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Peter Montgomery (mathematician)

From Wikipedia, the free encyclopedia

Peter L. Montgomery
Peter Montgomery in July 2009 at Microsoft Research.
Born(1947-09-25)September 25, 1947
DiedFebruary 18, 2020(2020-02-18) (aged 72)
NationalityAmerican
Alma mater
Known for
Scientific career
FieldsMathematics
Institutions
ThesisAn FFT extension of the elliptic curve method of factorization (1992)
Doctoral advisorDavid G. Cantor

Peter Lawrence Montgomery (September 25, 1947 – February 18, 2020) was an American mathematician who worked at the System Development Corporation and Microsoft Research. He is best known for his contributions to computational number theory and mathematical aspects of cryptography, including the Montgomery multiplication method for arithmetic in finite fields, the use of Montgomery curves in applications of elliptic curves to integer factorization and other problems, and the Montgomery ladder, which is used to protect against side-channel attacks in elliptic curve cryptography.[1][2]

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Transcription

Are we the only living thing in the entire universe? The observable universe is about 90,000,000,000 light years in diameter. There are at least 1,000,000,000 galaxies Each with 100,000,000,000 to 1,000,000,000,000 stars. Recently, we've learned that planets are very common too And there are probably trillions and trillions of habitable planets in the universe Which means there should be lot of opportunities for life to develop and exist, right? But where is it? Shouldn't the universe be teeming with spaceships? Let's take a step back. Even if there are aliens civilisations in other galaxies, there is no way we'll ever know about them. Basically, everything outside of our direct galactic neighborhood, the so called, "Local Group" is pretty much out of our reach forever, because of the expension of the universe. Even if we had really fast spaceships it would literally take billions of years to reach these places, travelling throught the emptiest areas in the universe. So, let's focus on the Milky Way. The Milky Way is our own galaxy, it consists of up to 4 hundred billions stars. That's a lot of stars, roughly 10 thousands for every grain of sand on earth. There are about 20 billions sun-like stars in the Milky Way and estimates suggest that a fivth of them have an earth-sized planet in its habitable zone, the area with conditions that enable life to exist. If only 0.1% of those planets harbored life, there would be 1 million planets with life in the Milky Way. But wait, there's more. The Milky Way is about 13 billion years old. In the beginning, it would not have been a good place for life because things exploded a lot, but after 1 to 2 billion years, the first habitable planets were born. Earth is only 4 billions years old, so there have probably been trillions of chances for life to develop on other planets in the past. If only a single one of them had developed into a space travelling super civilization we would have noticed by now. What would such a civilization look like? There are 3 categories. A Type 1 civilization would be able to access the whole energy available on its planet. In case you are wondering, we are currently around 0.73 on the scale and we should reach Type 1 sometime in the couple hundred of years. Type II would be a civilization capable of harnessing all of the energy of its home star. This would require some serious science fiction, but it is doable in principle. Concepts like the Dyson sphere, a giant complex surrounding the Sun would be conceivable. Type III is the civilization that basically controls its whole galaxy and its energy an alien race this advanced would probably be godlike to us. But why should we be able to see such an alien civilization in the first place? If we were to build generations of spaceships that could sustain a population for around one thousand years we could colonize the galaxy in 2 million years. Sounds like a long time, but remember, the Milky Way is huge. So, if it takes a couple of million years to colonize the entire galaxy and there are possibly millions if not billions of planets that sustain life in the Milky Way and these other life forms have had considerably more time than we've had, then where are all the aliens? This is the Fermi Paradox, and nobody has an answer to it But we do have some ideas. Let's talk about filters. A filter in this context represents a barrier that is really hard for life to overcome. They come in various degrees of scary. One: There are Great Filters and we've passed them. Maybe it is way harder for complex life to develop than we think. The process allowing life to begin hasn't yet been completely figured out and the conditions required may be really complicated. Maybe in the past the Universe was way more hostile, and only recently things have cooled down to make complex life possible This would also mean that we may be unique, or at least one of the first, if not the first civilization in the entire Universe. Two: There are Great Filters and they are ahead of us. This one would be really really bad. Maybe life on our level exists everywhere in the Universe but it gets destroyed when it reaches a certain point, a point that lies ahead of us. For example, awesome future technology exists, but when activated, it destroys the planet. The last words of every advanced civilization would be "This new device will solve all of our problems once I push this button." If this is true, then we are closer to the end than to the beginning of human existence. Or maybe there is an ancient Type III civilization that monitors the Universe and once a civilization is advanced enough it gets eliminated, in an instant. Maybe there is something out there that it would be better not to discover. There is no way for us to know. One final thought: maybe we are alone. Right now, we have no evidence that there's any life besides us. Nothing. The Universe appears to be empty and dead. No one sending us messages no one answering our calls. We may be completely alone, trapped on a tiny moist mud ball in an eternal Universe. Does that thought scare you? If it does, you are having the correct emotional reaction. If we let life on this planet die, perhaps there would be no life left in the Universe. Life would be gone, maybe forever. If this is the case, we just have to venture to the stars and become the first Type III civilization to keep the delicate flame of life existing and to spread it until the Universe breathes its final breath and vanishes into oblivion. The Universe is too beautiful not to be experienced by someone. This video was made posible by your support. It takes at least 100 hours to make one of our videos, and thanks to your contributions on Patreon we are slowly able to do more and more of them. If you want to help us out and get your own personal bird for example, check out the Patreon page.

