# elliptic curve cryptography explained

Let Alice's key pair be $(d_A,Q_A)$ and Bob's key pair be $(d_B,Q_B)$. This method is called double and add. ECC appears to offer a better tradeoff: high security with short, fast keys. What background do you already have? Only time will show. Adding two points on the curve, A and B, is our Billiards shot. But how can they exchange this secret coordinate if the enemy is eavesdropping? Most of it is taken from the Wiki links suggested by D.W. Elliptic curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. Consequently, ECC is particularly appropriate for embedded devices and other systems where performance is at a premium. Factoring is a well-known problem and has been studied since antiquity like the Sieve of Eratosthenes for finding all prime numbers up to any given limit. Once upon a time, in a land far, far away, there lived two men by the name of Neal Koblitz and Victor S. Miller. Feedbacks are welcome. Here’s the value printed out from the program I wrote: As you can see, ab_c and a_bc are mostly identical. And we flip $$y$$ value $$8$$ to be $$-8$$ and mod by $$19$$ will get you $$(18, 11)$$. I assume that those who are going through this article will have a basic understanding of cryptography (terms like encryption and decryption). It was discovered by Victor Miller of IBM and Neil Koblitz of the University of Washington in the year 1985. How do we decide when a small sample is statistically significant or not? It turns out that the only way for Eve to discover that number d is by replaying the game shot after shot until the table reaches the same state. Elliptic Curve Cryptography Encryption Results. It was discovered by Victor Miller of IBM and Neil Koblitz of the University of Washington in the year 1985. One area of data protection that will be affected by quantum computing capabilities is encryption because current encryption algorithms will become obsolete since it could be deciphered in essentially less time. Think about this, Alice and Bob are traveling in the space, they are about to exchange the location of rebellion’s new HQ. This time we start from last point $$A + B$$ to another point $$C$$. To subscribe to this RSS feed, copy and paste this URL into your RSS reader. How it works depends on the cryptographic scheme you apply it to. Both ways lead to the same result. There are some widely used cryptographic algorithms which need a finite, cyclic group (a finite set of element with a composition law which fulfils a few characteristics), e.g. Following Cloudflare’s Nick Sullivan blog’s terminology, Elliptic Curve Cryptography (ECC) can be described as a bizzaro Billiards game. How many steps you jump, the value $$N$$ is your private key. That is the main reason for us in ZenGo to open source our cryptography algorithms implementations and have them audited. And Bob is jumping 3 lightyears a time, and he jumps 4 times, $$4 \times 3$$ or $$3 \times 4$$ they are all going to end up at $$12$$: For eavesdroppers, they need to find out $$N$$ or $$M$$ to be able to get the same coordinate. Now we know how to jump forward on the elliptic curve in warp speed, with that, we can jump forward bazillion times easily. The ECC schemes are probably faster. While multiplying two prime numbers together is easier than factoring the product into its component parts, when the prime numbers start to get very long, even just the multiplication step can take some time on a low powered device. This is not a sustainable situation for mobile and low-powered devices that have limited computational power. What is so special about elliptic curves? In the end, I didn't find an article that really explains it from end-to-end in an institutive way. Limitations of Elliptic Curve Cryptography? Seriously, though, the following explanation requires that you have a basic understanding of finite fields. site design / logo © 2020 Stack Exchange Inc; user contributions licensed under cc by-sa. Do you want to know how the mathematics works? (The coordinates here are to be chosen from a fixed finite field of characteristic not equal to 2 or 3, or the curve equation will be somewhat more complicated.) To lose your enemy, you are taking a straight shortcut off the route to another point on the route, bounce to the other side of the route sharply once you hit the third point. Also, involved mathematics are a bit harder than with RSA, and there have been patents, so implementers are a bit wary. On the other hand, ECC is newer than some other well-known alternatives, and there is a bit of a patent minefield surrounding some kinds of elliptic-curve cryptography, so ECC hasn't seen as much deployment as classic RSA/DSA/El Gamal -- but ECC is used in the wild in some systems. Suppose Alice wants to establish a shared key with Bob, but the only channel available for them may be eavesdropped by a third party. Basically, $$y^{2}\pmod{p}$$ will be equal to $$x^{3} + a x + b\pmod{p}$$ only on specific integer points. You can think of this process is some kind of space travel. Each party must have the other party's public key (an exchange must occur). I wanted to add a couple of very handy references. Which theoretical propulsion system has the highest specific impulse? Elliptic Curve Cryptography as a Billiards Game. With that in mind, I would like to write a post explaining Elliptic Curve Cryptography, cover from the basics to key exchange, encryption, and decryption. Suddenly, they realized there are drones from the empire are tailing them and intercepting the communication between their spacecraft. This is it, since we can double a point easily, and the order of adding it doesn’t matter. Open source and radically transparent. That’s it, now you know how Elliptic Curve Cryptography works. I recommend that you start by reading the description of elliptic curve cryptography in Wikipedia, and then let us know what you'd like to know: What didn't you understand? To make it easier to understand, personally I really like the presentation of it in 3D space as a donut shape, like the one from Trustica video series about ECC: But given that I am writing an article, it’s easier to present it in a simple 2D image. As a result, Bob was able to verify that the message was indeed generated by Alice based only on public data. Note, that with this new base-point, Eve now has a new private key d’ that corresponds to the original public key Q, i.e Q = d×G but also Q = d’×G’ = d’×Q/d’=Q. Here’s the critical one, remember how we get to $$3P$$ via $$P + 2P$$? The attacker can then perform man-in-middle attack. given a starting point P and final point NP, can you find what is the N value? We also don't want to dig too deep into the mathematical rabbit hole, I only want to focus on getting the sense of how it works essentially. In other words, unlike with factoring, based on currently understood mathematics, there doesn't appear to be a shortcut that is narrowing the gap in a Trapdoor Function based around this problem. Recently, I am learning how Elliptic Curve Cryptography works. The text for provides an excellent theoretical definition of the algorithm but I'm having a hard time understanding all of the theory involved in ECC. And please don’t forget to patch your vulnerable Windows machines! add a Point to itself (A+A). At first, let’s see doing the calculation in a silly way, just keep adding $$P$$. A curve equation, usually $y^2 = x^3 + ax +b$, where $a$ and $b$ are constant values from the finite field. For every public-key cryptosystem you already know of, there are alternatives based upon elliptic curve cryptography (ECC). Simply speaking, the bigger the spread between the difficulty of going one direction in a Trapdoor Function and going the other, the more secure a cryptographic system based on it will be. Let’s try to add up to a slightly bigger number, say $$227P$$, let’s first break it down to binary so that we can get its power of two composition.