Fastest way to calculate a 128-bit integer modulo a 64-bit integer

Question

I have a 128-bit unsigned integer A and a 64-bit unsigned integer B. What\'s the fastest way to calculate A % B - that is the (64-bit) remainder from dividing A

Answer 1

The solution depends on what exactly you are trying to solve.

E.g. if you are doing arithmetic in a ring modulo a 64-bit integer then using Montgomerys reduction is very efficient. Of course this assumes that you the same modulus many times and that it pays off to convert the elements of the ring into a special representation.

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To give just a very rough estimate on the speed of this Montgomerys reduction: I have an old benchmark that performs a modular exponentiation with 64-bit modulus and exponent in 1600 ns on a 2.4Ghz Core 2. This exponentiation does about 96 modular multiplications (and modular reductions) and hence needs about 40 cycles per modular multiplication.