A Novel Computational Technique for Fast Modular Inversion

The computation of a multiplicative inverse modulo an integer is a fundamental operation in modular arithmetic and has important applications in number theory, cryptography, and computer science. The Extended Euclidean Algorithm (EEA) is one of the most widely used general methods for computing modular inverses. Although mathematically efficient, its conventional procedure may require a relatively large number of computational steps, particularly when the operands become large.This paper presents a new approach to the computation of multiplicative inverses modulo an integer. The work begins with a simple and efficient manual method designed primarily for small integers, in which a set of computational techniques is used to obtain the modular inverse with fewer and simpler operations than conventional approaches. During the development and analysis of this method, a fundamental technique was identified that can be generalized beyond the original manual procedure.Based on this technique, a new algorithm for modular inversion is derived. The resulting algorithm has a structure related to the Extended Euclidean Algorithm, but follows a substantially different computational progression and requires considerably fewer principal computational steps to reach the modular inverse. In particular, the proposed approach eliminates a significant portion of the repetitive operations encountered in the conventional Extended Euclidean procedure through the introduction of a simple decision condition.