The Portlucio and Calvo sculptor turned into an architect must have attracted some curious looks while establishing a complicated painting team in the corner of a Renaissance era. He planted his instrument, which involved a easel, a mirror and a wire frame, near the then Cathedral of Florence of Florence in Italy, a cathedral whose monumental dome would soon design.
It was called Filippo Brunelleschi, and was using the team to create a baptism painting near the cathedral. It is said that this demonstration of its recently discovered perspective laws occurred at some point between 1415 and 1420, if your biographers are right. The use of perspective laws amazed spectators, altered the course of Western art for more than 450 years and, more recent, led to mathematical discoveries that allow cryptography of the elliptical curve. This is the security scheme that supports Bitcoin and other cryptocurrencies and has also become a rapidly growing encryption method on other Internet platforms.
But how did the art of life lead to mathematics that modern government cryptography? The story covers six centuries and two continents and touches the infinity itself. His characters include a French war prisoner and two mathematicians attacked at his best, one by illness and the other by the gun of a duelist.
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Merging perspective and geometry
The first steps on the way from Brunelleschi to Bitcoin involved connecting visual geometry within the rules of perspective to Euclidean geometry, the nesting kingdom of lines and points that teach us in school.
The French mathematician Girard Desargues, who investigated the geometry of the perspective in the seventeenth century, was the first taxpayer. His findings, however, were expressed in a fairly dark language and fought to find a audience. His key contributions were included in a book that had an impression of 50 copies, even for that time, and many of those copies were possible bought by the editor and destroyed. Duration of unarggest life, only his French mathematical partner Blaise Pascal became a burning disciple of his work. Pascal contributed his own theorem to the study of what was known as “projective geometry.”
Despite the darkness of disassembly, he made a revolutionary advance by adding the concept of points and lines in 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 the disarranger system, each The pair of lines meets exactly at one point, without special exceptions for parallel lines. In addition, the parables and hyperbolas are equivalent to the ellipses, with the addition of one or two points in the infinity, respectively.
These ideas, designed valuable, would languish in the dark for more than 100 years. When they traveled, it was not because the unattended work was rediscovered. Rather, a different French mathematician, Gaspard Monge, began working on the same questions and obtained similar results.
A mathematician in war
However, the most complete work on projective geometry at this time occurred in the nineteenth century of the French and mathematician engineer Jean-Victor Poncelet, in quite intentional circumstances.
Poncelet attended the prestigious Polytechnique Ecole of France, graduating in 1810. Then he joined the Corps of Military Engineers of France as a lieutenant and was ordered to what is now Belarus to support Napoleon for the invasion of Russia in 1812. He and his companions of the troops oppose a burned and abandoned Moscow in September in September in September in September in September in September in September in September. Russians referred to Russians referred to the people who make people who make the people who make people russians, Russia, when Russians referred to the peace of the city, Russians, Russia, Russia, Russia, Russia. Poncelet was with Napoleon when the army left Moscow and the return to France.
Poncelet remained with the French army until the battle of Krasnoye in Russia, where he was separated from his unity and possible went by dead. After the battle, he was collected by the Russian army and marched to Saratov, Russia, more than 700 miles from Krasnoyo and more than 2,000 miles from his house in Metz, France.
Althegh Poncelet did not limit himself to a prison, he was “deprived of books and comforts of all kinds”, 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 reconstruct all the mathematics he had learned until that time. I could not carry out this plan, however, saying that I was “distressed above all by the misfortune of my country and my own luck.”
Instead, it expanded essentially in the work of Monge and recreates the work of disarrk independently. In retrospect, perhaps it is not surprising that a prisoner of war to thousands of miles from home and insecure when, or even if it were repatriated, he would focus his efforts on understanding the points in infinity, a distance that he could have in a powerful situation.
After the war that had caused bone for that invasion ended, Poncelet returned to France and his work of two volumes on projective geometry, published in 1822, was much rather retired and widely read than the work of the disassembly.
Integral and curves
Around the same time, Poncelet was finishing his book on projective geometry, Norwegian mathematician Niels Henrik Abel was studying elliptical integrals. These integrals are quite difficult expressions that began as parts of an attempt to measure the circumference of an ellipse. Abel discovered that they are certain circumstances in which the inverse of these elliptical integrals could be used, which are called elliptical curves. It turned out that curves are much easier to work. However, additional research on elliptical curves would be left to others; Abel died of tuberculosis at age 26 in 1829, only months after publishing an important article on the subject.
At the beginning of the 1830s, the French mathematician Évarist Galois laid the foundations for a new mathematics field. Galois would be so tragic but also stubbornly in a duel at age 20, but before his death he exposed 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 would fall before a German mathematician, August Möbius (or Fame of Möbius Strip) to discover how to merge projective geometry with the family of the Cartesian coordinate system. The system that developed, which are called homogeneous coordinates, plays a fundamental role in the cryptography of the elliptical curve.
Several decades later, in 1901, another French mathematician, Henri Poincaré, realized that points with rational coordinates, is that, points with coordinates that can be represented as fractions in the graphic of an elliptical curve composed of a group. What Poincaré realized is that if he defined an operation (typically called “addition”) that touched two rational points in the graph of the curve and produced a third, the result was always the rational point in the curve. This process only used the homogeneous coordinates discovered by Möbius that include a point in Infinity, Howver. It is important to note that the elliptical curve groups turned out to be Abelianas, whose order is that the order in which these addition operations were carried out did not matter.
This is where the affairs were until the mid -1980s, some Victor S. Miller, then IBM researcher, and Neal Koblitz, from the University of Washington, realized regardless of whether a private public key could be made.
Encryption keys
Public-private key encryption, which is how almost all Internet traffic is ensured, is based on two encryption keys. The first key, a private one, does not share with anyone; It remains safely on the sender’s device. The second key, the public, consists of the private key, and this key is the “clear” sentence, which means that anyone can intercept and read it. It is important to note that both keys must decipher the message.
In the cryptography of the elliptical curve, each part agrees in a certain curve, and then each one performs a random number of addition operations that begin from the same point in the same curve. Each part sends a number corresponding to the point that have reached the other. These are public keys. The other part, then, performs the same addition operations that used the first time in the new number they recovered.
Because the groups of elliptical curves are commutative, which means that it does not matter in which order the addition is carried out, both parties will reach a number corresponding to the same final point in the curve, and this is the number and the data.
The cryptography of the elliptical curve is a relative recruitment of the encryption game. The first set of tools did not appear until 2004, too late to become a standard for the web, but early enough to adopt Bitcoin inventors, which was launched in 2009.
His status as the de facto standard for cryptocurrencies made people more familiar and a more comfortable implementation, although RSA’s encryption is still behind, the standard method in use today, by a wide margin.
However, cryptography of the elliptical curve has different advantages in RSA cryptography: it provides stronger security for bit and is faster than RSA. A cryptographic key of the elliptical curve of only 256 bits is approximately as safe as a 3,072 -bit RSA key and considerably safer than the 2,048 -bit keys that are commonly used. These shorter keys allow a faster page representation for web traffic, and there is less processor load on the server side. The principles of cryptography of the elliptical curve are being used to try to develop cryptographic systems that are more resistant to quantity.
If continuous trends, mathematics behind the escape point discovered by the Renaissance artists 600 years can be a fundamental part of the Internet encryption in the future.
