9.6. Existence of Multiplicative Inverse

Now, using the notion of GCD and Bezout’s coefficient, we will define what we call as multiplicative inverse modulo N. Let us first see the definition and then we will see how exactly we can find out the multiplicative inverse using extended Euclid’s algorithm.

I define operation multiplication modulo N that is denoted by this notation ( . ) . So, this notation is for multiplication modulo N, that means you multiply a and b and then take the remainder. That is our definition.

Now, I say an integer b to be the multiplicative inverse of another integer a modulo N if a times b modulo N gives me 1, that means, if you multiply a with b and then take the modulo N and if the value is 1, then I will say that value b is the inverse of a (multiplicative inverse), why it is inverse because typically in the regular arithmetic, when I say inverse of a is 1 / a, the interpretation there is that if I multiply a with 1 / a, then I get 1.

This chunk introduces the concept of the multiplicative inverse in modular arithmetic. The multiplicative inverse of an integer a modulo N is an integer b such that when you multiply a by b and then take the result modulo N, you get 1. This can be likened to finding a number which 'undoes' the multiplication of a in the modular system, similar to how dividing by a number returns to 1 in standard arithmetic.

Imagine you have a special kind of key (a) that can only lock a box (the operation). To unlock it, you need another unique key (b), which when used in conjunction with the lock (the multiplication modulo N), will open the box such that the outcome is a 'successful unlock' (which we consider as 1). Just like having a lock and key, you can't unlock the box unless you have the exact key to do so.

Now, another interesting fact here is the following if b is the multiplicative inverse of a modulo N then any number of the form b plus minus any multiple of your modulus N, k ± N for all k ∈ ℤ, will also be an inverse of a and you can verify that. So what will be the result of multiplying a with this number b plus minus kN and then taking modulo N? Well, this will be the same as this.

This shows that if at all you have multiplicative inverse, then they are infinite in numbers that means, once you find 1 multiplicative inverse for a you are guaranteed to have infinite number of multiplicative inverse.

This chunk highlights a key property of multiplicative inverses in modular arithmetic: if you find one inverse b for a, you can generate an infinite number of inverses by adding or subtracting multiples of the modulus N. This is due to the periodic nature of modular arithmetic, which creates a repeating cycle.

Think of a clock as your modular system, where the hour hand completes a full circle (your modulus) every 12 hours. If the hour hand is on 3 o'clock (your a), it has a specific opposite point (your b) on the clock face that refers back to the 12 o'clock position (1). But if I say you can also check back at 15 o'clock, 27 o'clock, etc., those all still effectively give you the same point back to '1 o'clock' (the outcome). Thus, just like every position wraps around, every infinite point created by multiples of 12 ultimately leads you back to 1 o'clock.

But now, the interesting question is when can we say that the multiplicative inverse modulo N for a number a exists - is it the case that for every a for every modulus N I can say that the multiplicative inverse modular and exist? Unfortunately the answer is no, there is only under certain conditions I can say that multiplicative inverse modular and exist.

So, the theorem statement is the following: If you are given some number integer a and a modulus N then the claim is that the multiplicative inverse of a exists if and only if a is co-prime to N namely the GCD of a and N is 1.

This chunk addresses a fundamental aspect of modular arithmetic: the conditions for the existence of a multiplicative inverse. Specifically, a multiplicative inverse exists for a number a modulo N only if a and N are co-prime, meaning their greatest common divisor (GCD) is 1. If they share any common factors, then an inverse cannot be defined in the modular framework.

Picture a two-piece jigsaw puzzle, where each piece must have unique edges to fit correctly together. If a piece (a) shares an edge with another piece (N) (indicating a GCD greater than one), it can't form the correct picture because they can't combine properly. However, if both pieces are unique in their design sharing no common notches (GCD of 1), they will fit together perfectly, forming the complete picture (the multiplicative inverse).

Now, I want to prove the necessity condition. Namely, I want to show that if at all the multiplicative inverse of a exists then it implies that the number a is co-prime to your modulus. So, assume you have an a for which you can find out the multiplicative inverse how you find out I do not care, but it exists. And suppose the multiplicative inverse of a is b, I have to show; my goal is to show that a is co-prime to N.

In this portion, the necessity condition for the existence of a multiplicative inverse is established: If an inverse exists, then a must be co-prime to N. This is shown by asserting that if the GCD of a and N is greater than one, then a product involving those numbers cannot yield a result of 1 modulo N, ultimately proving that no inverse can exist under those conditions.

Consider again a pair of puzzle pieces. If you can't fit two pieces together to complete a design (meaning there's a common edge), this indicates they can't create the final outcome—we can't have a perfect puzzle picture (no multiplicative inverse). Conversely, if two pieces fit perfectly without overlapping parts (GCD of 1), that signifies they complement each other, enabling the completed design, just like how a multiplicative inverse functions.

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