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Elliptic Curve Cryptography ECC is one of the most powerful but least understood types of cryptography in wide use today. Fundamentally, we believe it's important to be able to understand the technology behind any security system in order to trust it. To that end, we looked around to find a good, relatively easy-to-understand primer on ECC in order to share with our users. Finding none, we decided to write one ourselves. That is what follows. In other words, settle in for a bit of an epic because there's a lot to cover.
If you just want the gist, the TL;DR is: ECC is the next generation of public key cryptography and, based on currently understood mathematics, provides a significantly more secure foundation than first generation public key cryptography systems like RSA.
If you're worried about ensuring the highest level of security while maintaining performance, ECC makes sense to adopt. If you're interested in the details, read on. The history of cryptography can be split into two eras: The turning point between the two occurred in , when both the RSA algorithm and the Diffie-Hellman key exchange algorithm were introduced.
These new algorithms were revolutionary because they represented the first viable cryptographic schemes where security was based on the theory of numbers; it was the first to enable secure communication between two parties without a shared secret. Cryptography went from being about securely transporting secret codebooks around the world to being able to have provably secure communication between any two parties without worrying about someone listening in on the key exchange.
Whitfield Diffie and Martin Hellman. Modern cryptography is founded on the idea that the key that you use to encrypt your data can be made public while the key that is used to to decrypt your data can be kept private. As such, these systems are known as public key cryptographic systems.
The first, and still most widely used of these systems, is known as RSA — named after the initials of the three men who first publicly described the algorithm: What you need for a public key cryptographic system to work is a set of algorithms that is easy to process in one direction, but difficult to undo.
In the case of RSA, the easy algorithm multiplies two prime numbers. If multiplication is the easy algorithm, its difficult pair algorithm is factoring the product of the multiplication into its two component primes. Algorithms that have this characteristic — easy in one direction, hard the other — are known as Tra door Functions.
Finding a good Trapdoor Function is critical to making a secure public key cryptographic system. The RSA algorithm is the most popular and best understood public key cryptography system.
Its security relies on the fact that factoring is slow and multiplication is fast. What follows is a quick walk-through of what a small RSA system looks like and how it works. In general, a public key encryption system has two components, a public key and a private key.
Encryption works by taking a message and applying a mathematical operation to it to get a random-looking number. Decryption takes the random looking number and applies a different operation to get back to the original number.
Encryption with the public key can only be undone by decrypting with the private key. Computers don't do well with arbitrarily large numbers.
We can make sure that the numbers we are dealing with do not get too large by choosing a maximum number and only dealing with numbers less than the maximum.
We can treat the numbers like the numbers on an analog clock. Any calculation that results in a number larger than the maximum gets wrapped around to a number in the valid range. In RSA, this maximum value call it max is obtained by multiplying two random prime numbers. The public and private keys are two specially chosen numbers that are greater than zero and less than the maximum value, call them pub and priv.
To encrypt a number you multiply it by itself pub times, making sure to wrap around when you hit the maximum. To decrypt a message, you multiply it by itself priv times and you get back to the original number. It sounds surprising, but it actually works. This property was a big breakthrough when it was discovered.
To create a RSA key pair, first randomly pick the two prime numbers to obtain the maximum max. Then pick a number to be the public key pub. As long as you know the two prime numbers, you can compute a corresponding private key priv from this public key. This is how factoring relates to breaking RSA — factoring the maximum number into its component primes allows you to compute someone's private key from the public key and decrypt their private messages.
Let's make this more concrete with an example. Take the prime numbers 13 and 7, their product gives us our maximum value of Let's take our public encryption key to be the number 5. Then using the fact that we know 7 and 13 are the factors of 91 and applying an algorithm called the Extended Euclidean Algorithm , we get that the private key is the number You can take a number and multiply it by itself 5 times to encrypt it, then take that number and multiply it by itself 29 times and you get the original number back.
