The Mathematics of Communication: Keeping Secrets

Stephen J. Greenfield, Department of Mathematics, Rutgers, The State University of New Jersey, New Brunswick, New Jersey

A 2004 SENCER Model


For thousands of years, people have tried to communicate secretly and securely. Cryptography is the field of mathematics dedicated to exploring schemes to conceal messages and to verifying the difficulty of “breaking” these schemes-that is, revealing the hidden message without the consent or knowledge of those communicating. This course discusses some of the mathematical and social issues related to cryptography. Historically, governments did most cryptographic investigation and these efforts were rarely publicized. They were massive: the largest single employers of mathematically trained people in the United States and the former Soviet Union have been the government agencies with cryptographic responsibilities. There’s been an enormous increase in public cryptographic work in the last quarter century, and in the accompanying controversies. This increase has been caused by the easy availability of computers and their interconnections (via the Internet and the web) and by the development of new ideas, such as public key cryptography, which allow for secure communication between parties who have made no previous commitment to each other. Every person who has used an ATM (automatic teller machine), made a phone call using a cellular phone, or had their health records transmitted among caregivers or insurers should be concerned about secure communication. Social issues include the conflict between the right to privacy and the desire of some government agencies to have assured access to certain communications, and the difficulty and propriety of preserving intellectual property rights over a collection of bits.

This course covers a number of important topics in the mathematics and theoretical computer science discovered within the last quarter century including: modular arithmetic, algorithms and factoring, cubing and cube inquiry, Diffie-Hellman key exchange and RSA protocols, binary arithmetic, and randomness. The pedagogy has been adapted from strategies commonly used in the humanities and social sciences in an effort to make the course more appealing to students whose interests are in those fields. Active learning techniques included class discussions, group reports, debates and position papers. This requires some adjustment for instructors who may be most familiar with traditional teaching strategies in the mathematical sciences.

Learning Goals

What basic science is covered?

Concepts from mathematics and theoretical computer science will be introduced on an “as needed” basis, motivated by trying to address the social problems mentioned previously. Different topics may need differing mathematical underpinning, changing the concepts introduced. Here’s a minimal list of the scientific concepts which can be covered:

  • Shamir’s secret sharing scheme (polynomial interpolation, Lagrange interpolation)
  • Modular arithmetic
  • Fermat’s little theorem
  • Euclidean algorithm for solving ax = 1 mod p
  • Euler’s extension of Fermat’s result to a product of two primes
  • RSA encryption and digital signatures
  • Attacks on RSA: factoring
  • Diffie-Hellman encryption
  • The difficulty of arithmetic
  • How to exponentiate efficiently; P versus NP; PRIMES is in P.
  • Binary notation and arithmetic
  • Hashing
  • Digital watermarks; introduction to steganography
  • AES (in the original course, this was DES!)
  • Randomness and one-time pads; pseudo-randomness
  • Elements of probability
  • What is an algorithm?

In addition, students should be able to experiment actively with the computational ideas of the course. Thus, in Rutgers implementations, Maple was introduced early in the course, with very little discussion of the program. Maple was used as a calculator with larger capacity than those owned and used by students. Such calculators typically can’t display meaningful examples of the topics discussed in this course.

How does this course advance institution-wide objectives?

Two important objectives were addressed with this course.

#1: Show students who are not majoring in science/engineering/technology the relevance of mathematics and science to their lives.

This was done by displaying the variety and power of the solutions that mathematics can offer to the “civic questions or problems” listed as part of the response to, “Why is this course a SENCER model?”

Active learning allowed students to test the mathematical techniques developed, either in class or in homework. Class discussions and papers further showed the relevance of the material. For example, there were several memorable class discussions about medical record privacy, with students taking and intelligently discussing very different positions. Teams prepared presentations about the crypto policies of various governments and the positions of the FBI and the EFF (Electronic Frontier Foundation). Students became intensely involved with the positions they were representing. In addition, the class discussions on intellectual property were fascinating, and many students were well-prepared.

