Exponentiation by squaring

Exponentiation by squaring is a smart way to quickly find big powers of numbers, like figuring out what 2 raised to the 100th power is without multiplying 2 by itself 100 times! This method is very useful in mathematics and computer programming. Instead of doing many multiplications, you use a trick called "squaring" — which means multiplying the number by itself — and then combining the results in a clever way.

This technique works not just for regular numbers, but also for more complicated things like polynomials and square matrices. It is especially helpful when you need to do calculations in modular arithmetic, which is important in areas like cryptography. In some areas, like with elliptic curves, this method is sometimes called "double-and-add."

Because it reduces the number of steps needed, exponentiation by squaring is popular in computer science and math. It helps solve big problems much faster and easier, whether you're working with simple numbers or more advanced math concepts.

Basic method

This method helps us find large powers of a number faster. It uses a smart way to break down the problem. If we want to find x raised to the power n, we look at whether n is odd or even.

If n is even, we square x (multiply it by itself) and then find the power of half of n. If n is odd, we multiply x by the result of squaring x and finding the power of one less than half of n. We keep doing this until we reach a simple case, like raising x to the power of 0, which is always 1. This saves many steps compared to multiplying x by itself n times.

Computational complexity

This method uses fewer steps to find big powers of numbers than just multiplying them again and again. It needs about ⌊ log₂ n ⌋ squarings and the same number of multiplications. For numbers bigger than 4, this is quicker than multiplying the base over and over.

When we square a number, the number of digits about doubles each time. If multiplying two numbers with d digits takes O(d^k) steps, the total steps to compute xⁿ can be described as O((n log ⁡ x)^k).

2^k-ary method

This algorithm helps find the value of xⁿ by breaking down the exponent into parts based on base 2^k. It was first suggested by Brauer in 1939. The algorithm uses a special function to organize the exponent and then combines powers of x in a smart way to save steps.

For example, to calculate the exponent 398, the algorithm can choose different groups of powers of x to make the calculation faster and use fewer multiplications. This method is useful for quickly working out big powers of numbers or other elements in mathematics.

Montgomery's ladder technique

Some ways to calculate big powers can let others see secret numbers by watching the steps used. This can be a problem when keeping numbers safe, like in special math used for security.

A method called "Montgomery's ladder" helps keep these numbers secret. It always does the same steps, no matter what the secret number is, so it's harder to discover it by watching.

However, this method can still be partly seen by how fast the computer works, so newer ways add extra steps to keep it even safer.

Fixed-base exponent

There are different ways to calculate xⁿ when the base number stays the same and only the exponent changes. Preparing some values ahead of time helps make these calculations faster.

Yao’s method is one way to do this. It breaks the exponent into smaller parts and uses precomputed values to speed up the math. The Euclidean method is another approach. It uses a step-by-step process to simplify the problem. Both methods are useful when you need to calculate big exponents quickly, like in computer programs or advanced math.

Further applications

This method can help us work with very big numbers in special parts of math. For example, it can help us find the remainder after dividing a huge power by another number. This is important in cryptography, where we often use big numbers.

For example, finding the remainder when 13789⁷²²³⁴¹ is divided by 2345 would take a long time the usual way. But with this special method, we can do it in just 27 steps, which is much quicker.

This method can also help us work with groups of numbers and for calculating powers of matrices. It works in many different math systems.

Signed-digit recoding

Sometimes, it can be quicker to use negative numbers and the inverse of the base, especially if flipping the base is easy or already known. For example, when finding x^2k−1, the usual method needs k−1 multiplications and k−1 squarings. But you could do k squarings to get x^2k and then multiply by x⁻¹ to get x^2k−1.

Exponentiation by squaring is a way to compute big powers quickly. It works by doubling the exponent (squaring) and sometimes multiplying by the original number. But there is a smarter way called addition-chain exponentiation. This method lets you use any earlier results to save steps. For example, finding x¹⁵ can be done in fewer steps if you reuse earlier powers like x³.

Finding the best way to do this for any exponent is very hard, so computers usually use simpler methods that are good enough. These methods don’t change the fact that the number of steps grows slowly, just a little better than the squaring method.