I have to write a program to calculate a**b % c
where b
and c
are both very large numbers. If I just use a**b % c
, it's really slow. Then I found that the built-in function pow()
can do this really fast by calling pow(a, b, c)
.
I'm curious to know how does Python implement this? Or where could I find the source code file that implement this function?
Power Python: pow() Method. Python includes a built-in function that can be used to calculate powers: pow(). pow() accepts three parameters: a base number, an exponent to which the base is raised, and a modulo operator.
Given two numbers base and exponent, pow() function finds x raised to the power of y i.e. xy. Basically in C exponent value is calculated using the pow() function. pow() is function to get the power of a number, but we have to use #include<math.
The function pow() is used to calculate the power raised to the base value. It takes two arguments. It returns the power raised to the base value. It is declared in “math.
If a
, b
and c
are integers, the implementation can be made more efficient by binary exponentiation and reducing modulo c
in each step, including the first one (i.e. reducing a
modulo c
before you even start). This is what the implementation of long_pow()
does indeed. The function has over two hundred lines of code, as it has to deal with reference counting, and it handles negative exponents and a whole bunch of special cases.
At its core, the idea of the algorithm is rather simple, though. Let's say we want to compute a ** b
for positive integers a
and b
, and b
has the binary digits b_i
. Then we can write b
as
b = b_0 + b1 * 2 + b2 * 2**2 + ... + b_k ** 2**k
ans a ** b
as
a ** b = a**b0 * (a**2)**b1 * (a**2**2)**b2 * ... * (a**2**k)**b_k
Each factor in this product is of the form (a**2**i)**b_i
. If b_i
is zero, we can simply omit the factor. If b_i
is 1, the factor is equal to a**2**i
, and these powers can be computed for all i
by repeatedly squaring a
. Overall, we need to square and multiply k
times, where k
is the number of binary digits of b
.
As mentioned above, for pow(a, b, c)
we can reduce modulo c
in each step, both after squaring and after multiplying.
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