When we think of prime numbers, the first thing that we tend to associate them with is randomness. Prime numbers are scattered all over the number line and there is no fixed formula that can tell you when the next one is going to occur. This has been used heavily by mathematicians and cryptographers to develop security systems for internet, banking, communication, and so on. Coming to the topic at hand, we are going to talk about prime divisors of a given number. Prime divisors of a number are divisors of that number that happen to be prime numbers. Big surprise, right? Alright, what’s so interesting about them? Continue reading

# Tag Archives: Prime numbers

# What’s So Interesting About The Prime Counting Function?

Mathematicians have obsessed over prime numbers for centuries, and will continue to do so for the foreseeable future. Prime numbers are so enigmatic and fascinating that mathematicians just can’t stop thinking about them! Prime counting function is probably one of the most famous problems in this domain. This function simply counts the number of prime numbers less than or equal to a given number. Pretty straightforward! But why is this of such great importance? What are we going to do with this information? Continue reading

# How Do We Know That There Are Infinitely Many Prime Numbers?

There is a very famous theorem which says that there are infinitely many prime numbers. For people who are new to this, a prime number is a number that doesn’t have any divisors except for 1 and itself. For example, 11 is a prime number because it doesn’t have any divisors apart from 1 and 11. On the other hand, 12 is not a prime number because it is divisible by 1, 2, 3, 4, 6, and 12. Now how do we know that there are infinitely many primes? As numbers get bigger, they tend to have more divisors. So may be at some point, all the numbers can possibly start being composite and they will have a lot of divisors, right? We can delve into a deep mathematical proof to prove this, but let’s take a different route. Let’s see if we can prove this with logic, shall we? Continue reading

# 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 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 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 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

# 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

# What Do You Know About Infinity?

What do you think of the term “infinity”? In layman’s terms, infinity is something that has no end, something that is bigger than all the things we know. This is what most people think! Many people who have not seen advanced mathematics do not know about the fact that there are many different types of infinity, some of which are bigger than others. Now how is that possible? Isn’t there just one single infinity which quantifies the essence of uncountability? Continue reading

# P vs NP: The Epic Saga

P vs NP problem is one of the great unsolved problems in theoretical computer science. This problem has become broadly recognized in the mathematical community as a mathematical question because it is fundamental, important and beautiful. It is in fact one of the seven Millennium Prize Problems. If you solve this problem, you get $1 million and become really famous among mathematicians and computer scientists. If you are evil, then you can use your proof to become richer than God, then publish your proof, reject the prize money and become extremely well respected in the mathematics community! Wait a minute, really? How can I use this to become rich? Before we answer that, let’s see what exactly is the difficulty in solving the problem. Shall we? Continue reading