mathematics

Peter Montgomery (mathematician)

Peter Montgomery (mathematician) is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Peter Montgomery (mathematician) rather than just read about it. In short: 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…

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Reference excerpt

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.

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, 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. He then returned to academia in 1987, earning his PhD in mathematics from UCLA in 1992 under the supervision of David Cantor. He joined the cryptography group at Microsoft Research in 1998, where he worked until his retirement in 2014. 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.

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. 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. 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. Bibcode:1985MaCom..44..519M. doi:10.1090/S0025-5718-1985-0777282-X. MR 0777282. Peter L. Montgomery (1987). "Speeding the Pollard and elliptic curve methods of factorization". Mathematics of Computation. 48 (177): 243–264. Bibcode:1987MaCom..48..243M. doi:10.1090/S0025-5718-1987-0866113-7. MR 0866113. Peter L. Montgomery (1995), "A block Lanczos algorithm for finding dependencies over GF(2)", Advances in cryptology—EUROCRYPT '95 (Saint-Malo, 1995), Lecture Notes in Computer Science, vol. 921, Springer-Verlag, pp. 106–120, doi:10.1007/3-540-49264-X_9, ISBN 978-3-540-59409-3, MR 1367513

References

External links Incomplete list of Montgomery's papers In Memoriam: Peter L. Montgomery (1947–2020) (Notices - American Mathematical Society, April 2021, Volume 68, Number 4, Joppe W. Bos and Kristin E. Lauter)

Illustrations

Worked examples

Example 1 — a first encounter with Peter Montgomery (mathematician)

Start with the simplest possible case. Write down what Peter Montgomery (mathematician) claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Peter Montgomery (mathematician) before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Peter Montgomery (mathematician) ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Peter Montgomery (mathematician)

In research: Peter Montgomery (mathematician) appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Peter Montgomery (mathematician) in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Peter Montgomery (mathematician) is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1947 births, 2020 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Peter Montgomery (mathematician) outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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Frequently asked questions

What is Peter Montgomery (mathematician) in simple terms?

Why does Peter Montgomery (mathematician) matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Peter Montgomery (mathematician)?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Peter Montgomery (mathematician).