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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.
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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.
Rather than allow any value for the points on the curve, we restrict ourselves to whole numbers in a fixed range. If we pick the maximum to be a prime number, the elliptic curve is called a prime curve and has excellent cryptographic properties. Here's the plot of the same curve with only the whole number points represented with a maximum of This hardly looks like a curve in the traditional sense, but it is. It's like the original curve was wrapped around at the edges and only the parts of the curve that hit whole number coordinates are colored in.
You can even still see the horizontal symmetry. In fact, you can still play the billiards game on this curve and dot points together. The equation for a line on the curve still has the same properties. Moreover, the dot operation can be efficiently computed. You can visualize the line between two points as a line that wraps around at the borders until it hits a point. It's as if in our bizarro billiards game, when a ball hits the edge of the board the max then it is magically transported to the opposite side of the table and continues on its path until reaching a point, kind of like the game Asteroids.
With this new curve representation, you can take messages and represent them as points on the curve. You could imagine taking a message and setting it as the x coordinate, and solving for y to get a point on the curve. It is slightly more complicated than this in practice, but this is the general idea. An elliptic curve cryptosystem can be defined by picking a prime number as a maximum, a curve equation and a public point on the curve.
A private key is a number priv , and a public key is the public point dotted with itself priv times. Computing the private key from the public key in this kind of cryptosystem is called the elliptic curve discrete logarithm function.
This turns out to be the Trapdoor Function we were looking for. The elliptic curve discrete logarithm is the hard problem underpinning elliptic curve cryptography. Despite almost three decades of research, mathematicians still haven't found an algorithm to solve this problem that improves upon the naive approach. 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.
This means that for numbers of the same size, solving elliptic curve discrete logarithms is significantly harder than factoring. Since a more computationally intensive hard problem means a stronger cryptographic system, it follows that elliptic curve cryptosystems are harder to break than RSA and Diffie-Hellman.
Nieuwe artikelen van deze auteur. The gap between the difficulty of factoring large numbers and multiplying large numbers is shrinking as the number i. Then pick a number to be the public key pub. Moreover, the dot operation can be efficiently computed. 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.
To visualize how much harder it is to break, Lenstra recently introduced the concept of " Global Security. This is a kind of cryptographic carbon footprint. By this measure, breaking a bit RSA key requires less energy to than it takes to boil a teaspoon of water. Comparatively, breaking a bit elliptic curve key requires enough energy to boil all the water on earth.
For this level of security with RSA, you'd need a key with 2,bits. With ECC, you can use smaller keys to get the same levels of security. Small keys are important, especially in a world where more and more cryptography is done on less powerful devices like mobile phones. 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. While you could likely continue to keep RSA secure by increasing the key length that comes with a cost of slower cryptographic performance on the client.
ECC appears to offer a better tradeoff: Elliptic curves in action After a slow start, elliptic curve based algorithms are gaining popularity and the pace of adoption is accelerating. De gecombineerde citaties worden alleen voor het eerste artikel geteld. Het aantal in de tabel 'Geciteerd door' omvat citaties van de volgende artikelen in Scholar.
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