On Calculating Multiplicative Inverses Modulo $2^{m}$

A need to calculate modular multiplicative inverses that involve a power of 2 arises in various number theory and cryptography applications. Known algorithms that perform these calculations involve a number of steps, linearly proportional to the number of characters in a given number. The character can be a single bit or a binary word of any standard size. This letter presents an efficient methodology for a character-based computation of modular multiplicative inverses, efficient at any given base, reducing these calculations to a logarithmic number of steps. Hence, the amount of calculations involved and the computational resources are dramatically decreased from an O(m) complexity to a O(logm) complexity, where m denotes the number of characters in the operand.