7.4.3. Arithmetic Rules of Modular Arithmetic

So there are some interesting arithmetic rules which your modular arithmetic follows. So imagine that a modulo N is a’ and b modulo N is b’. Then a + b modulo N will be the same as a’ + b’ modulo N, a - b modulo N will be same as a’ - b’ modulo N so on and similarly, a multiplied by b modulo N is same as a’ into b’ and then you take modulo N.

In modular arithmetic, there are specific rules that we can follow when performing operations like addition, subtraction, and multiplication. When you take the modulo of two numbers, the rules indicate that you can first reduce those numbers modulo N before performing the operation. This simplification helps in reducing the complexity of calculations, especially when dealing with large numbers.

Think of this as a game of music notes on a circular scale (like a clock). For instance, if you have two notes playing at different pitches (a and b), you can first determine where they land on the scale (their identities modulo N). Then, it's easier to add, subtract, or multiply them to find out what new note is produced, rather than trying to stick to their original positions.

So, let us prove these rules, we will just prove it for the case of addition, for the other operations you can prove in the similar way.

To prove the addition rule, you start with two numbers, a and b. When we say that a modulo N is a’ and b modulo N is b’, we can express a as a = q_a * N + a’ and b as b = q_b * N + b’, where q_a and q_b are some quotients. When adding a and b together and taking the modulo N, the N parts (cancelling out during division) do not affect the remainder, which results in the same output as simply adding the reduced forms of the numbers.

Imagine you have two bags of apples that are sorted and returned to a common basket (mod N). The total number of apples you put back into the basket is the same whether you first count the apples from each bag or measure how many go back together.

So, what will be the interpretation of these arithmetic rules the interpretation here will be the following that you can first reduce the operands namely a and b modulo N and then you can perform the plus operation, subtraction operation, multiplication operation and if again required and you can do a modulo instead of first adding and then taking a modulo.

This means that the computation can be simplified significantly. You do not have to compute the full value of a and b before applying the operations. Instead, if you reduce them to their remainders first, the arithmetic becomes easier, and you only need to operate with smaller numbers.

Using the previous analogy of apples, if you only focus on how many apples are left after putting them in a basket instead of all apples carried from several bags, you can instantly find the total without having to worry about the larger quantities.

Now, we have seen the rules for addition, subtraction, multiplication, what about division? So, imagine a modulo N is a’ and b modulo N is b’. Of course, a’ and b’ are in the range 0 to N - 1. Now, what can I say about a over b modulo N and a’ over b’ modulo N. Can I say that these 2 expressions will be same? Well the answer is no because at the first place the value a over b modulo N may not be well defined.

In modular arithmetic, dividing by another number is not as straightforward as addition, subtraction, or multiplication. This is because division can lead to fractions or remainders that might not be integers. Therefore, you cannot always directly relate a/b modulo N to a'/b' modulo N since a division in modular arithmetic does not guarantee that the relationship holds.

Consider sharing pizza among friends. If you have 5 slices and 3 friends, you can easily say each gets more than 1 slice with some leftover, but if you had 5 slices and wanted to share 2 slices with 3 friends, how much do they get would be a complex division, as a part of a slice can be a fraction that is difficult to handle conceptually in this context.

Now, what will be the complexity measurement? How exactly we judge whether our given algorithm that we designed for performing this modular arithmetic operations are efficient or not.

When determining how efficient a modular arithmetic algorithm is, we look at how many operations the algorithm needs as the size of the input numbers, represented in bits, increases. Ideally, we want algorithms that operate in polynomial time relative to the number of bits in the numbers involved because this ensures that even for large inputs, we would get results relatively quickly.

Imagine trying to pack boxes for a move. If your packing process is quick and takes a manageable time depending on the number of boxes, you have an efficient system. However, if it takes exponentially longer as you add more boxes (like having 10 times the number leads to 1,000 times the effort), that’s inefficient.

The naive algorithm will require you to perform b times polynomial in n number of operations, but my b itself could be as large as 2 to the power 1024 bit number, those many, this is an enormously large quantity, you cannot even imagine how big this number is.

If one were to follow a simple approach of calculating the exponent by multiplying a with itself many times until reaching the value ‘b’, it would require an exorbitant number of operations, which is not practical for large values of b. This illustrates the necessity of advancing to smarter algorithms such as the square-and-multiply approach, which greatly reduces the computational requirements.

Think of trying to climb a mountain by continuously walking step after step without any strategy. If the mountain is enormous (like calculating a large exponent), progress can feel overwhelmingly slow. Instead, if you take large strides (like using square-and-multiply), you can ascend much quicker and reach the summit much more efficiently.

So 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 technique works by expressing the exponent in binary representation and progressively calculating powers of the base number, squaring when moving to higher powers and multiplying only when necessary. This not only makes computations faster but also ensures fewer operations by effectively skipping unnecessary calculations.

Imagine using a shortcut to get to a destination, where instead of walking every step of the way individually, you drive to significant landmarks and only walk the remaining distance. This strategy minimizes effort and maximizes speed, much like the square-and-multiply method.

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