This met not only Rutgers objectives, but fully justified the NSF funding which substantially aided the creation of this course. The NSF stated:

The role of science and technology in American society is undergoing dramatic change. In an increasingly technology-oriented society, a basic understanding of science and mathematics is essential not only for those who pursue careers in scientific and technical fields but for all people. At present, however, not all students have access to quality instruction in these areas, and most adults have limited opportunities to develop a better understanding of the role of science. This nation needs a population that is well prepared to fulfill the needs of a technically competent work force and that exercises their full rights and responsibilities of citizenship in a modern democracy.

#2: (An objective of the instructor, shared by many) Give students the chance to write about and present complex issues, issues having substantial technical and mathematical content stemming from important social, political, and legal questions.

Students at a large institution like Rutgers have an incredibly wide range of educational opportunities. But the chance to write and speak about important technical issues (and to have feedback about these efforts) is not common. For example, writing in undergraduate math and computer science courses is rarely extensive. This course allowed students in the humanities and social sciences to display some of their talents in written and spoken advocacy in a mathematics course, which many had not imagined was possible. I believe that they gained a measure of confidence and perhaps even pleasure in discussing some of the applications of mathematics to social and legal problems.

Linking Mathematics and Social Issues

Current social, political, and legal problems related to crypto

The Web is a wonderful and monstrous construct. The social, political, and legal problems related to this course (which I’ll label more briefly here as “social problems”) typically have enormous numbers of links available for exploration. For example, today (7/18/2004) Google reports over 335 million links in response to the query, copyright. Students and instructors both have huge jobs trying to locate useful information, especially about controversies which develop rapidly. Each time I gave this course, I thought carefully about the social topics I wanted to investigate. I looked for appropriate links (and texts, of course!). I found that 6 months later, if I wished to pursue the same topics, almost a quarter of the links I gathered no longer existed or were off-topic (perhaps a form of what’s called link rot). More positively I can report that students found and submitted interesting and relevant links on virtually every social problem discussed in the course.

One book I recommend for the social problems related to classical crypto is Privacy on the Line: The Politics of Wiretapping and Encryption by Whitfield Diffie and Susan Landau, 352 pp paperback, MIT Press, 1998, $22. Several recent books by Lawrence Lessig are interesting and relevant. The quote in the SENCER abstract for this course is from Code and Other Laws of Cyberspace by Lawrence Lessig, 297 pp paperback, Basic Books, 2000, $12.

What capacious civic questions or problems are addressed in the course?

Some social issues have special resonance in a technical society. A few of these issues are listed and briefly discussed below. The beginning of an intellectual and physical infrastructure which may cope with such issues has only occurred within the last 20 years. New results in mathematics and theoretical computer science are at the heart of these developments.

Students and all participants in our digital society should know what is possible, even as they help to decide what the aims of laws and practices should be.

  • Security and privacy of email

Governments assert that widespread use of secure cryptographic communication would be a great hindrance to capturing terrorists, criminals including drug traffickers, and spies. At the same time, “open” email can typically be read by dozens or hundreds of people involved in the storage and transmission of the messages. Can there be a suitable compromise to the varying desires of stakeholders?

  • Confidentiality of medical records

Certain laws and practices essentially open many medical records to police inspection and outside auditing. The stated reasons for these policies are to decrease fraudulent billing and to simplify the transactions of participants. But there exist ways of securing privacy and “sharing” information access which can be implemented with little overhead.

  • Privacy of cell phones

Evidence has been presented that cell phones broadcasts, which carry an increasing portion of electronic conversations, are easily intercepted and quite insecure. Should users be permitted to specify levels of security and privacy? Should such choices be well-known?

