I have been studying this link from wikipedia of modulo of a large number, Here is the pseudocode.
function modular_pow(base, exponent, modulus)
result := 1
while exponent > 0
if (exponent mod 2 == 1):
result := (result * base) mod modulus
exponent := exponent >> 1
base = (base * base) mod modulus
return result
I don't understand the explanation given in wiki.Why I have to check if exp%2 is even or odd. also why I am doing the three operations?
This algorithm is a combination of the Exponentiation by Squaring algorithm and modulo arithmetic.
To understand what's going on, first consider a situation when exponent
is a power of 2
. Then, assuming that exponent = 2 ^ k
, the result could be computed by squaring the result k
times, i.e.
res = (...((base ^ 2) ^2 ) ... ) ^2))
---------------------
k times
When exponent
is not a power of 2
, we need to make additional multiplications. It turns out that if we can divide exponent
by 2 without remainder, we can square the base, and divide the exponent. If, however, there is a remainder, we must additionally multiply the intermediate result by the value of the current base
.
What you see is the same exponentiation by squaring applied to modulo multiplication. The algorithm denotes integer division by two using the exponent >> 1
operation, which is identical to floor(exponent / 2)
.