7.5.1. Modular Addition, Subtraction and Multiplication

So, imagine you are given a value a, which is an integer and you are given a modulus N. So, N will be called as the modulus where the modulus is positive, then a modulo N is r, where r will be a value in the range 0 to N – 1, such that a is equal to some quotient times the modulus plus your r. So basically r is the remainder which you obtain after dividing a by N where the remainder is in the range 0 to N - 1.

Modular arithmetic is a way of handling integers where we only focus on the remainders after division. In this context, we take an integer 'a' and a positive integer 'N' known as the modulus. We calculate 'a modulo N', denoted as r, which is the remainder after dividing 'a' by 'N'. Interestingly, this remainder 'r' must always be between 0 and N-1. For example, if we calculate 5 modulo 4, we find that the remainder is 1 because 5 divided by 4 gives a quotient of 1 with a remainder of 1. Conversely, if we consider -11 modulo 3, the result is also 1, not -2, because we must ensure our remainder lies within the prescribed range.

Think of a clock as a real-life analogy for modular arithmetic. A clock has numbers from 0 to 11 (or 1 to 12), representing hours. If it's 10 o'clock now and you add 5 hours, you'll end up at 3 o'clock. That’s because 15 hours (10 + 5) modulo 12 equals 3. The clock helps us visualize how modular arithmetic works: once you reach the maximum (12 hours), you wrap around to the beginning.

Now, let us define what we call as congruence with respect to modulo. If you have a modulus N, then we say that a is congruent to b modulo N if the remainder that we obtained by dividing a by N is the same as the remainder that we obtained by dividing b by N.

Congruence in modular arithmetic is a relation that indicates when two numbers yield the same remainder when divided by a modulus 'N'. If 'a' gives the same remainder as 'b' when divided by 'N', we write this as a ≡ b (mod N). This relationship helps simplify processes in arithmetic and number theory, particularly when solving equations or determining equivalences between numbers.

Consider two friends sharing candies. If one friend has 10 candies and the other has 22, both can be divided into groups of 6 (the modulus) to find out how many are left after sharing. 10 candies modulo 6 gives a remainder of 4, and 22 candies modulo 6 gives the same remainder of 4. So, we can say that both friends have the same leftover candies when shared in groups of 6.

So there are some interesting arithmetic rules which your modular arithmetic follows. If 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, and a multiplied by b modulo N is the same as a′ into b′ and then you take modulo N.

In modular arithmetic, there are several rules that allow us to simplify calculations. These rules state that operations like addition, subtraction, and multiplication on numbers under a modulus can be performed first on reduced values and still produce the same result as if we performed the operations first then took the modulus. This streamlines many calculations, especially with larger integers.

Imagine you’re making batches of cookies and can only use up to a certain amount of dough (the modulus). Suppose you have 12 cups of flour (the total amount you can handle) but mixing it all at once is tricky. You could first use 10 cups and then add 5 more later. Even though those additions happen separately, the total still needs to fit within your 12-cup constraint. Thus, whether you mix all at once or in parts, the results will be equivalent if you respect your limits.

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′. Can I say that a over b modulo N is the same as a′ over b′ modulo N? The answer is no.

While addition, subtraction, and multiplication in modular arithmetic maintain their properties across reduced values, division does not. Unlike the other operations, dividing by a number in modular arithmetic can lead to ambiguous or undefined results. Specifically, if the denominator isn't invertible within the modulus, the division can't be performed reliably. This highlights the importance of understanding the limitations of operations in modular arithmetic.

Imagine you've got 10 cookies to divide between 3 friends. You can easily distribute cookies equally (this is like addition and multiplication). But if I ask how many cookies each friend would receive if you were to split just one cookie (using division), you can’t exactly do that without cutting the cookie—and even then, depending on how you choose to share, the results can vary, leaving you with ambiguity. This is analogous to the complications that arise when attempting division in modular arithmetic.

So, now, let us discuss some algorithms for modular arithmetic. And we will be seeing algorithms for addition, multiplication, subtraction and modular exponentiation. So, this is addition, this is subtraction, this is multiplication, and this is called modular exponentiation.

To efficiently perform operations in modular arithmetic, particularly when working with large numbers, we can rely on several algorithms. These algorithms enable us to calculate sums, products, and differences, as well as modular exponentiation, which is crucial in fields like cryptography. The focus is on maintaining efficient running times, typically polynomial in terms of the number of bits needed to represent the integers involved.

Think of these algorithms like cooking recipes where each recipe is tailored for specific types of dishes (or operations). Just as you wouldn’t use the same method to bake a cake and fry an egg, different modular algorithms serve specific purposes, ensuring efficiency whether you're combining numbers, raising them to powers, or otherwise manipulating them.

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 method provides a highly efficient way to compute modular exponentiation. Instead of naively multiplying a number by itself repeatedly, this method takes advantage of the binary representation of the exponent. By squaring the base and multiplying it selectively, based on the bits in the exponent, you can significantly reduce the number of multiplications required, leading to a polynomial time complexity instead of the exponential time of the naive method.

Imagine you’re a library assistant who needs to find a specific book among millions. Instead of checking every shelf one by one (which would take forever), you use a search algorithm that quickly narrows down your options—first eliminating half the shelves, then half of those, and so on. This is akin to the square and multiply method, which eliminates unnecessary operations and leads to a much faster solution.

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