Have you heard of elliptic curves before? They are used extensively in number theory and cryptography. The reason elliptic curve cryptography is gaining popularity is because it’s fundamentally much stronger than the RSA algorithm, the algorithm that we all love and adore. If you don’t know what elliptic curves are, just google it and see what they look like. You are reading this sentence without googling it, aren’t you? Okay I’m going to assume that you know what elliptic curves look like. Do they look anything like ellipses? No! So why are they called “elliptic” curves? Continue reading

# Tag Archives: Cryptography

# 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

# Quantum Encryption And Black Holes – Part 2/2

In the previous post, we discussed about the concepts of quantum encryption and black holes. We also talked about how we do cryptography in the subatomic world. This blog post is a continuation of that discussion. As the title suggests, the overarching theme is the relationship between quantum encryption and black holes. Let’s continue talking about it then. Although quantum encryption looks extremely robust in theory, how practical is it? What do we know about its security and how is it related to black holes? We know that nothing can escape from black holes, so we need a way to understand more about the black holes. Continue reading

# Quantum Encryption And Black Holes – Part 1/2

Is that really the title? It looks like two random things mashed up together. Doesn’t make much sense, right? Well, recent research suggests that quantum encryption and black holes may be related. A proposed mathematical proof outlines the way in which information behaves in coded messages, and this may have implications for black holes. The proof basically suggests that the radiation spit out by black holes may retain information about them. The research not only focuses on encoding communications in quantum mechanical systems, but also addresses a long-standing question for theoretical physicists: What exactly happens to all the stuff that falls into a black hole? Is it possible to retrieve any information about the black hole? Continue reading

# Image Steganography

As discussed in my previous post, steganography is the art of hiding the fact that communication is taking place. We achieve this by hiding original information inside other information known as carrier files. Many different carrier file formats can be used, but digital images are the most popular because of their frequency of occurrence on the internet. For hiding secret information in images, there exists a large variety of steganographic techniques, some are more complex than others, and all of them have respective strong and weak points. Different applications have different requirements of the steganography technique used. For example, some applications may require absolute invisibility of the secret information, while others require a larger secret message to be hidden. How do we achieve this? How robust is it? Continue reading

# Steganography

Let’s say that we want to communicate with someone secretly. We prefer that only the intended recipient have the ability to decode the contents of the communication. We obviously want to keep the message secret. Sounds familiar? A common solution to this problem is to use encryption. An encryption scheme takes a message and transforms it into an unreadable format so that an eavesdropper can’t read it. Now what if we don’t want anyone to find out that there is communication going on? As in, if the attackers don’t know that something is going on, then there are lesser chances of getting attacked right? How do we achieve this? Are there techniques to that allow us the hide this information? Continue reading