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

FACULTY OF SCIENCE
     Department of Mathematics and Statistics

My Unusual Career as a
Computer Scientist

Hugh Williams, Professor Emeritus
hwilliam@ucalgary.ca
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
    - Mersenne prime: Mn=2n-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-Dy2=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)pn±1 (2≤p≤107),
   where p is a prime and p≠3.

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

   -Several other forms e.g. 102n-10n-1

A Wacky Example

99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999998999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999999999999999
99999999999999999999999999999999
             is prime (=10506-10253-1)

Another Example Problem

 Solve the Diophantine equation
                 x3+Dy3+D2z3-3Dxyz=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 Finalé

 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)