Montgomery Arithmetic

Related Concepts

Modular Arithmetic; Modular Exponentiation

Definition

Suppose a machine performs arithmetic on words of w bits. Let a, b, and n be cryptographically sized integers represented using s such words. Then the Montgomery modular product of a and b modulo n is $ab{r}^{-1}$ (mod n) where $r = {2}^{sw}$. This is computed at a word level using a particularly straightforward and efficient algorithm. Compared with the normal “school book” method, for each word of the multiplier the reduction modulo n is performed by adding rather than subtracting a multiple of n, only a single digit is used to decide on this multiple, and the accumulating product is shifted down rather than up.

Background

The modular reduction u (mod n) is typically computed on a word-based machine by repeatedly taking several leading digits from u and n, obtaining the leading digit of their quotient, and using that multiple of n to reduce u. This takes a number of clock cycles on a general processor, and...