FACULTY OF SCIENCE Department of Mathematics and Statistics My Unusual Career as a Computer Scientist

Hugh Williams, Professor Emeritus [email protected] Jan 21, 2021 A Lifetime Achievement Award High School (1957-62)

Burlington High School is where I began to understand and love mathematics. The school had an excellent cadre of mathematics teachers who encouraged my studies. I was very fortunate.

3 Inspiration(1)

. Primes n - Mersenne prime: Mn=2 -1

- “According to Lucas, a number Mn, where n is greater than 2 is a prime if and only if it divides the (n-1)st term of a series in which the first number is 4; the second, the square of the first minus 2; the third, the square of the second minus 2; and so on– in other words, 4, 14, 194, 37634, and so on.” Constance Reid, From Zero to Infinity , 1955 Mersenne Primes

. In 1876 M127 is prime (39 digits)

. In 1955 M2281 is prime (687 digits)

. In 2018 M82589933 is prime (24862048 digits) Inspiration(2)

. Diophantine Equations - Equations whose solutions are constrained to be whole numbers (integers) - The Pell Equation: x2-Dy 2=1 e. g. if D=7 , then x=8 and y=3 is a solution -”When D=1620 , the [least positive] value of x has 3 figures [ x=161, y=4 ]; when D = 1621 , it has 76 figures.”R. D. Carmichael, Diophantine Analysis , 1915 My Questions

Why does Lucas’ test work? Why are the solutions to the Pell equations so different? How does one even compute solutions for these equations? These questions, along with the pure amazement and beauty the results convey, is what captured my attention and brought me into a world of numbers that are more than just black and white images on a page. University of Waterloo (1962-69)

. Completed my undergraduate and graduate degrees at the same institution. Don’t do this. . The U of Waterloo was a great place, mainly because it was new, small and not overly rigid. That has changed.

University of Waterloo (1961) The IBM 1620 Computer

. In 1962 I made my first contact with a computer. . I was utterly smitten.

Doug Lawson And Wes Graham at the console of the IBM 1620 (1961) The Cattle Problem of Archimedes

. An unsolved problem dating to ca. 220 BCE . Essentially, the problem was to solve a Pell equation with D=410286423278424 . In 1965, Gus German, Bob Zarnke and I recognized that computers had advanced to the point that this ancient problem could be solved . Were able to find the solution by using the university’s IBM 7040 and 1620 . Wes Graham generously allowed us the use of these machines The Solution

. The solution involved a number of over 206000 decimal digits

Hugh Williams, Gus German and Bob Zarnke (1965) Degrees Awarded at U of W

. B. Sc. (Hons) 1966 . M. Math 1967 . Ph.D. 1969 . Thesis: A Generalization of the Lucas Functions . Earliest Ph. D (along with Byron Ehle) awarded in the Dept. of Applied Analysis and Computer Science (“Applied Analysis” was dropped from the name in 1975.) My Ph.D. Supervisor: Ron Mullin

. Ron is the recipient of the first earned degree awarded by the U of W (M.A. 1960)(Ph.D. 1964)

Ron Mullin and Scott Vanstone Personal Matters

. Married Lynn Gilbert in 1967 . Two children: Helen b. 1971, Cassandra b. 1973

Lynn, Helen and Cassandra Research

. Number Theory is the branch of mathematics that deals with the properties and relationships of numbers, especially the positive integers. . A computational number theorist studies how computational devices can be enlisted to solve problems arising in number theory through the development and implementation of efficient and provably correct algorithms. . My research lies within the ambit of computational number theory. The University of Manitoba (1970-2001)

Department of Computer Science . Assistant Professor 1970-1972 . Associate Professor 1972-1979 . Full Professor 1979-2001 . Killam Research Fellowship 1983-84 . Associate Dean (Research Development) 1994-2001 . Retired 2001 . Professor Emeritus 2004-present Some Simple Example Problems

. Primality testing -Necessary and sufficient tests for primes of special forms such as (p-1)p n±1 (2≤p≤10 7), where p is a prime and p≠3 .

-Primality of (10 n-1)/9 for n=317, 1031

-Several other forms e.g. 10 2n -10 n-1 A Wacky Example

99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999998999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999999999999999 99999999999999999999999999999999 is prime (=10 506 -10 253 -1) Another Example Problem

. Solve the Diophantine equation x3+Dy 3+D 2z3-3Dxyz=1 . -use Voronoi’s algorithm (1896) -how can a computer determine efficiently that an irrational number is larger than 0? Who Cares?