  • Trust and electronic commerce

What are the intellectual constructs which make it possible for a customer in Omaha to buy books from Amazon? The two participants must exchange information such as credit card numbers and book titles which both may prefer to keep confidential. In earlier times, a buyer and seller might meet privately to satisfy such confidentiality desires. In cyberspace, no prearrangements have been made, and only information can be exchanged. Why should the methods currently used be trusted?

  • Intellectual property in a digital age

The term “intellectual property” can refer to ideas protected by copyright, patent, trademark, or trade secret. The Digital Dilemma, a publication of the National Academy of Sciences, states, “…A printed book can be accessed by one or perhaps two people at once, people who must, of course, be in the same place as the book. But make that same text available in electronic form, and there is almost no technological limit to the number of people who can access it simultaneously, … For publishers and authors, the question is, How many copies of the work will be sold (or licensed) if networks make possible planet-wide access? Their nightmare is that the number is one. …”

Even without what could be hyperbole, we all recognize that musicians have traditionally received only a small percentage of the revenue from the final sale of a record or a compact disc or DVD. With the Web, such items could be sold directly, at a lower price and possibly yielding more revenue to the performers. Should these changes be controlled for the benefit of previous profiteers? What technological controls are possible and/or desirable?

  • Timestamping without revealing

How can one timestamp a document? For example, two parties may enter into a congtract whose details they may prefer to keep private, but whose execution may reasonably be foreseen to involve the possibility of disagreement or even litigation. Techniques related to crypography provide an opportunity to record the document publicly in a condensed and secret fashion, with a very high likelihood that neither party can later deny the agreement or its details. These techniques are now in use. Interestingly, early excavations in what was Mesopotamia reveal that such problems were recognized even then, and solutions with then-current “technology” were attempted.

  • Auctions: secret bids in the open

Suppose parties must submit a bid to an auction by a desired date, and the transactions may be vulnerable to a corrupt official, who may open a sealed bid to disclose information to competitors. Real cryptographic procedures can make detection of such cheating very likely.

  • Digital cash

The money in your pocket is difficult to trace. As the world moves to a digital economy, bank transfers may become easier to trace resulting in a decrease of the privacy of financial transactions. Can anonymous forms of digital cash be created? What are the desired properties and problems of such cash?

  • Electronic voting

This topic was not discussed in the original course, but most of the ideas and protocols mentioned then apply to the questions of electronic voting. Many cryptographers are currently investigating how to guarantee the accuracy, integrity, and security of electronic voting.

The Course


A syllabus, per se, is not available for this course. Please see instructor’s Web site for course-related documents and information.

Course Design

Each time I’ve taught the subject, the topics and their order have changed somewhat, so certainly what’s there is a (considered!) snapshot of the course dated 5/27/2000. Lectures and other material I have prepared for the New Jersey Governor’s School of Engineering and Technology is also relevant because some of the topics are the same.

Some guiding principles might be:

  1. Bring in the math to discuss and “solve” social problems.
  2. Investigate the math, maybe simplify, but don’t lie.
  3. Ask students to do the math: construct simple examples and engage in simple contests.
  4. Have students write papers and give class presentations. Organize class discussions. These activities may encourage students whose past encounters with mathematical instruction have not been happy.

Beginning the course has its own difficulties. Nervous students who historically have had weak performance and little interest in mathematics courses need to be encouraged, even inveigled. In 2000, I began with some discussion of the history of cryptography and some simple examples (modular arithmetic was introduced here, along with some other constructions). More recently, I have begun with secret sharing, the material of lecture #3 in my Web outline.