In order to represent a message mathematically we have to turn the letters into numbers. A common representation of the Latin alphabet is UTF Each character corresponds to a number. Each of these digits are smaller than our maximum of 91, so we can encrypt them individually.
Let's start with the first letter. We do that by dividing by 91 and taking the remainder. Voila, we're back to This works with the rest of the digits, resulting in the original message. The takeaway is that you can take a number, multiply it by itself a number of times to get a random-looking number, then multiply that number by itself a secret number of times to get back to the original number. RSA and Diffie-Hellman were so powerful because they came with rigorous security proofs.
The authors proved that breaking the system is equivalent to solving a mathematical problem that is thought to be difficult to solve.
Factoring is a very well known problem and has been studied since antiquity see Sieve of Eratosthenes. Any breakthroughs would be big news and would net the discoverer a significant financial windfall. That said, factoring is not the hardest problem on a bit for bit basis.
Specialized algorithms like the Quadratic Sieve and the General Number Field Sieve were created to tackle the problem of prime factorization and have been moderately successful. These algorithms are faster and less computationally intensive than the naive approach of just guessing pairs of known primes. These factoring algorithms get more efficient as the size of the numbers being factored get larger. The gap between the difficulty of factoring large numbers and multiplying large numbers is shrinking as the number i.
As the resources available to decrypt numbers increase, the size of the keys need to grow even faster. This is not a sustainable situation for mobile and low-powered devices that have limited computational power. The gap between factoring and multiplying is not sustainable in the long term. All this means is that RSA is not the ideal system for the future of cryptography. In an ideal Trapdoor Function, the easy way and the hard way get harder at the same rate with respect to the size of the numbers in question.
We need a public key system based on a better Trapdoor. Building blocks of a better Trapdoor After the introduction of RSA and Diffie-Hellman, researchers explored other mathematics-based cryptographic solutions looking for other algorithms beyond factoring that would serve as good Trapdoor Functions. In , cryptographic algorithms were proposed based on an esoteric branch of mathematics called elliptic curves. But what exactly is an elliptic curve and how does the underlying Trapdoor Function work?
Unfortunately, unlike factoring — something we all had to do for the first time in middle school — most people aren't as familiar with the math around elliptic curves. The math isn't as simple, nor is explaining it, but I'm going to give it a go over the next few sections. If your eyes start to glaze over, you can skip way down to the section: What does it all mean. An elliptic curve is the set of points that satisfy a specific mathematical equation. The equation for an elliptic curve looks something like this:.
There are other representations of elliptic curves, but technically an elliptic curve is the set points satisfying an equation in two variables with degree two in one of the variables and three in the other.
An elliptic curve is not just a pretty picture, it also has some properties that make it a good setting for cryptography. One of these is horizontal symmetry. Any point on the curve can be reflected over the x axis and remain the same curve. A more interesting property is that any non-vertical line will intersect the curve in at most three places.
Let's imagine this curve as the setting for a bizarre game of billiards. Take any two points on the curve and draw a line through them, it will intersect the curve at exactly one more place.
In this game of billiards, you take a ball at point A, shoot it towards point B. When it hits the curve, the ball bounces either straight up if it's below the x-axis or straight down if it's above the x-axis to the other side of the curve.
We can call this billiards move on two points "dot. It turns out that if you have two points, an initial point "dotted" with itself n times to arrive at a final point, finding out n when you only know the final point and the first point is hard. To continue our bizzaro billiards metaphor, imagine one person plays our game alone in a room for a random period of time. It is easy for him to hit the ball over and over following the rules described above.
If someone walks into the room later and sees where the ball has ended up, even if they know all the rules of the game and where the ball started, they cannot determine the number of times the ball was struck to get there without running through the whole game again until the ball gets to the same point.
Easy to do, hard to undo: This simplified curve above is great to look at and explain the general concept of elliptic curves, but it doesn't represent what the curves used for cryptography look like. For this, we have to restrict ourselves to numbers in a fixed range, like in RSA.