. During the early stages of my career, very few people were interested in these types of problems, which were considered to be impractical and not really part of what was considered to be “mainstream” computer science. . In the mid 70s, the NRC (now NSERC) Grant Selection Committee for computer science decided I was not a computer scientist, froze my grant for a year and transferred my application to the Pure and Applied Mathematics Committee. This decision initiated a very difficult time for me. Was I a computer scientist or a mathematician? Number theory and Cryptography

. In the late seventies, it was discovered that primes could be utilized in the production of “unbreakable” secret codes. More importantly, they could be recruited in solving the problem of authenticating messages. This would be become pivotal in the establishment of internet commerce. . All of a sudden, computational number theorists were in demand for the development, implementation and evaluation of secure cryptosystems for computers. Personal Consequences

. My research direction changed, but not very much

. My NSERC grant (From Pure and applied Math) increased

. I also began to look at the possibility of using modern technology to build special purpose computers, called sieving devices, for solving certain problems.

. I was able to attract graduate students Number Sieve Devices

Eugene Carrisan’s number sieve 1919 Richard Lukes’ Number sieve 1993 Cryptography at U of Manitoba

. Developed new courses . Revised my research program (somewhat) . Organized and attended several professional meetings . Published a number of research and expository papers . Started research collaborations with international partners — Co-founded, with Johannes Buchmann, the area of algebraic number theory cryptography Tucson (1987)

Johannes Buchmann and I

Johannes Buchmann and I University of Calgary (2001-2016)

. Invited in 2001 to take the iCORE (Alberta informatics Circle of Research Excellence, now AITF) Chair in Algorithmic Number Theory and Cryptography (ICANTC) . Lots of money for visitors, graduate students, post docs, secretarial and administrative assistance . Chair-dedicated academics in the mathematics and computer science departments . Acquired funding for state-of-the-art computer hardware . Co-founded (with Rei Safavi-Naini) ISPIA, The Institute for Security, Privacy and Information Assurance The Initial Team at Crypt Lake (2003)

Mike Jacobson, Renate Scheidler, Hugh Williams and Mark Bauer The ICANTC Vision

. ICANTC was focused on research and training excellence in cryptology and information security . Originally mandated to conduct research in fundamental algorithmic number theory and mathematical cryptography, ICANTC broadened its reach beyond theoretical research into the rapidly growing area of applied information security . ISPIA is the vehicle by which ICANTC conducted these expanded activities ICANTC and Others (2009)

The Gang and significant others Legacy of ICANTC (2001-2013)

. ISPIA continues to maintain a strong, multi-faceted membership cluster that includes academics in mathematics, computer science, engineering, physics, commerce and law, as well as professionals from the public and private sectors. . The Calgary Number Theory Research Group. Research interests include algebraic number theory, algorithmic and computational number theory, arithmetic dynamics, arithmetic geometry, and cryptography. Legacy of ICANTC (2)

. The strong Information Security component of the U of C’s Department of Computer Science. This includes concentrations at both the graduate and undergraduate level . Our many graduates and post docs, who have gone on to successful careers in government, business and a variety of academic institutions The Communications Security Establishment (CSE)

. In 2008, I was the successful applicant for the position of Director of the Cryptologic Research Institute (CRI), a classified research institute within CSE. . This was a secondment from the University of Calgary, which still remained my employer. . The original, somewhat vague, mandate of the CRI, which at the time was little more than an idea, was “to bring together talented Canadian mathematicians from various disciplines to conduct fundamental research in areas of mathematics of interest to CSE.” My Tenure at CRI (2009-2015 )

. Had to get used to security and acronyms . Had to design and then get the new accommodations built. . Research staff had to be selected . Academic contractors had to be recruited . To attract membership the Institute had to be rebranded. In 2011, the CRI became TIMC, the Tutte Institute for Mathematics and Computing . An internal library had be to established Challenges of the Directorship

. Navigating the bureaucracy of the federal government in general and that of CSE in particular . Procuring funding . Creating programs . Establishing and maintaining the profile of TIMC, both within CSE and beyond . Overcoming prejudices concerning TIMC within CSE Career Final é

. While very demanding, my time at TIMC was very special, due to the quality of the people I was privileged to lead. I enjoyed a most interesting, fulfilling and informative 6 years

. I retired from the directorship of TIMC in 2015 and subsequently retired from the University of Calgary in 2016, after 46 years in the academy Reflections

. In my own, clearly biased, view I had a wonderful career--I was paid to do what I loved and, in spite of that, even managed to eke out a few achievements. I was also very lucky to have been active during the beginning of the computer revolution. . It really doesn’t get much better than that for any academic. To use the bikers’ term, I had a great ride. . Nevertheless, I have always been mindful of an observation attributed to the great mathematician and physicist, Sir Isaac Newton… Isaac Newton (1632-1727)

"I do not know what I may appear to the world, but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me. " Lessons Learned

. Stay where your interests take you . Keep several balls in the air . Just keep working . Write it all down . Go to professional meetings . Take on students . Do not be overly concerned by the opinions of others— concentrate on pleasing yourself . Do not pass up opportunities—they are rare . Maintain your curiosity and sense of wonder . Stay physically active . Do not spend all your time working. Get a hobby My Hobby: Outdoor Photography

Queen of Sheba Orchid, Western Australia Roseate Spoonbill, Florida Back to Where It Began

The IBM 1620 in the Computer History Museum, Mountain View CA (2003)