Lecture Schedule

  1. Introduction & Caesar cipher
  2. More basic crypto & meeting Maple
  3. Secret sharing
  4. Medical record privacy
  5. Modular arithmetic
  6. More modular arithmetic
  7. The difficulty of arithmetic
  8. Algorithms and hardness
  9. Guest lecture by J. Reeds
  10. P vs. NP and experimentation
  11. Diffie-Hellman key exchange
  12. Fermat’s little theorem
  13. RSA
  14. Working with RSA
  15. Attacking Diffie-Hellman; digital signatures & trust
  16. Beginning binary
  17. Randomness & one-time pads
  18. The first crypto policy presentations
  19. Bitstreams & xor
  20. More policy presentations
  21. Beginning authentication
  22. Hashing
  23. Intellectual property
  24. Protecting digital intellectual property
  25. A discussion about DES
  26. Enigma on videotape
  27. Enigma discussion & review for the final
  28. Conclusion & evaluation

Pedagogical Methodologies

Pedagogy from across the river

I teach at a Rutgers campus with some serious geographic problems. Chunks of the campus are separated by the Raritan River. Many of the humanities and social science departments are headquartered across the river, in New Brunswick. Most of the science departments and the Engineering School are on the Piscataway side of the river.

I decided to try the pedagogy of the disciplines “across the river.” In particular, I wanted students to write essays. Then I needed to learn how to read essays. I’ve certainly read mathematics written by students earlier in my career and tried, with both undergraduate and graduate students, to encourage clear statements and good writing. In this course, I might have students writing about medical record privacy or the cryptological policies of the French government. I might read papers which attacked or supported positions on complex issues. I sought help, and found it by consulting Professor Kurt Spellmeyer of the English Department, who is also Director of the Writing Program. The Writing Program instructs almost every student at Rutgers. Professor Spellmeyer very kindly spent time with me discussing how to help students write (more directly: how to grade essays usefully!) and he supplied me with the material he uses to help his instructional staff. This was truly helpful. I should report that my first attempts to grade 25 two-page essays took much more time than grading 100 calculus exams would have!

We also talked about managing class discussions. I had little experience with this instructional skill. I now know that presiding over a class discussion, encouraging participation by students, and learning how to elicit various points of view is an additional teaching tool. I haven’t mastered it but at least I now recognize it more clearly!

Outside speakers: Bring the world in

I have invited one or two outside speakers each time I’ve given this course. I want to get people who work in areas directly affected by the controversies discussed in the course. Such people are available everywhere. You can look for lawyers dealing with intellectual property. You can look in your own institution. In the biological sciences, there might well be faculty members who have strongly held opinions about who owns the information in “my” DNA. You can look at the staff of your own institution. In computer services, there’s bound to be people who handle hacker attacks, or who service electronic mail. One of the most important Rutgers computer services people was very interested in speaking. He told me about a pet horror: students who send their social security numbers over “open” e-mail.

Computational help

Certainly cryptography can be done “by hand” as it was during most of human history (although helpful devices like Jefferson’s wheel have been used for hundreds of years). Realistic examples of what’s used for Web commerce, for instance, need help that’s electronic and fast. I’ve seen some effective programs on graphic calculators which show various cryptographic protocols. But, for the most part, graphing calculators don’t have the storage, the programs, and the speed to be useful here.

See (link) for one example. I divided a class into groups to prepare presentations on crypto policies of various organizations. I also used those groups to “break” RSA messages which were titles of some of the Sherlock Holmes short stories. I used numbers which were 42 decimal digits long. I don’t know currently available, reasonably priced hand-held devices which can cope with arithmetic on such integers. Also, experimentation showing the difficulty of factoring and discrete logs only really begins to show results with “big” numbers.

I strongly recommend that students use programs like Maple or Mathematica whenever possible. I used Maple because Rutgers has a campus-wide license for it, and the program is present on various systems that students use. I did not ask or require that students become Maple programmers, just that they be able to use the program as a large calculator. I did create some simple programs which helped me construct examples. I wrote these programs in Maple but similar routines certainly can be written in other languages.

Evaluating Learning

Student assessment

Students were assessed by short quizzes in class. Several papers were graded: two individual efforts and a group effort. Group presentations were made, and the instructor devoted some effort to assessing contributions to the presentations. The instructor also attempted to assess contributions to class discussions.

