Public Keys, One Way Functions and Hard Problems

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Lesson synopsis

Public KeySecurity has always been a major focus of computer science research, and with the explosion of Internet use by commerce, the need for secure transactions has taken on more urgency. Most recently, cyber-thieves demonstrated that true security on the Internet is going to require a new level of understanding of how to protect personal data, and more importantly, financial transactions. This lesson introduces two important concepts: public key encryption and one-way functions. It provides an opportunity for students to understand the underpinnings of almost all Internet security: they will come to appreciate that any lock can be eventually broken, and that theoretical computer scientists study ‘hard’ problems to lengthen the time it will take to break a lock. Note that this is not a lesson in encryption, but in how mathematics is used to secure information.

Age Levels

11 - 14 years

Objectives

Introduce students to:
The concept of a public key
How the modulo function is a one-way function
How the Diffie-Hellman key exchange uses a one-way function
What computer scientists mean by ‘hard’ problems

Anticipated learner outcomes

Students will be able to
Practice creating public keys with the classic color model.
Exchange information with the Diffie-Hellman method using modulo arithmetic.
Explain why no lock can be completely secured, and that given time, any mathematical ‘lock’ can be broken.
Use exponentiation and modulo arithmetic to create cyber-keys.

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Punch card from a COBOL program
Jean Sammet

Jean E. Sammet was one of the first developers and researchers in programming languages. During the 1950’s - 1960’s she supervised the first scientific programming group for Sperry Gyroscope Co. and served as a key member of the original COBOL (COmmon Business-Oriented Language) committee at Sylvania Electric Products. She also taught one of the first graduate programming courses in the country at Adelphi College. After joining IBM in 1961, she developed and directed the first FORMAC (FORmula MAnipulation Compiler). This was the first widely used general language and system for manipulating nonnumeric algebraic expressions. In 1979 she began handling Ada activities for IBM’s Federal Systems Division. Ada is a structured, object-oriented high-level computer programming language, designed for large, long-lived applications, where reliability and efficiency are paramount. Jean has a B.A. from Mount Holyoke College and an M.A. from the University of Illinois, both in Mathematics. She received an honorary D.Sc. from Mount Holyoke (1978).

Turing machine
Alan Mathison Turing
Alan Mathison Turing

Did you know that computing has been used in military espionage and has even influenced the outcome of major wars? Alan Mathison Turing designed the code breaking machine that enabled the deciphering of German communications during WWII. As per the words of Winston Churchill, this would remain the single largest contribution to victory. In addition, he laid the groundwork for visionary fields such as automatic computing engines, artificial intelligence and morphogenesis. Despite his influential work in the field of computing, Turing experienced extreme prejudice during his lifetime regarding his sexual orientation. There is no doubt that computers are ubiquitously part of our lives due to the infusion of Turing’s contributions.

Gordon and SenseCam QUT
Gordon Bell
Gordon and SenseCam QUT

Gordon Bell is a pioneering computer designer with an influential career in industry, academia and government. He graduated from MIT with a degree in electrical engineering. From 1960, at Digital Equipment Corporation (DEC), he designed the first mini- and time-sharing computers and was responsible for DEC's VAX as Vice President of R&D, with a 6 year sabbatical at Carnegie Mellon University. In 1987, as NSF’s first, Ass't Director for Computing (CISE), he led the National Research Network panel that became the Internet. Bell maintains three interests: computing, lifelogging, and startup companies—advising over 100 companies. He is a Fellow of the, Association of Computing Machinery, Institute of Electrical and Electronic Engineers, and four academies. He received The 1991 National Medal of Technology. He is a founding trustee of the Computer History Museum, Mountain View, CA. and is an Researcher Emeritus at Microsoft. His 3 word descriptor: Computing my life; computing, my life.

Cursor
James Dammann

If you have used a word processor today, moved your mouse on your laptop, dragged an object around on your smartphone, or highlighted a section of text on your tablet, you can thank Jim Dammann. In 1961 during his second year at IBM and just one year after completing his PhD, Jim created the concept of what today we all take for granted -- the cursor. This idea he documented in utilizing the cursor within word processing operations.

After retiring from IBM, Jim went on to inspire future generations of software engineers at Florida Atlantic University. His work there too demonstrated his creativity for he spent considerable effort enhancing their software engineering program by integrating ideas and feedback from local industries into the University curricular. Today, Jim lives in the Westlake Hills west of Austin Texas and spends most of his time in his art studio. He wrote and published The Opaque Decanter, a collection of poems about art, which provided a new view at part of art history.

First computer mouse
Douglas Engelbart
Douglas Engelbart

In 1967, Douglas Engelbart applied for a patent for an "X-Y position indicator for a display system," which he and his team developed at the Stanford Research Institute (SRI) in Menlo Park, California. The device, a small, wooden box with two metal wheels, was nicknamed a "mouse" because a cable trailing out of the one end resembled a tail.

In addition to the first computer mouse, Engelbart’s team developed computer interface concepts that led to the GUI interface, and were integral to the development of ARPANET--the precursor to today’s Internet. Engelbart received his bachelor’s degree in electrical engineering from Oregon State University in 1948, followed by an MS in 1953 and a Ph.D. in 1955 both from the University of California, Berkeley.

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