The portly, balding sculptor-turned-architect must have drawn a few curious gazes as he set up a complicated painting apparatus in the corner of a Renaissance-era piazza. He planted his instrument, which involved an easel, a mirror and a wire framework, near the then unfinished cathedral of Florence in Italy—a cathedral whose monumental dome he would soon design.
His name was Filippo Brunelleschi, and he was using the apparatus to create a painting of the baptistry near the cathedral. This demonstration of his recently discovered laws of perspective is said to have occurred sometime between 1415 and 1420, if his biographers are correct. The use of the laws of perspective amazed bystanders, altered the course of Western art for more than 450 years and, more recently, led to mathematical discoveries that enable elliptic curve cryptography. This is the security scheme that underpins Bitcoin and other cryptocurrencies and has become a fast-growing encryption method on other Internet platforms as well.
But how did Renaissance art lead to the mathematics that govern modern cryptography? The tale spans six centuries and two continents and touches on infinity itself. Its characters include a French prisoner of war and two mathematicians struck down in their prime—one by illness and the other by a duelist’s pistol.
On supporting science journalism
If you’re enjoying this article, consider supporting our award-winning journalism by subscribing. By purchasing a subscription you are helping to ensure the future of impactful stories about the discoveries and ideas shaping our world today.
Merging Perspective and Geometry
The first steps in the path from Brunelleschi to Bitcoin involved connecting the visual geometry within the rules for perspective to Euclidean geometry, the orderly realm of lines and points that we’re taught in school.
French mathematician Girard Desargues, who researched the geometry of perspective in the 17th century, was the first contributor. His findings, however, were couched in rather obscure language and struggled to find an audience. His key contributions were included in a book that had a print run of 50 copies, small even for that era, and many of those copies were eventually bought back by the publisher and destroyed. During Desargues’s lifetime, only fellow French mathematician Blaise Pascal became an ardent disciple of his work. Pascal contributed his own theorem to the study of what became known as “projective geometry.”
Despite Desargues’s obscurity, he made a revolutionary breakthrough by adding the concept of points and lines at infinity to Euclidean geometry. By including those points, projective geometry could be merged with Euclidean geometry in a way that was consistent for both systems.
In Desargues’s system, every pair of lines meets at exactly one point, with no special exceptions for parallel lines. Furthermore, parabolas and hyperbolas are equivalent to ellipses, with the addition of one or two points at infinity, respectively.
These insights, though valuable, would languish in obscurity for more than 100 years. When they reappeared, it was not because Desargues’s work was rediscovered. Rather, a different French mathematician, Gaspard Monge, began to work on the same questions and derived similar results.
A Mathematician at War
The most comprehensive work on projective geometry in this era, however, came in the 19th century from French engineer and mathematician Jean-Victor Poncelet, under rather trying circumstances.
Poncelet attended France’s prestigious École Polytechnique, graduating in 1810. He then joined France’s corps of military engineers as a lieutenant and was ordered to what is now Belarus to support Napoleon’s invasion of Russia in 1812. He and his fellow troops overran a burned out and abandoned Moscow in September of that year, and when the Russians refused to sue for peace after losing the city, Poncelet was with Napoleon when the army left Moscow and began the return to France.
Poncelet remained with the French army right up to the Battle of Krasnoye in Russia, where he was separated from his unit and possibly left for dead. After the battle, he was scooped up by the Russian army and marched to Saratov, Russia, more than 700 miles from Krasnoye and more than 2,000 miles from his home in Metz, France.
Although Poncelet was not confined to a prison, he was “deprived of books and comforts of all sorts,” according to an English translation of his introduction to his first book on projective geometry. As a coping mechanism, he decided that he would try to redevelop all the math he had learned up to that point. He could not carry out this plan, however, saying that he was “distressed above all by the misfortune of my country and my own lot.”
Instead he essentially expanded on Monge’s work and recreated Desargues’s work independently. In hindsight, it is perhaps not surprising that a prisoner of war thousands of miles from home and unsure of when, or even if, he would be repatriated would focus his efforts on understanding points at infinity—a distance that might have seemed quite intelligible to someone in Poncelet’s situation.
