7.5. Algorithms for Modular Arithmetic

So, let us begin with modular arithmetic. 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 involves taking the remainder of a number when divided by another number, known as the modulus. If we denote this operation as 'a modulo N' or 'a mod N', it gives us a result 'r' that is the remainder when 'a' is divided by 'N'. For instance, when calculating '5 mod 4', we divide 5 by 4, which gives us a quotient of 1 and a remainder of 1. Thus, '5 mod 4' equals 1. In general, the result 'r' must always fall within the range of 0 to N-1. This is a key aspect of modular arithmetic, allowing it to behave similarly to clock arithmetic.

Think of modular arithmetic like reading a clock. If it's 5 o'clock now and you add 4 hours, you don’t end up at 9 o'clock—it wraps around to 9 o'clock. Similarly, in modular arithmetic, when you exceed the modulus (like going past 12 on a clock), it wraps back around. So if you think about '14 mod 12', it's like saying 14 hours from now on a 12-hour clock, which gives you 2 o'clock.

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

In modular arithmetic, two integers 'a' and 'b' are said to be congruent modulo 'N' if they leave the same remainder when divided by 'N'. This relationship is denoted as 'a ≡ b mod N'. An important related fact is that this congruence holds true if and only if the difference (a-b) is divisible by N. This means that 'a' and 'b' lie on the same position when we think of numbers as points that wrap around after reaching 'N'.

Imagine walking around a circular track marked from 0 to N. If you start at position 2 and walk to position 5, you are effectively congruent to starting at position 5 and being at position 2. In this sense, if you walk around the track (the modulus) many times, your final position only depends on your last move—this is the essence of congruence.

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.

In modular arithmetic, operations like addition, subtraction, and multiplication can be performed before or after the modulus operation without affecting the outcome. This means that if you take the modulus of two separate numbers first and then add them, you will get the same result as if you added the numbers first and then took the modulus. This property makes calculations much simpler because you can typically work with smaller numbers.

Think of having 200 candies but only being able to fit 50 in a box. If you want to divide them among friends, first counting them in smaller groups (modulus) makes it easier. You can quickly count out groups of 50, rather than counting them all out at once and having to keep track of the 'overflow'. The principle of simplifying first then adjusting with modulus closely mimics this practical task.

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.

Division in modular arithmetic is not straightforward like addition or multiplication. The quotient 'a/b' does not always yield a remainder that fits within the confines of the modulus. For example, if 'a' is smaller than 'b', the division yields a fraction, which cannot be directly represented in integer modular arithmetic. Therefore, one cannot assume that if the products 'ac' and 'bc' are congruent modulo 'N', the same is true for 'a' and 'b' unless certain conditions are met.

Consider you have 4 apples and want to share them among 6 friends. Each friend may receive a fraction of an apple, and discussing it in whole numbers (modular arithmetic) doesn't apply. This illustrates why division doesn't align neatly in modular arithmetic—it produces fractions, which don't follow the integer rules of counting used in modular mathematics.

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 is a highly efficient way to compute powers in modular arithmetic. Instead of performing a potentially large number of multiplications for a number raised to an exponent, this approach cleverly reduces the number of operations by taking advantage of the binary representation of the exponent. This method allows us to square the result at each step and multiply only when necessary, drastically reducing computation time, especially with large numbers.

Think of trying to fold a piece of paper multiple times. Instead of folding it once for every single fold, each fold you make can double the thickness of the paper—exponentially increasing its size with fewer actions. Similarly, the square and multiply method allows us to quickly reach large powers with fewer multiplication steps, showcasing an efficient process where we effectively manage our operations.

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