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sslotin merged 1 commit into from
Nov 12, 2022
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typo #191

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3 changes: 2 additions & 1 deletion content/english/hpc/number-theory/montgomery.md
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---
title: Montgomery Multiplication
weight: 4
published: true
---

Unsurprisingly, a large fraction of computation in [modular arithmetic](../modular) is often spent on calculating the modulo operation, which is as slow as [general integer division](/hpc/arithmetic/division/) and typically takes 15-20 cycles, depending on the operand size.
Expand Down Expand Up @@ -287,6 +288,6 @@ int inverse(int _a) {
}
```

While vanilla binary exponentiation with a compiler-generated fast modulo trick requires ~170ns per `inverse` call, this implementation takes ~166ns, going down to ~158s we omit `transform` and `reduce` (a reasonable use case is for `inverse` to be used as a subprocedure in a bigger modular computation). This is a small improvement, but Montgomery multiplication becomes much more advantageous for SIMD applications and larger data types.
While vanilla binary exponentiation with a compiler-generated fast modulo trick requires ~170ns per `inverse` call, this implementation takes ~166ns, going down to ~158ns we omit `transform` and `reduce` (a reasonable use case is for `inverse` to be used as a subprocedure in a bigger modular computation). This is a small improvement, but Montgomery multiplication becomes much more advantageous for SIMD applications and larger data types.

**Exercise.** Implement efficient *modular* [matix multiplication](/hpc/algorithms/matmul).