After the war that had been sparked by that invasion ended, Poncelet returned to France and his two-volume work on projective geometry, published in 1822, was far more well-received and widely read than Desargues’s work.
Integrals and Curves
At around the same time that Poncelet was finishing his book on projective geometry, Norwegian mathematician Niels Henrik Abel was studying elliptic integrals. These integrals are rather difficult expressions that started off as parts of an attempt to measure the circumference of an ellipse. Abel discovered that there are certain circumstances where the inverse of these elliptic integrals—what are called elliptic curves—could be used instead. The curves, it turned out, are much easier to work with. Further research into elliptic curves would be left to others, however; Abel died from tuberculosis at age 26 in 1829, mere months after publishing an important paper on the subject.
In the early 1830s French mathematician Évariste Galois laid the groundwork for a new field of mathematics. Galois would die tragically but also stubbornly in a duel at age 20, but before his death he laid out the principles of group theory, in which mathematical objects and operations that follow certain rules constitute a group.
The French had managed to unite projective geometry with Euclidean geometry, but it would fall to a German mathematician, August Möbius (of Möbius strip fame) to figure out how to merge projective geometry with the Cartesian coordinate system familiar to algebra students as a means of graphing equations. The system he developed, which uses what are called homogeneous coordinates, play a pivotal role in elliptic curve cryptography.
Several decades later, in 1901, another French mathematician, Henri Poincaré, realized that points with rational coordinates—that is, points with coordinates that can be represented as fractions on the graph of an elliptic curve—composed a group. What Poincaré realized is that if you defined an operation (typically called “addition”) that took two rational points on the graph of the curve and yielded a third, the result was alwaysanother rational point on the curve. This process onlyworked if you used the homogeneous coordinates discovered by Möbius that include a point at infinity, however. Importantly, elliptic curve groups turned out to be Abelian, which meant that the order in which those addition operations were performed didn’t matter.
This is where matters stood until the mid-1980s, when Victor S. Miller, then a researcher at IBM, and Neal Koblitz of the University of Washington independently realized that you could build a public-private key cryptographic system based off elliptic curve groups.
Encryption Keys
Public-private key encryption, which is how almost all traffic on the Internet is secured, relies on two encryption keys. The first key, a private one, is not shared with anyone; it is kept securely on the sender’s device. The second key, the public one, is composed from the private key, and this key is sent “in the clear,” meaning that anyone can intercept it and read it. Importantly, both keys are required to decrypt the message being sent.
In elliptic curve cryptography, each party agrees on a certain curve, and then each performs a random number of addition operations that start from the same point on the same curve. Each party then sends a number corresponding to the point they’ve arrived at to the other. These are the public keys. The other party then performs the same addition operations they used the first time on the new number they received.
Because elliptic curve groups are commutative, meaning that it doesn’t matter in what order addition is carried out, both parties will arrive at a number corresponding to the same final point on the curve, and this is the number that will be used to encrypt and decrypt the data.
Elliptic curve cryptography is a relative latecomer to the encryption game. The first suite of tools did not appear until 2004, far too late to become a standard for the Web but early enough to adopted by the inventors of Bitcoin, which launched in 2009.
Its status as the de facto standard for cryptocurrencies made people more familiar with it and more comfortable implementing it, although it still lags behind RSA encryption, the standard method in use today, by a wide margin.
Yet elliptic curve cryptography has distinct advantages over RSA cryptography: it provides stronger security per bit and is faster than RSA. An elliptic curve cryptographic key of just 256 bits is roughly as secure as a 3,072-bit RSA key and considerably more secure than the 2,048-bit keys that are commonly used. These shorter keys allow for faster page rendering for Web traffic, and there’s less processor load on the server side. Principles from elliptic curve cryptography are being used to try to develop cryptographic systems that are more quantum-resistant.
If trends continue, the mathematics behind the vanishing point discovered by Renaissance artists 600 years ago may turn out to be a fundamental part of Internet encryption in the future.
