Exponentiation by Squaring

• is a method used for computing the power of a number to a given exponent
• the method is based on the observation that if we square a number and halve the exponent, the result is the same as if we just raised the number to the original exponent. This means that we can compute the power of a number using fewer operations than if we just used repeated multiplication

General Algorithm

The exponentiation by squaring algorithm works as follows:

• If the exponent is negative, return 1/base
• If the exponent is 0, return 1
• If the exponent is 1, return the base
• If the exponent is even, compute the result of raising the base to the exponent/2 using exponentiation by squaring, and then square the result
• If the exponent is odd, compute the result of raising the base to (exponent - 1)/2 using exponentiation by squaring, and then multiply the result by itself and the base

By using this algorithm, we can compute the power of a number to a given exponent in logarithmic time, which is much faster than using repeated multiplication. This makes it particularly useful for large exponents or for modular exponentiation in cryptography

public number calculate(number base, number exponent) {
	if (exponent < 0)
		return calculate(1/base, -exponent);
	if (exponent == 0)
		return 1;
	if (exponent == 1)
		return base;
	if (exponent is even)
		return calculate(base * base, exponent/2);
	if (exponent is odd)
		return base * calculate(base * base, (exponent-1)/2);
}