Abstract: Modular reduction and modular multiplication for large numbers are required operations in public key cryptography. Moreover, efficient execution of these two operations is important to achieve high performance levels in cryptographic engines and processes. The present invention uses multiplication and addition instead of using division and subtraction to perform modular arithmetic. The present invention also achieves some of its advantages through processing which begins with the high order bits coupled with judicious observations pertaining to circumstances under which carry output signals from addition operations are generated. These carry output signals are used to provide corrections which thus enable the use of the higher order bits and the efficiencies that such use engenders. Additionally, the method of the present invention recognizes special circumstances that are employed to speed up processing.