Did you get the joke in the picture to the left? If not, you will do so in a few minutes. I was recently reading an article and I came across the terms mentioned in the title. From the looks of it, they are very close to each other, right? In many fields within mathematics, we talk about objects and the maps between them. Now you may ask why we would want to do that? Well, transformation is one of the most fundamental things in any field. For example, how do we transform a line into a circle, or fuel into mechanical energy, or words into numbers? There are infinitely many types of transformations that can exist. Obviously, we cannot account for every single type of transformation that can possibly exist. So we limit ourselves to only the interesting ones. So what exactly is it all about? How does it even relate to the title of this blog post? Continue reading “Homomorphism vs Homeomorphism”

# Category: Mathematics

# A Beginner’s Look At Lambda Calculus

Back in school, you must remember studying differential and integral calculus. Now what on earth is lambda calculus? Well, lambda calculus is basically a simple notation for functions and applications in mathematics and computer science. It has a significant impact in the field of programming language theory. It forms the basis for all the modern functional programming languages like Haskell, Scala, Erlang, etc. The main idea here is to apply a function to an argument and forming functions by abstraction. The good thing about lambda calculus is that the syntax is quite sparse, which makes it an elegant notation for representing functions. Here, we get a well-formed theory of functions as rules of computation. We will discuss this further soon. Even though the syntax of lambda calculus is sparse, it is really flexible and expressive. This feature makes is particularly useful in the field of mathematical logic. So what exactly is lambda calculus? How do we understand it? Continue reading “A Beginner’s Look At Lambda Calculus”

# Elliptic Curve Cryptography: Part 4/4 – How Do We Use Elliptic Curves?

In the previous blog post, we discussed about elliptic curves and saw what they look like. We also looked at some of their special properties that enable it to be a good trapdoor function. But this is still very mathematical, right? The curves are great to look at and we understand general concept of elliptic curves, but how do we use them in real life? The curves denoted by those equations don’t represent the curves that are used in cryptography. In the real world, things are digitized. We need to convert things into bits and move them around quickly. So how do we do it? Continue reading “Elliptic Curve Cryptography: Part 4/4 – How Do We Use Elliptic Curves?”

# Elliptic Curve Cryptography: Part 3/4 – What Is An Elliptic Curve?

In the previous blog post, we discussed why RSA will not be sufficient anymore. We looked at how machines are getting stronger, and that we cannot rely on factorization as our primary mathematical foundation. We also talked a bit about what we are looking for in our new system, and then said a quick hello to elliptic curves. In this blog post, we will see what elliptic curves are. As always, we will keep the equations to a minimum, and instead, try to understand the underlying concept. Let’s go ahead, shall we? Continue reading “Elliptic Curve Cryptography: Part 3/4 – What Is An Elliptic Curve?”

# Elliptic Curve Cryptography: Part 2/4 – Why Do We Need It?

In our previous blog post, we discussed public key cryptography and how it works in general. The RSA is so powerful because it comes with rigorous mathematical proofs of security. The authors basically proved that breaking the system is equivalent to solving a very difficult mathematical problem. When we say “difficult”, what it means is that it would take trillions of years to break it even if all the supercomputers in the world were to work in parallel. By the way, this is just one encoded message! The problem in question here is prime number factorization. It is a very well-known problem and has been studied for a very long time. So if it’s such a difficult problem, why do we need something new? What’s the problem here? Continue reading “Elliptic Curve Cryptography: Part 2/4 – Why Do We Need It?”

# Elliptic Curve Cryptography: Part 1/4 – Why Should We Talk About It?

Elliptic curve cryptography is one of the most powerful techniques used in the field of modern cryptography. Just to be clear, elliptic curves have nothing to do with ellipses. I agree that the name can be slightly misleading, but once we discuss what elliptic curves are, it will become much clearer. So as you may have already guessed, elliptic curve cryptography (ECC) is based on the properties of elliptic curves. ECC is very useful in internet security and it is being deemed as the successor to the almighty RSA crypto system. If that didn’t make sense to you, don’t worry! We are going to discuss everything in detail. A lot of of websites are increasingly using ECC to secure everything like customer data, connections, passing data between data centers, etc. So what makes ECC so special? Is it strong enough to take over the world? Continue reading “Elliptic Curve Cryptography: Part 1/4 – Why Should We Talk About It?”

# What Is A Markov Chain?

If you have studied probability theory, then you must have heard Markov’s name. When we study probability and statistics, we tend to deal with independent trials. What this means is that if you conduct an experiment a lot of times, we assume that the outcome of one trial doesn’t influence the outcome of the next trial. For example, let’s say you are tossing a coin. If you toss the coin 5 times, you are bound to get either heads or tails with equal probability. If the outcome of the first toss is heads, it doesn’t tell us anything about the next trial. But what if we are dealing with a situation where this assumption is not true? If we are dealing with something like estimating the weather, we cannot assume that today’s weather is not affected by what happened yesterday. If we go ahead with the independence assumption here, we are bound to get wrong results. How do we formulate this kind of model? Continue reading “What Is A Markov Chain?”

# What Is Maximum Likelihood Estimation?

Let’s say you are trying to estimate the height of a group of people somewhere. If the group is small enough, you can just measure all of them and be done with it. But in real life, the groups are pretty large and you cannot measure each and every person. So we end up having a model which will estimate the height of a person. For example, if you are surveying a group of professional basketball players, you may have a model which will be centered around 6’7″ with a variance of a couple of inches. But how do we get this model in the first place? How do we know if this model is accurate enough to fit the entire group? Continue reading “What Is Maximum Likelihood Estimation?”

# Dissecting The Riemann Zeta Function

The Riemann Zeta function is an extremely important function in mathematics and physics. It is intimately related to very deep results surrounding the prime numbers. Now why would we want to care about prime numbers? Well, the entire concept of web security is built around prime numbers. Most of the algorithms for banking security, cryptography, networking, communication, etc are constructed using these prime numbers and the related theorems. The reason we do this is because of the inherently sporadic nature of prime numbers. You never know where the next one is going to appear on the number line! So what does that have to do with the Reimann zeta function? If prime numbers are random, what’s the point of looking into them? Continue reading “Dissecting The Riemann Zeta Function”

# What Is A Singularity?

You must have heard the term “singularity” being used in various contexts. We see it in mathematics, physics, circuit design, mechanics, fluid dynamics, etc. The word “singular” means something that is extraordinary, unique, and strange. When we talk about singularity in mathematics, we usually refer to the uniqueness of mathematical objects. In particular, singularities refer to the points where the mathematical objects are not well-behaved i.e. we can’t define them for those points. In higher mathematics, we usually define functions to explain the behavior of any system. These functions are the mathematical objects under consideration, and we will see how to understand their singularities. But why do we care about singularities? Is it just a mathematical concept or do we ever see it in real life? Continue reading “What Is A Singularity?”