Posts Tagged ‘cryptography’
“If geometry is dressed in a suit coat, topology dons jeans and a T-shirt”*…
Paulina Rowińska on how, in the mid-19th century, Bernhard Riemann conceived of a new way to think about mathematical spaces, providing the foundation for modern geometry and physics…
Standing in the middle of a field, we can easily forget that we live on a round planet. We’re so small in comparison to the Earth that from our point of view, it looks flat.
The world is full of such shapes — ones that look flat to an ant living on them, even though they might have a more complicated global structure. Mathematicians call these shapes manifolds. Introduced by Bernhard Riemann in the mid-19th century, manifolds transformed how mathematicians think about space. It was no longer just a physical setting for other mathematical objects, but rather an abstract, well-defined object worth studying in its own right.
This new perspective allowed mathematicians to rigorously explore higher-dimensional spaces — leading to the birth of modern topology, a field dedicated to the study of mathematical spaces like manifolds. Manifolds have also come to occupy a central role in fields such as geometry, dynamical systems, data analysis and physics.
Today, they give mathematicians a common vocabulary for solving all sorts of problems. They’re as fundamental to mathematics as the alphabet is to language. “If I know Cyrillic, do I know Russian?” said Fabrizio Bianchi, a mathematician at the University of Pisa in Italy. “No. But try to learn Russian without learning Cyrillic.”
So what are manifolds, and what kind of vocabulary do they provide?…
[Rowińska explains manifolds and the history of the development of our understanding of them, concentrating on the pivotal role of Riemann…]
… Manifolds are crucial to our understanding of the universe… In his general theory of relativity, Einstein described space-time as a four-dimensional manifold, and gravity as that manifold’s curvature. And the three-dimensional space we see around us is also a manifold — one that, as manifolds do, appears Euclidean to those of us living within it, even though we’re still trying to figure out its global shape.
Even in cases where manifolds don’t seem to be present, mathematicians and physicists try to rewrite their problems in the language of manifolds to make use of their helpful properties. “So much of physics comes down to understanding geometry,” said Jonathan Sorce, a theoretical physicist at Princeton University. “And often in surprising ways.”
Consider a double pendulum, which consists of one pendulum hanging from the end of another. Small changes in the double pendulum’s initial conditions lead it to carve out very different trajectories through space, making its behavior hard to predict and understand. But if you represent the configuration of the pendulum with just two angles (one describing the position of each of its arms), then the space of all possible configurations looks like a doughnut, or torus — a manifold. Each point on this torus represents one possible state of the pendulum; paths on the torus represent the trajectories the pendulum might follow through space. This allows researchers to translate their physical questions about the pendulum into geometric ones, making them more intuitive and easier to solve. This is also how they study the movements of fluids, robots, quantum particles and more.
Similarly, mathematicians often view the solutions to complicated algebraic equations as a manifold to better understand their properties. And they analyze high-dimensional datasets — such as those recording the activity of thousands of neurons in the brain — by looking at how those data points might sit on a lower-dimensional manifold.
Asking how scientists use manifolds is akin to asking how they use numbers, Sorce said. “They are at the foundation of everything.”…
“What Is a Manifold?” from @quantamagazine.bsky.social.
Apposite: Rowińska in conversation with Ira Flatow on Science Friday: “How Math Helps Us Map The World.”
* David S. Richeson, Euler’s Gem: The Polyhedron Formula and the Birth of Topology (Riemann’s work was an advance on the foundation that Euler laid in his 1736 paper on the Seven Bridges of Königsberg, which led to his polyhedron formula)
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As we get down with geometry, we might spare a thought for John Wallis; he died on this date in 1703. A clergyman and mathematician, he served as chief cryptographer for Parliament (decoding Royalist messages during the Civil War) and, later (as Savilian Chair of geometry at Oxford after the hostilities), for the the royal court. Wallis is credited with introducing the symbol ∞ to represent the concept of infinity, and used 1/∞ for an infinitesimal… which earned him (along with his contemporaries Isaac Newton and Gottfried Wilhelm Leibniz) a share of the credit for the development of infinitesimal calculus. He was a founding member of the Royal Society and one of its first Fellows.
“Make visible what, without you, might perhaps never have been seen”*…
Lawrence Weschler is no stranger to controversy. In 2000 he published an article in The New Yorker, recounting a theory that David Hockey had shared with him, that ignited a fire storm in the art world– and that burns (or at least smolders) to this day.
