7.5.3. Modular Exponentiation

Now, we will see a very nice method, which is a polynomial time algorithm for performing modular exponentiation and this is called as the square and multiply approach.

The square and multiply approach is a method used to calculate modular exponentiation efficiently. Instead of directly multiplying the base multiple times, this method takes advantage of the binary representation of the exponent to reduce the number of multiplications needed.

Think of making multiple copies of a document. Instead of printing one copy at a time, you can print two copies, then double that to four copies, and so on, which is much faster. Similarly, the square and multiply method allows us to quickly calculate large powers by squaring instead of multiplying repeatedly.

Suppose I want to compute a^53 modulo N. The naive approach will be you multiply a to itself and then take mod N.

The naive approach for calculating a^53 involves multiplying a by itself 52 times and then taking the modulus after each multiplication. This would result in a large number of operations, especially for large exponents, making it inefficient. In contrast, the square and multiply method allows us to compute the result with significantly fewer multiplications.

If you were to climb stairs where each step doubles the height you cover, reaching a high point this way would be much faster than taking one small step at a time. Similarly, the square and multiply approach leverages exponents to reduce the effort needed in calculation.

The idea here will be the following, I will treat my exponent in binary form.

To understand how the square and multiply method works, we first convert the exponent into its binary form. Each bit in the binary representation tells us whether to include a certain power of the base in our final product. For example, the binary of 53 is 110101, which means we include the powers corresponding to the bits that are 1.

Consider a treasure map that only shows you the key points for digging. Only some of the points are marked, and you only dig at those spots. Similarly, the 1s in the binary representation identify which powers of the base to 'dig up' or multiply.

In each iteration, I will compute the next higher power of a by squaring the current power.

In the square and multiply method, we start with the base and square it to obtain higher powers iteratively. For each bit of the exponent, if it is 1, we multiply the current result (accumulator) by the current power of the base. This is repeated until all bits have been processed.

It's like building a tower. You start with the first block and keep stacking additional blocks by doubling the height based on the received instructions (the binary bits). Each block you add increases the height according to the instructions provided by the bits.

So, in the worst case, what can happen you will be definitely doing n number of compulsory squaring where n is the number of bits of your exponent.

The efficiency of the square and multiply algorithm lies in its ability to keep the number of operations manageable. In the worst-case scenario, it requires about 2n multiplications, which is efficient relative to the naive approach that could require exponential time due to the number of multiplications based on the size of the exponent.

Imagine you are hosting a party and need to set tables. Instead of setting one table at a time, you double the tables each time based on the number of guests you have, making it much quicker to accommodate a large crowd.

Let me go through the pseudocode 1 step at a time.

The pseudocode provides a detailed step-by-step guide to implementing the square and multiply method. It outlines how to initialize the variables and adjust them as you process the bits of the exponent, showing the iterative squaring and conditional updates necessary for finding the modular exponentiation.

Think of this pseudocode as a recipe in the kitchen. Each step tells you exactly what to do, from gathering your ingredients (initializing variables) to cooking (performing operations) until you finally serve the dish (arriving at the answer). Following these steps ensures you don’t miss anything important.

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