7.1. Discrete Mathematics

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.

Number theory is a branch of pure mathematics that deals with the properties and relationships of numbers, particularly integers. In this context, we'll focus on modular arithmetic, which is crucial for understanding algorithms used in computer science, particularly in cryptography. While number theory is vast, this lecture targets specific applications that can be employed in practical scenarios like cryptography.

Think of number theory like a toolbox for building secure communications. Just like you need certain tools to craft something, in computer science, various number theory concepts allow us to construct secure systems for exchanging sensitive information.

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.

Modular arithmetic involves calculations where numbers 'wrap around' after reaching a certain value. In this case, 'a' is an integer, while 'N' is the modulus. The result of 'a modulo N' gives a remainder 'r', which must be less than 'N'. This means if you divide 'a' by 'N', the integer part of the division (the quotient) is discarded, and you keep the remainder. For example, when you compute 5 modulo 4, you divide 5 by 4, which gives a quotient of 1 (because 4 goes into 5 one time) and a remainder of 1 (since 5 - 4*1 = 1). Similarly, for -11 modulo 3, you find the equivalent positive remainder, which is also 1.

Imagine a clock with 12 hours. If it's 8 o'clock and you add 5 hours, you will not say it's 13 o'clock, but 1 o'clock. This is essentially what modular arithmetic does: it resets after reaching a certain number (in this case, 12).

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.

Congruence in modular arithmetic means that two numbers give the same remainder when divided by a specific modulus. This is represented as 'a ≡ b (mod N)'. For example, if both 8 and 3 yield a remainder of 2 when divided by 6, we can write 8 ≡ 2 (mod 6), and 3 ≡ 2 (mod 6), indicating both 8 and 3 are congruent to 2 with respect to modulus 6.

Think of congruence like two friends getting the same score on a test, but from different questions. They both might arrive at the same answer (the same score modulo another number), indicating that while their paths might differ (the questions they answered), they still reached the same conclusion.

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, addition, subtraction, and multiplication can be performed first on the reduced numbers (the remainders) instead of the original values. For example, if you have a = 7 and b = 5, instead of computing (a + b) % N directly, you can find a' = 7 mod N and b' = 5 mod N first. Performing the addition of a' and b' and taking the result modulo N will yield the same outcome. This approach helps simplify calculations, especially when dealing with large numbers

Imagine you are making a salad with different ingredients, but you have a limited bowl size (the modulus). Instead of trying to fit all the ingredients (numbers) into one big salad, you can pre-measure (reduce the operands) and then combine them in the bowl, which helps manage space (operations) better.

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 these 2 expressions will be same? Well the answer is no...

Unlike addition, subtraction, and multiplication, division in modular arithmetic is more complex and not always possible. The division of two numbers modulo N does not result in the same relationship as the other operations. If we say a * c ≡ b * c (mod N), it does not mean a ≡ b (mod N), unless 'c' is not divisible by N. This points to the fact that you cannot just 'cancel' numbers the way you might in regular arithmetic.

If you think of it in terms of sharing food; if you have a certain amount of pizza (a) and you try to divide it equally among friends (b), the total depends significantly on how many friends there are (c) and sometimes you can't share evenly, leading to leftovers (remainder).

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

Algorithms used in modular arithmetic are crucial for efficiency, especially with large numbers. Addition, subtraction, and multiplication operations can be designed to run in polynomial time in relation to the number of bits of the integers involved. For instance, when adding two integers, the algorithm can be structured to process one bit at a time (like the addition process in binary), making it much more efficient than operating on full numbers directly. This efficiency is critical in applications such as cryptography, where computations can involve very large integers.

Think of it like a digital lock. Instead of trying to guess the full number one digit at a time, you can use clues to narrow down to a manageable number, making it easier to find the correct combination quickly.

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 algorithm is an efficient method for computing large powers modulo N. Instead of directly multiplying a number by itself repeatedly, which can be computationally expensive, the algorithm breaks it down using binary representation of the exponent. By squaring intermediate values and conditionally multiplying based on the bits of the exponent, the number of multiplications required is drastically reduced, thus enhancing performance. This is especially beneficial in cryptographic applications where high powers are common.

Imagine trying to find 8 cubed (888), instead of multiplying 8 by itself two times directly, you can calculate the result of 8 squared and use that value to do a simple multiplication instead. This way, you make fewer calculations and save time - just like optimizing the effort in a marathon by pacing oneself carefully!

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