[phobos] Issue6244, implementing 'powmod'

[Andrei Alexandrescu]

A possible solution is to implement mul128 using primitives for 32- and 
64-bit numbers. A possible signature would be:

void mul128(ulong a, ulong b, out ulong lo, out ulong hi);

The technique is to do the multiplication "by hand" as if you had two 
numbers having two digits each. You obtain a 4-digit number.

Consider the two-digit decimal numbers ab and cd, for example 43 and 65. 
Then the multiplication "by hand" is:

(10a + b) * (10c + d) = 100ac + 10(ad + bc) + bd

Since a, b, c, d are single digits it follows we only need 1-digit 
multiplication and addition with carry. Now of course our base is not 10 
but 2^^32, so we multiply two numbers, each having two 32-bit "digits" 
and we get 128 bits worth of result.

mul128() would be a generally useful function to put in druntime. I 
think someone on the general forum has implemented it already, you may 
want to ask.


Andrei

On 10/2/17 12:48 PM, Alex Jercaianu via phobos wrote:
> Hello,
> 
> I am trying to implement the 'powmod' functionality as described here [1].
> 
> The issue that I am having is that the algorithm uses multiplications 
> which can cause overflow.
> If the base is 32 bits, then I can use 64 bit variables to handle the 
> result of multiplications, however the problem arises if the base is 64 
> bit.
> 
> The pseudocode would look the following:
> 
> while (exponent > 0)
> {
>      if (exponent & 1)
>      {
>          result = mulmod(result, base, modulus);
>      }
> 
>      base = mulmod(base, base, modulus);
>      exponent >>= 1;
> }
> 
> return result;
> 
> The problem that I am facing is with the 'mulmod' function which should 
> do multiplication and modulo of the result without overflow problems.
> 
> Do you think that it would be a good idea to limit the base to 32 bits?
> Or does D have any facility similar to 'mulmod'?
> 
> Thanks,
> Alex
> 
> [1] - 
> https://en.wikipedia.org/wiki/Modular_exponentiation#Right-to-left_binary_method 
> 
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