Exponentiation by squaring

Exponentiation by squaring is a clever way to quickly calculate 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 all those multiplications, you use a trick called "squaring" — which means multiplying the number by itself — and then combining the results in a smart 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 fields where addition is more common, such as with elliptic curves, this method is sometimes called "double-and-add."

Because it cuts down the number of steps needed, exponentiation by squaring is a favorite in computer science and math. It makes solving big problems much faster and easier, whether you're working with simple numbers or more advanced math concepts.

Basic method

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

If n is even, we can 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. This process continues until we reach a simple case, like raising x to the power of 0, which is always 1. This way, we save many steps compared to multiplying x by itself n times.

Computational complexity

This method uses fewer steps to calculate large powers of numbers compared to simple multiplication. It needs about ⌊ log₂ n ⌋ squarings and the same number of multiplications. For numbers larger than 4, this is faster than multiplying the base over and over again.

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 large powers can let others see secret numbers by watching what steps are 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, called precomputations, helps make these calculations faster.

Yao’s method is one way to do this. It works by breaking the exponent into smaller parts and using precomputed values to speed up the math. The Euclidean method is another approach. It uses a step-by-step process to simplify the problem, making it easier to compute large powers efficiently. Both methods are useful in situations where you need to calculate big exponents quickly, like in computer programs or advanced math.

Further applications

This method can also help us work with big numbers in special math areas, such as when we want to find the remainder after dividing a very large power by another number. This is important in cryptography, where we often need to work with big numbers in a special kind of math system.

For example, finding the remainder when 13789⁷²²³⁴¹ is divided by 2345 would take a lot of time and space using regular methods. But with this special way of calculating, we can do it with just 27 steps, which is much faster.

This method can also be used for working with groups of numbers and for calculating powers of matrices. It works in many different math systems where the way we combine numbers has certain properties.

Signed-digit recoding

In some calculations, it can be faster to allow negative numbers and use the inverse of the base, if flipping the base is quick 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.