Barrett reduction

In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett.

A naive way of computing

${\displaystyle c=a\,{\bmod {\,}}n\,}$

would be to use a fast division algorithm. Barrett reduction is an algorithm designed to optimize this operation assuming ${\displaystyle n}$ is constant, and ${\displaystyle a<n^{2}}$, replacing divisions by multiplications.

Historically, for values ${\displaystyle a,b<n}$, one computed ${\displaystyle ab\,{\bmod {\,}}n\,}$ by applying Barrett reduction to the full product ${\displaystyle ab}$. Recently, it was shown that the full product is unnecessary if we can perform precomputation on one of the operands.