A three-hour closed book final exam was given. The exam and the grades on the exam are available here. Student performance on the exam was very good. The grade distribution was higher than in almost every calculus course I’ve taught. Course grades are also discussed on the Web page cited. The grades given should be considered in company with the stated majors of the students. This information is available here.

Course assessment

The standard student evaluation forms, “The Student Instructional Ratings Survey,” primarily request information about the students’ perceptions of the instructor’s teaching effectiveness. These surveys were given near the end of the course. In addition, a series of questions about both the mathematical and social aspects of the course were asked at both the first meeting and the last meeting. The questions and results of all three assessments, for two semesters, can be seen here. These assessments were answered anonymously.

The evaluation results are displayed and discussed at the Web link just cited. Instructional experiments frequently result in good student evaluations (a sort of Hawthorne effect?). The students liked the course. The differences in the pre- and post- test are interesting. Happily, scores on the “objective” questions (testing some math knowledge) increased sharply. There were changes in the scores on the additional questions (about some of the social issues discussed in the course) but these are harder to interpret.

My thoughts

I enjoyed teaching this course. I previously avoided courses directly concerned with quantitative skills because I didn’t like the material usually taught and thought most students didn’t like the courses. I still believe that’s generally true, but teaching this course has encouraged me. Challenging material can be taught with some success if its relevance to students is clearly demonstrated. “Even” liberal arts students satisfying quantitative skills requirements can find learning math interesting and even enjoyable, and I can feel the same about my teaching them.

Sustaining change is problematic

Continuing this course in the manner I created it has been nearly impossible with other instructors. I spent large amounts of time on the course and did not follow a text. Other instructors have since taught the course, and I have been told several times, “I covered much more [math] than you did.” The point of the course to me was knitting together math and society, not heroically pulling students along into more cryptography. Below is a quote from my Web notes on lecture #28, which reveals my ambitions for the students:

I told the students that I could imagine they were trapped (in an elevator? at a cocktail party?) with a professional cryptographer and with a staff member from a Congressional committee on communication, and that they were to try to understand and contribute to a conversation about the issues raised in this course.

The people who taught this course other than me, and the person who is scheduled to teach it in fall 2004, don’t want to grade essays and monitor class discussions. There’s little desire to wade through Web pages covering copyright or the privacy of email. Following a textbook is much easier than trying to blend social and technical ideas. I have audited some of the classes given by my successors, and, sure enough, these are math classes: lecture, with less student involvement than I think is optimal, especially for the type of students in this course. But, as I wrote above, at a major state research university, “hiring and promotions are primarily based on scholarly promise and achievement, with appropriate attention given to teaching and service.” Thus, especially for junior faculty, the amount of time and attention which can sensibly be devoted to teaching is limited. Also, I have observed that mathematicians are generally among the most conservative academics pedagogically. So … more work is needed.

Background and Context


Stephen J. Greenfield
Department of Mathematics
Rutgers, The State University of New Jersey
Stephen Greenfield is a Professor of Mathematics at Rutgers. He is a member of DIMACS, a national center for discrete mathematics and theoretical computer science. His research interests are complex analysis and partial differential equations. He has also worked in some areas of discrete mathematics and cryptography.

He is interested in educational innovation and sustaining improvements to instruction. He has taught about 35 different math courses in his career, ranging from precalculus to advanced graduate courses. He jointly taught an undergraduate course with a physicist. He was in charge of changing various undergraduate calculus sequences at Rutgers. These changes included appropriate use of technology (involving both graphing calculators and Maple), the use of workshops with peer mentors (small groups of students working on nonroutine problems in class with written reports graded for both content and presentation), and Web support for instruction. He has advised undergraduate research efforts, and led seminars for undergraduates.

He has served as Undergraduate Vice-Chair of the Mathematics Department and as Graduate Director. He obtained significant internal and external funding for educational activities affecting both undergraduate and graduate students.

