# Modular Reduction ## Introduction To calculate modular addition, $$(A+B) \mod q$$, we can simply use a piecewise function: $$ (A + B) \mod q = \begin{cases} A + B &\text{if }A + B < q \\ A + B − q &\text{if }A + B ≥ q \end{cases} $$ While modular adder is simple and easy to implement, modular multipliers are trickier. The standard algorithm for modular multiplication uses [trial division](https://en.wikipedia.org/wiki/Trial_division), which is inefficient, not scalable, and difficult to implement in hardware architecture. The most popular workaround uses [Barrett](https://app.gitbook.com/o/vlMwaGV8Ukn8S4gDaGYp/s/rwz1ZAZJtK5FHz4Y1esA/~/changes/55/primitives/modular-multiplication/barrett-reduction) or [Montgomery](https://app.gitbook.com/o/vlMwaGV8Ukn8S4gDaGYp/s/rwz1ZAZJtK5FHz4Y1esA/~/changes/55/primitives/modular-multiplication/montgomery-reduction) reduction based multiplication algorithm. There is also some specialized multiplication algorithms like Karatsuba-Barrett Algorithm etc.