7.2. Lecture - 55: Modular Arithmetic

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.

This chunk is an introduction to the topic of modular arithmetic within the broader field of number theory. It highlights the importance of number theory for computer science, particularly its relevance to cryptography and other algorithms. The lecturer emphasizes that while number theory is extensive, this session will focus on essential elements that directly apply to practical computing applications.

Consider the concept of number theory as a vast library. Modular arithmetic is just one important book on its shelves, but it holds keys to many real-world applications, such as securing your online transactions.

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. So, for instance, 5 modulo 4 is 1 and -11 modulo 3 is 1.

This chunk introduces the concept of the modulo operation. When dividing an integer 'a' by a positive integer 'N', the result can be expressed in terms of a quotient and a remainder. The remainder 'r' is what we refer to as 'a modulo N'. It is important to note that 'r' must be a non-negative integer that falls within the range from 0 to N-1. For instance, the calculation of '5 modulo 4' gives a remainder of 1, as does '-11 modulo 3', illustrating the concept using both a positive and a negative number.

Imagine you are sharing candies among friends; if you have 5 candies and each friend gets 4, you will have 1 candy left over. Similarly, if you have a negative amount of candies (like owing candies), you're still determining how many you have left over after sharing as much as you can.

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, that means if a modulo and b modulo N are same then I will say that a is congruent to b modulo N and the notation that we use is the following : [a ≡ b mod N].

This chunk focuses on the concept of congruence in modular arithmetic. Two integers 'a' and 'b' are said to be congruent modulo 'N' if they give the same remainder when divided by 'N'. This is mathematically represented as 'a ≡ b mod N'. This concept is crucial because it helps us establish equivalences between numbers in modular systems.

Think of two train schedules that represent time. If both trains arrive at station N at the same minute, we say they are synchronized in terms of their schedule. In modular arithmetic, two numbers can be considered synchronized (congruent) if they leave the same remainder when divided by a specific modulus.

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.

This section covers fundamental arithmetic operations applicable in modular arithmetic. The primary insight is that we can simplify complex operations by first reducing the operands under the modulus before executing the operation. This means we can find the modulo of each part of an expression and then perform the operation, thereby returning the same result as if we performed the operation first and then took the modulo.

Imagine you are carrying out a series of tasks. Instead of juggling all tasks at once (which can be complicated), you tackle reduced versions of each task separately, cataloging the outcomes so they remain manageable and similar to the overall result.

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.

This chunk introduces the challenges associated with division in modular arithmetic. Unlike addition, subtraction, and multiplication, division is not straightforward and may not always yield a well-defined result in modular contexts. It's crucial to recognize that just because the products are congruent modulo N, it does not imply that the numbers being divided are congruent as well.

If you're trying to fairly split a pizza among two friends, and there's an odd slice left over, division becomes complicated. In modular arithmetic, similar complexity arises when trying to divide numbers, as it's not merely a matter of splitting evenly.

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. Why it is called square and multiply? It will be clear soon so let me demonstrate the algorithm.

This section introduces the square and multiply technique, which is an efficient algorithm for computing powers modulo N, particularly useful in cryptography. By converting the exponent into binary form, this method can exponentially reduce the number of multiplicative operations needed, thus speeding up the calculations significantly compared to naive methods.

Think of the square and multiply method as using a smart way to navigate through a maze: rather than taking one step at a time, you leap ahead to key points, checking only where you need to make adjustments to reach your destination quickly.

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