Learning the binary exponentiation (exponentiation by squaring) method for efficient power calculation.
Calculating large powers of a number, such as `a^b`, can be computationally expensive if done naively by multiplying `a` by itself `b` times. This approach has a time complexity of O(b). Fast exponentiation, also known as binary exponentiation or exponentiation by squaring, is a much more efficient algorithm that achieves this in O(log b) time. The core idea is to leverage the binary representation of the exponent `b`. The algorithm relies on two key observations. First, if `b` is an even number, then `a^b = a^(2 * b/2) = (a^2)^(b/2)`. Second, if `b` is an odd number, then `a^b = a * a^(b-1)`. We can combine these ideas into a simple recursive or iterative algorithm. In the iterative approach, we initialize a result to 1. We traverse the bits of the exponent `b` from right to left. If the current bit is 1, we multiply our result by the current value of `a`. In every step, we square `a`. This process effectively computes the power by combining the powers of two that sum up to `b`. This technique is not only faster but also crucial when combined with modular arithmetic to compute `(a^b) mod m` for large `a`, `b`, and `m`, as it keeps all intermediate results small, preventing overflow.