mathematics

Hugh C. Williams

Hugh C. Williams 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 Hugh C. Williams rather than just read about it. In short: Hugh Cowie Williams (born 23 July 1943) is a Canadian mathematician. He deals with number theory and cryptography.

Key takeaways

Hugh C. Williams belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Hugh C. Williams to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Hugh C. Williams from memory before moving on to harder problems.

Reference excerpt

Hugh Cowie Williams (born 23 July 1943) is a Canadian mathematician. He deals with number theory and cryptography.

Early life Williams studied mathematics at the University of Waterloo (bachelor's degree 1966, master's degree 1967), where he received his doctorate in 1969 in computer science under Ronald C. Mullin and Ralph Gordon Stanton (A generalization of the Lucas functions). He was a post-doctoral student at York University.

Career In 1970 he became assistant professor at the University of Manitoba, where in 1972 he attained associate professor status and professor in 1979. In 2001 he became a professor at the University of Calgary, and professor emeritus since 2004. Since 2001 he has held the "iCore Chair" in Algorithmic Number Theory and Cryptography. Together with Rei Safavi-Naini he heads the Institute for Security, Privacy and Information Assurance (ISPIA) - formerly Centre for Information Security and Cryptography - at Calgary. Between 1998 and 2001 he was an adjunct professor at the University of Waterloo. He was a visiting scholar at the University of Bordeaux, at Macquarie University and at University of Leiden. From 1978 to January 2007 he was associate editor of the journal Mathematics of Computation. Among other things Williams dealt with primality tests; Williams primes were named for him. He developed custom hardware for number-theoretical calculations, for example the MSSU in 1995. In cryptography, he developed in 1994 with Renate Scheidler and Johannes Buchmann a method of public key cryptography based on real quadratic number fields. Williams developed algorithms for calculating invariants of algebraic number fields such as class numbers and regulators. Williams deals with math history and wrote a book about the history of primality tests. In it, he showed among other things that Édouard Lucas worked shortly before his early death on a test similar to today's elliptic curve method. He reconstructed the method that Fortuné Landry used in 1880 (at the age of 82) to factor the sixth Fermat number (a 20-digit number). Together with Jeffrey Shallit and François Morain he discovered a forgotten mechanical number sieve created by Eugène Olivier Carissan, the first such device from the beginning of the 20th century (1912), and described it in detail.

Publications The influence of computers in the development of number theory. In: Computational Mathematics with Applications. Band 8, 1982, S. 75–93. Factoring on a computer. Mathematical Intelligencer, 1984, Nr. 3. with Attila Pethö, Horst-Günter Zimmer, Michael Pohst (Hrsg.): Computational Number Theory. de Gruyter 1991. with J. O. Shallit: Factoring integers before computers. In: W. Gautschi (Hrsg.): Mathematics of computation – 50 years of computational mathematics 1943–1993. Proc. Symposium Applied Math., Band 48. American Mathematical Society, 1994, S. 481–531. Édouard Lucas and primality testing. Wiley 1998. (Canadian Mathematical Society Series of Monographs and Advanced Texts. Band 22.) with M. J. Jacobson: Solving the Pell Equation. Springer 2008.

References

External links Literature by and about Hugh C. Williams in the German National Library catalogue Hugh C. Williams Archived 2011-08-27 at the Wayback Machine at the website of the University of Calgary Profile of Hugh C. Williams Archived 2014-01-12 at the Wayback Machine at the faculty with links to publications Williams references at the Prime Pages

Illustrations

Worked examples

Example 1 — a first encounter with Hugh C. Williams

Start with the simplest possible case. Write down what Hugh C. Williams 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 Hugh C. Williams 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 Hugh C. Williams 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 Hugh C. Williams

In research: Hugh C. Williams 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 Hugh C. Williams 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: Hugh C. Williams is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1943 births, 20th-century Canadian mathematicians, 21st-century Canadian mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Hugh C. Williams 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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How to study Hugh C. Williams in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what Hugh C. Williams means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain Hugh C. Williams out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Hugh C. Williams in simple terms?

Hugh Cowie Williams (born 23 July 1943) is a Canadian mathematician. He deals with number theory and cryptography.

Why does Hugh C. Williams 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 Hugh C. Williams?

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 Hugh C. Williams.