python, experimental testing and analysis of the results. The research result is the creation of an algorithm for performing arithmetic operations in the modular arithmetic within quadratic range, revealing a significant performance gain compared to positional number systems, confirmed experimentally. Therefore, this study proves that modular arithmetic can improve task productivity that requires speed and resources.