And he’s at it again…
A few months back—in the lee of the Rijksmuseum’s epic Vermeer show and Ren’s [Wechsler’s] controversial Atlantic magazine article (featured in our Issue #39) on Vermeer and Benjamin Binstock’s intriguing contention that eight of the thirty-four paintings conventionally attributed to the Delft master were in fact by his daughter Maria—the eminent curator Helen Molesworth invited Ren and Claudia Swan (the historian behind Rarities of These Lands and other classics on the Dutch Golden Age) to engage in a conversation evaluating both that show and Binstock’s thesis for an episode of her ongoing Dialogues podcast, out of the David Zwirner Gallery. And indeed, that half-hour episode dropped yesterday—and we thought you might enjoy hearing it here. Spoiler alert: Two of the top people in the field seem decidedly open to Binstock’s theory…
Fascinating: “Vermeer’s Daughter?”
* Robert Bresson
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As we argue over attribution, we might send grateful birthday greetings to Leon Battista Alberti; he was born on this date in 1404. The archetypical Renaissance humanist polymath, Alberti was an author, artist, architect, poet, priest, linguist, philosopher, cartographer, and cryptographer. Indeed, with Johannes Trithemius, he is considered the father of cryptography. And he collaborated with Toscanelli on the maps used by Columbus on his first voyage.
But he is surely best remembered as the man who “wrote the book” on perspective: he authored of the first general treatise– De Pictura (1434)– on the the laws of perspective, which built on and extended Brunelleschi’s work to describe the approach and technique that established the science of projective geometry… and fueled the progress of painting, sculpture, and architecture from the Greek- and Arabic-influenced formalism of the High Middle Ages to the more naturalistic (and Latinate) styles of Renaissance.
“There are two types of encryption: one that will prevent your sister from reading your diary and one that will prevent your government”*…
… But sometimes the encryption you think will work against governments won’t even deter your sister. Joesph Cox on the recently-uncovered vulnerabilities in TETRA, the encryption standard used in radios worldwide…
A group of cybersecurity researchers has uncovered what they believe is an intentional backdoor in encrypted radios used by police, military, and critical infrastructure entities around the world. The backdoor may have existed for decades, potentially exposing a wealth of sensitive information transmitted across them, according to the researchers… The end result, however, are radios with traffic that can be decrypted using consumer hardware like an ordinary laptop in under a minute…
The research is the first public and in-depth analysis of the TErrestrial Trunked RAdio (TETRA) standard in the more than 20 years the standard has existed. Not all users of TETRA-powered radios use the specific encryption algorithim called TEA1 which is impacted by the backdoor. TEA1 is part of the TETRA standard approved for export to other countries. But the researchers also found other, multiple vulnerabilities across TETRA that could allow historical decryption of communications and deanonymization. TETRA-radio users in general include national police forces and emergency services in Europe; military organizations in Africa; and train operators in North America and critical infrastructure providers elsewhere.
Midnight Blue [presented] their findings at the Black Hat cybersecurity conference in August. The details of the talk have been closely under wraps, with the Black Hat website simply describing the briefing as a “Redacted Telecom Talk.” That reason for secrecy was in large part due to the unusually long disclosure process. Wetzels told Motherboard the team has been disclosing these vulnerabilities to impacted parties so they can be fixed for more than a year and a half. That included an initial meeting with Dutch police in January 2022, a meeting with the intelligence community later that month, and then the main bulk of providing information and mitigations being distributed to stakeholders. NLnet Foundation, an organization which funds “those with ideas to fix the internet,” financed the research.
The European Telecommunications Standards Institute (ETSI), an organization that standardizes technologies across the industry, first created TETRA in 1995. Since then, TETRA has been used in products, including radios, sold by Motorola, Airbus, and more. Crucially, TETRA is not open-source. Instead, it relies on what the researchers describe in their presentation slides as “secret, proprietary cryptography,” meaning it is typically difficult for outside experts to verify how secure the standard really is.
…
Bart Jacobs, a professor of security, privacy and identity, who did not work on the research itself but says he was briefed on it, said he hopes “this really is the end of closed, proprietary crypto, not based on open, publicly scrutinised standards.”…
The veil, pierced: “Researchers Find ‘Backdoor’ in Encrypted Police and Military Radios,” from @josephfcox in @motherboard. (Not long after this article ran– and after the downfall of Vice, Motherboard’s parent), Cox and a number of his talented Motherboard colleagues launched 404 Media. Check it out.)
Remarkably, some of the radio systems enabling critical infrastructure are even easier to hack– they aren’t even encrypted.
* Bruce Schneier (@schneierblog)
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As we take precautions, we might recall that it was on this date in 1980 that the last IBM 7030 “Stretch” mainframe in active use is decommissioned at Brigham Young University. The first Stretch was was delivered to Los Alamos National Laboratory in 1961, giving the model almost 20 years of operational service.
The Stretch was famous for many things, but perhaps most notably it was the first IBM computer to use transistors instead of vacuum tubes; it was the first computer to be designed with the help of an earlier computer; and it was the world’s fastest computer from 1961 to 1964.
