Airy zeta function

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

$c=amodn.$

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

$amodn=a-\lfloor am\rfloor n$ 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 $n{2}^{-k}$. We precompute m such that $m=\lfloor 4^k/n\rfloor$. Then m represents the fixed-point number $m{2}^{-k}\approx (n{2}^{-k}{)}^{-1}$.

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

Clearly $a\equiv r(\mathrm{mod}n)$, and if $a<n^2$ then $r<2n$. So

$amodn=\{\begin{array}{ll}r& \text{if }r<n\\ r-n& \text{otherwise}\end{array}$

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