power and modulo on the fly for big numbers

Question

I raise some basis b to the power p and take the modulo m of that.

Let\'s assume b=55170 or 55172 and m=3043839241 (which happens to be the square of 55171). The li

Answer 1

Ok fellows, it took me some time, and finally destroyed a long but never proven assumption, which was, that if you multiply two 64-bit-positive integral values (aka: Longs, and practically only 63-bit, after all), you could overrun, and get negative values, but not get an overrun to reach positive (but wrong) values again.

So I had tried to put a guard into my code, to calculate my value with BigInt, it too big, but the guard was insufficient, because I tested for res < 0. res < pm1 && res < pm2 isn't sufficient too.

To increase the speed I used a mutable.HashMap, and now the code looks like this:

val MVL : Long = Integer.MAX_VALUE
var modPow = new scala.collection.mutable.HashMap [(Long, Int, Long), Long ] () 

def powMod (b: Long, pot: Int, mod: Long) : Long = { 
      if (pot == 1) b % mod else modPow.getOrElseUpdate ((b, pot, mod), {
    val pot2= pot/2
    val pm1 = powMod (b, pot2, mod)             
    val pm2 = powMod (b, pot-pot2, mod)
    val res = (pm1 * pm2) 
    // avoid Long-overrun
    if (pm1 < MVL && pm2 < MVL)
        res % mod else {
            val f1: BigInt = pm1
            val f2: BigInt = pm2
            val erg = (f1 * f2) % mod
            erg.longValue 
        }
      })
}

You might ask yourself, whether the Long-declared MVL is really needed, or whether a

if (pm1 < Integer.MAX_VALUE && ...

would have worked too. No. It wouldn't. :) Another trap to avoid. :)

Finally it is pretty fast and correct and I learned two lessons about overruns and MAX_VALUE - comparision.

Answer 2

Not familiar with Scala, but...

def powMod (b: Long, pot: Int, mod: Long) : Long = {  
      if (pot == 1) b % mod else { 
          val pot2 = pot/2 
          val pm1 = powMod (b, pot, mod)              
          val pm2 = powMod (b, pot-pot2, mod)            
          (pm1 * pm2) % mod  
      }  
} 

Did you mean

          val pm1 = powMod (b, pot2, mod) 

Notice the pot2 instead of pot.

Strangely, it seems that this should loop forever/overflow the stack, but who knows what Scala is doing.

Answer 3

I think the answer is here:

scala> math.sqrt(Long.MaxValue).toLong < 3043839241L
res9: Boolean = true

That means you can have a long overflow even for numbers which are less than that particular module value. Let's try to catch it:

scala> def powMod (b: Long, pot: Int, mod: Long) : Long = {
     |       if (pot == 1) b % mod else {
     |           val pot2 = pot/2
     |           val pm1 = powMod (b, pot2, mod)
     |           val pm2 = powMod (b, pot-pot2, mod)
     |           val partial = ((pm1 % mod) * (pm2 % mod)).ensuring(res =>
     |             res > pm1 % mod && res > pm2 % mod, "Long overflow multiplying "+pm1+" by "+pm2)
     |           partial % mod
     |       }
     | }
powMod: (b: Long,pot: Int,mod: Long)Long

scala> powMod (55170, 5606, 3043839241L)
java.lang.AssertionError: assertion failed: Long overflow multiplying 3042625480 by 3042625480

There you have it.