Education and career

Montgomery began his undergraduate career at the University of California, Riverside, in 1965 and transferred to Berkeley in 1967, earning a BA in mathematics in 1969 and an MA in mathematics in 1971,[2] He joined the System Development Corporation (SDC) in 1972, where he worked for many years as a programmer implementing algorithms for the CDC 7600 and PDP series of computers, including the implementation of algorithms for multi-precision arithmetic that led to the invention of what is now known as Montgomery multiplication.[1][3] He then returned to academia in 1987, earning his PhD in mathematics from UCLA in 1992 under the supervision of David Cantor.[2][4] He joined the cryptography group at Microsoft Research in 1998, where he worked until his retirement in 2014.[1]

On February 28, 2020, an 829-bit (RSA-250) RSA key was successfully factorised. The team dedicated the computation to Peter Montgomery, who died on the 18th of the same month.[5]

Contributions

Montgomery is particularly known for his contributions to the elliptic curve method of factorization, which include a method for speeding up the second stage of algebraic-group factorization algorithms using FFT techniques for fast polynomial evaluation at equally spaced points. This was the subject of his dissertation, for which he received his Ph.D. in 1992 from the University of California, Los Angeles.[4]

He also invented the block Lanczos algorithm for finding nullspace of a matrix over a finite field, which is very widely used for the quadratic sieve and number field sieve methods of factorization; he has been involved in the computations which set a number of integer factorization records.

He was a Putnam Fellow in 1967.[6] In that year, he was one of only two contestants, along with child prodigy Don Zagier of MIT, to solve all twelve of the exam problems.

Selected works

  • Peter L. Montgomery (1985). "Modular multiplication without trial division". Mathematics of Computation. 44 (170): 519–521. doi:10.1090/S0025-5718-1985-0777282-X. MR 0777282.

References

  1. ^ a b c "Peter Lawrence Montgomery, 1947-2020". International Association for Cryptologic Research. International Association for Cryptologic Research. Retrieved February 20, 2020.
  2. ^ a b c Bos, Joppe W.; Lenstra, Arjen K., eds. (2017), Topics in computational number theory inspired by Peter L. Montgomery, Cambridge University Press, doi:10.1017/9781316271575, ISBN 9781316271575
  3. ^ Baum, Claude (1981), The system builders: The story of SDC, Santa Monica, CA: System Development Corporation, ISBN 0916368025
  4. ^ a b Peter Montgomery at the Mathematics Genealogy Project
  5. ^ Zimmermann, Paul (February 28, 2020). "[Cado-nfs-discuss] Factorization of RSA-250". Retrieved 2021-01-04.
  6. ^ "Putnam Competition Individual and Team Winners". Mathematical Association of America. Retrieved December 13, 2021.

External links


This page was last edited on 21 December 2023, at 17:33
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