“The sciences of cryptography and mathematics are very elegant, pure sciences. I found that the ends for which these pure sciences are used are less elegant.”*…
Mary, Queen of Scots wrote 57 encrypted messages during her captivity in England; until recently, all but 7 of them were believed lost. Meilan Solly tells the tale of their discovery and decryption…
Over the course of her 19 years in captivity, Mary, Queen of Scots, wrote thousands of letters to ambassadors, government officials, fellow monarchs and conspirators alike. Most of these missives had the same underlying goal: securing the deposed Scottish queen’s freedom. After losing her throne in 1567, Mary had fled to England, hoping to find refuge at her cousin Elizabeth I’s court. (Mary’s paternal grandmother, Margaret Tudor, was the sister of Elizabeth’s father, Henry VIII.) Instead, the English queen imprisoned Mary, keeping her under house arrest for nearly two decades before ordering her execution in 1587.
Mary’s letters have long fascinated scholars and the public, providing a glimpse into her relentless efforts to secure her release. But the former queen’s correspondence often raises more questions than it answers, in part because Mary took extensive steps to hide her messages from the prying eyes of Elizabeth’s spies. In addition to folding the pages with a technique known as letterlocking, she employed ciphers and codes of varying complexity.
More than 400 years after Mary’s death, a chance discovery by a trio of code breakers is offering new insights into the queen’s final years. As the researchers write in the journal Cryptologia, they originally decided to examine a cache of coded notes housed at the National Library of France as part of a broader push to “locate, digitize, transcribe, decipher and analyze” historic ciphers. Those pages turned out to be 57 of Mary’s encrypted letters, the majority of which were sent to Michel de Castelnau, the French ambassador to England, between 1578 and 1584. All but seven were previously thought to be lost…
What they found and how they made sense of it: “Code Breakers Discover—and Decipher—Long-Lost Letters by Mary, Queen of Scots,” from @meilansolly in @SmithsonianMag.
* Jim Sanborn, the sculptor who created the encrypted Kryptos sculpture at CIA headquarters
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As we crack codes, we might spare a thought for a rough contemporary of Mary’s, a man who refused to communicate in code: Giordano Bruno. A Dominican friar, philosopher, mathematician, and astronomer whose concept of the infinite universe expanded on Copernicus’s model, he was the first European to understand the universe as a continuum where the stars we see at night are identical in nature to the Sun. Bruno’s views were considered dangerously heretical by the (Roman) Inquisition, which imprisoned him in 1592; after eight years of refusals to recant, on this date in 1600, he was burned at the stake.
“I was a peripheral visionary. I could see the future, but only way off to the side.”*…
As Niels Bohr said, “prediciton is hard, especially about the future.” Still, we can try…
While the future cannot be predicted with certainty, present understanding in various scientific fields allows for the prediction of some far-future events, if only in the broadest outline. These fields include astrophysics, which studies how planets and stars form, interact, and die; particle physics, which has revealed how matter behaves at the smallest scales; evolutionary biology, which studies how life evolves over time; plate tectonics, which shows how continents shift over millennia; and sociology, which examines how human societies and cultures evolve.
The far future begins after the current millennium comes to an end, starting with the 4th millennium in 3001 CE, and continues until the furthest reaches of future time. These timelines include alternative future events that address unresolved scientific questions, such as whether humans will become extinct, whether the Earth survives when the Sun expands to become a red giant and whether proton decay will be the eventual end of all matter in the Universe…
A new pole star, the end of Niagara Falls, the wearing away of the Canadian Rockies– and these are just highlights from the first 50-60 million years. Read on for an extraordinary outline of what current science suggests is in store over the long haul: “Timeline of the far future,” a remarkable Wikipedia page.
Related pages: List of future astronomical events, Far future in fiction, and Far future in religion.
* Steven Wright
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As we take the long view, we might send grateful birthday greetings to the man who “wrote the book” on perspective (a capacity analogically handy in the endeavor featured above), Leon Battista Alberti; he was born on this date in 1404. The archetypical Renaissance humanist polymath, Alberti was an author, artist, architect, poet, priest, linguist, philosopher, cartographer, and cryptographer. He collaborated with Toscanelli on the maps used by Columbus on his first voyage, and he published the the first book on cryptography that contained a frequency table.
But he is surely best remembered as the author of the first general treatise– De Pictura (1434)– on the the laws of perspective, which built on and extended Brunelleschi’s work to describe the approach and technique that established the science of projective geometry… and fueled the progress of painting, sculpture, and architecture from the Greek- and Arabic-influenced formalism of the High Middle Ages to the more naturalistic (and Latinate) styles of Renaissance.
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