Airy zeta function

In modular arithmetic, Barrett reduction is a reduction algorithm introduced in 1986 by P.D. Barrett. A naive way of computing

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

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

Naive idea

Let $m=1/n$ be the inverse of $n$ as a floating point number. Then

$a\,{\bmod {\,}}n=a-\lfloor am\rfloor n$ where $\lfloor x\rfloor$ denotes the floor function,

assuming m was computed with sufficient accuracy.

Barrett algorithm

Barrett algorithm is a fixed-point analog which expresses everything in terms of integers. Let k be the smallest integer such that $2^{k}>n$. Think of n as representing the fixed-point number $n2^{-k}$. We precompute m such that $m=\lfloor 4^{k}/n\rfloor$. Then m represents the fixed-point number $m2^{-k}\approx (n2^{-k})^{-1}$.

Let $q=\left\lfloor {\frac {ma}{4^{k}}}\right\rfloor$ and $r=a-qn$.

Clearly $a\equiv r{\pmod {n}}$, and if $a<n^{2}$ then $r<2n$. So

$a\,{\bmod {\,}}n={\begin{cases}r&{\text{if }}r<n\\r-n&{\text{otherwise}}\end{cases}}$

Barrett algorithm for polynomials

It is also possible to use Barrett algorithm for polynomial division, by reversing polynomials and using X-adic arithmetic.Template:Clarify

See also

Montgomery reduction is another similar algorithm.

References

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Sources

• Chapter 14 of Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. Handbook of Applied Cryptography. CRC Press, 1996. ISBN 0-8493-8523-7.
• Bosselaers, et al., "Comparison of Three Modular Reduction Functions," Advances in Cryptology-Crypto'93, 1993. [1]
• W. Hasenplaugh, G. Gaubatz, V. Gopal, "Fast Modular Reduction", ARITH 18