New Method of Computing Multiplicative Inverse

Abstract

In past, as people mentioned multiplicative inverse (inverse for short) in Numerical Theorem, they always think of Euler's Generation of Fermat's Little Theorem, Extended Euclidean Algorithm and reverse deduct from Euclid's algorithm. However, most people still use reverse deductive expression from GCD (Greatest Common Divisor) to the two original integers. The main reasons are (1) Euler's Generation of Fermat's Little Theorem adopting exponent expression is not good for practicing. (2) Since Extend Euclidean Algorithm tries to keep every division step related to the original two integers, it makes the concise Euclid's Algorithm complicated. Although reverse deductive expression from GCD to the two original integers is not easy to get the final inverse, there is no good method to support it. To improve the weakness, it could be solved the inverse into two stages. First, keep the concise property of Euclid Algorithm in intuitive way by only using one stack to record all quotients in every division step. Second, deduce an efficient algorithm and easily use it to get the final inverse.