Scala algorithm: Compute modulo of an exponent without exponentiation

Published

Algorithm goal

Calculate $$(b^e) \% m$$ without using exponentiation. This could be useful where the exponent is very big, potentially overflowing.

Test cases in Scala

assert(exponentModulo(base = 2, exponent = 3, modulo = 3) == 2)
assert(exponentModulo(base = 4, exponent = 3, modulo = 3) == 1)
assert(
exponentModulo(base = 5, exponent = 10, modulo = 7) ==
exponentModuloRaw(base = 5, exponent = 10, modulo = 7)
)


Algorithm in Scala

7 lines of Scala (compatible versions 2.13 & 3.0), showing how concise Scala can be!

Explanation

Note that $$(a*b) \% m = ((a \% m) * (b \% m)) \% m$$. Proof: if $$a = xm + y$$ and $$b = wm + z$$, then , $$a * b = xwm^2 + (w + x)m + yz$$ and $$(a * b) \% m = yz \% m$$; $$((a \% m) * (b \% m)) \% m = (y * z) \% m = yz \% m$$.

Then, $$(b^e) \% m = (((b ^ {e - 1}) \% m) * (b \% m)) \% m$$. (this is © from www.scala-algorithms.com)

Scala concepts & Hints

1. foldLeft and foldRight

A 'fold' allows you to perform the equivalent of a for-loop, but with a lot less code.

def foldMutable[I, O](initialState: O)(items: List[I])(f: (O, I) => O): O =
items.foldLeft(initialState)(f)

2. Pattern Matching

Pattern matching in Scala lets you quickly identify what you are looking for in a data, and also extract it.

assert("Hello World".collect {
case character if Character.isUpperCase(character) => character.toLower
} == "hw")

3. Range

The (1 to n) syntax produces a "Range" which is a representation of a sequence of numbers.

assert((1 to 5).toString == "Range 1 to 5")

assert((1 to 5).reverse.toString() == "Range 5 to 1 by -1")

assert((1 to 5).toList == List(1, 2, 3, 4, 5))

4. Stack Safety

Stack safety is present where a function cannot crash due to overflowing the limit of number of recursive calls.

This function will work for n = 5, but will not work for n = 2000 (crash with java.lang.StackOverflowError) - however there is a way to fix it :-)

In Scala Algorithms, we try to write the algorithms in a stack-safe way, where possible, so that when you use the algorithms, they will not crash on large inputs. However, stack-safe implementations are often more complex, and in some cases, overly complex, for the task at hand.

def sum(from: Int, until: Int): Int =
if (from == until) until else from + sum(from + 1, until)

def thisWillSucceed: Int = sum(1, 5)

def thisWillFail: Int = sum(1, 300)


Scala Algorithms: The most comprehensive library of algorithms in standard pure-functional Scala

Study our 89 Scala Algorithms: 6 fully free, 89 published & 0 upcoming

Fully unit-tested, with explanations and relevant concepts; new algorithms published about once a week.

Explore the 21 most useful Scala concepts

To save you going through various tutorials, we cherry-picked the most useful Scala concepts in a consistent form.

Register now (free)

Register with GitHub

This gives you access to free test cases & hints for all our 89 published algorithms as well as our installation-free Scala IDE.

You can then choose to subscribe to "Scala Algorithms Unlimited", which gets you access to every current and upcoming algorithm solution. We use Stripe so your data is safe.

We will e-mail you at most once per week with new algorithms and relevant content.
Your data is protected and unsubscribing is straightforward.