7.3. Introduction to Number Theory

Hello, everyone, welcome to this lecture. So, we will now shift our focus on number theory. And number theory in itself is a very fascinating topic we can have a full course on number theory. But we will be discussing only the relevant topics in number theory that is useful in the context of computer science namely modular arithmetic, properties of prime numbers, algorithm related to GCD and this will be very useful especially in cryptography.

In this lecture, we are introduced to number theory, which is an intriguing area of mathematics. It encompasses various topics, but for this course, we will focus on aspects that have practical applications in computer science. These include modular arithmetic, understanding prime numbers, and algorithms related to the Greatest Common Divisor (GCD), which are critically important for fields such as cryptography.

Think of number theory like a toolbox. While it has many tools (like modular arithmetic and prime numbers), we are only going to pick the tools that help us build secure systems in technology, similar to how a mechanic may only use certain tools from a larger set to fix a car.

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 is a way of working with numbers within a circular range defined by a modulus N. When you take a number 'a' and divide it by 'N', the result can be expressed as a quotient and a remainder. This remainder, which is denoted as 'a modulo N', must lie between 0 and N-1. Therefore, for positive integers, we can visualize a clock where each hour represents a different remainder.

Imagine a clock that only goes up to N hours. If it's 5 o'clock now, and you add 8 hours, you don't go to 13 o'clock; instead, you wrap around and it becomes 1 o'clock. In this case, we say that 5 + 8 modulo 12 = 1.

Now, let us define what we call as congruence with respect to modulo. So 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 same as, the remainder that we obtained by dividing b by N, that means if a modulo and b modulo N are same then I will say that a is congruent to b modulo N.

Congruence is a concept that helps us determine whether two numbers give the same remainder when divided by a modulus N. We write this relationship as 'a ≡ b (mod N)', indicating that when 'a' and 'b' are divided by 'N', they leave identical remainders. This relationship allows us to treat numbers as equivalent under modular arithmetic, significantly simplifying calculations.

Consider two people in a race who finish at 12:02 and 1:02 respectively. If we consider their finish times modulo 1 hour, both effectively finished at '2 minutes past the hour'. Hence, in terms of timing, they can be seen as congruent.

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.

The arithmetic operations of addition, subtraction, and multiplication can all be simplified when dealing with modular arithmetic. Specifically, you can first reduce the numbers being operated on (a and b) modulo N, perform the arithmetic operation, and then apply the modulo again to obtain the same result as if you had first performed the operation and then reduced the result. This property drastically reduces the complexity of calculations.

It’s like managing a budget. If you track your spending over a month and find that spending on groceries and dining out can both be simplified to just the balance you have left at the end of the month, you can focus on just the numbers that matter instead of individual transactions.

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