He has worked with high school teachers, both as part of an educational program which was a direct ancestor of the course presented here and as an enthusiastic participant over the last decade in the Advanced Placement Calculus program. He learned a lot about teaching from these activities. For the last four summers, he has given series of 14 lectures to 100 high school students in the New Jersey Governor’s School of Engineering and Technology.

His efforts have been recognized by awards including the Rutgers University Minority Advancement Program alumni award for encouraging minority graduate study in mathematics, the FAS Award for Distinguished Contribution to Undergraduate Education, the Warren I. Sussman Award for Excellence in Teaching at Rutgers (the university’s most important award for teaching), and, recently, the 2003 Mathematical Association of America-New Jersey Section’s Award for Distinguished College or University Teaching.

Mathematics is part of human culture. Its applications to engineering and the “hard” sciences have been assumed. It is interesting to learn and to teach about how some parts of mathematics have been used to support other needs of society.

Course History

Where is the course taught?

The course is taught on the New Brunswick/Piscataway campus of Rutgers University. Here is the official statement of the mission of Rutgers University:

As the sole comprehensive public research university in the New Jersey system of higher education and the state’s land-grant institution, Rutgers University has the mission of instruction, research, and service. Among the principles the university recognizes in carrying out this three-fold mission are the following:

  • Rutgers has the prime responsibility in the state to conduct fundamental and applied research, to train scholars, researchers, and professionals, and to make knowledge available to students, scholars, and the general public.
  • Rutgers should maintain its traditional strength in arts and sciences, while at the same time developing such new professional and career-oriented programs as are warranted by public interest, social need, and employment opportunities.
  • Rutgers will continually seek to make its educational programs accessible to an appropriately broad student body.
  • Rutgers is committed to extending its resources and knowledge to a variety of publics, and bringing special expertise and competence to bear on the solution of public problems.

The university itself has a unique history, as a combination of colonial college (founded in 1766), land-grant university (1864), and, finally, as the “flagship” public research university of New Jersey (1956). Several colleges and other schools with intricate histories of their own were consolidated into Rutgers at various times in the last century. The university has more than 50,000 students at campuses in Camden, Newark, and New Brunswick/Piscataway. The last-named campus, in the center of the state, is by far the largest, with more than 35,000 students. Most of the university’s Centers and Graduate Programs have their principal presences in New Brunswick/Piscataway. The university is a member of the Association of American Universities, “an association of 62 leading research universities in the United States and Canada,” and is also a member of the National Association of State Universities and Land-Grant Colleges an “association of public universities, land-grant institutions and many of the nation’s public university systems.”

New Brunswick/Piscataway has more than 25,000 undergraduates, affiliated with a wide variety of schools and colleges, with many majors. The diversity of the undergraduate population is remarkable. Slightly more than half of the students are female. Barely half of the undergraduates are classified as “White,” with significant percentages of African Americans (10%), Latinos (10%), and Asian Americans (20%). About 90% of the undergraduates are New Jersey residents. 30% of New Jersey’s population is African American, Latino, or Asian American. The first two segments are slightly underrepresented in the Rutgers student body, while the Asian American portion has a much higher proportion of undergraduate enrollment than its percentage of state population, which is about 7%. The mean SAT scores for “registered first-year, regular-admit enrolled students” in fall 2003 were 588 (verbal) and 615 (math). The most recent six-year graduation rate for Rutgers undergraduates is 70%. Many undergraduates do not have English as their family language, and many come from the first generation in their families to attend college.

In December, 2003, the Rutgers New Brunswick/Piscataway Mathematics Department wrote a self-study documents, primarily authored by its chair, Professor Richard Falk. Professor Falk further particularized the mission of the department in the following way:

The Department of Mathematics supports the University’s mission of instruction, research, and service in the following ways. The Undergraduate Program of the Mathematics Department provides professional training and liberal education, both to its majors and also to the wide variety of students in other disciplines who enroll in mathematics courses. The Graduate Program provides high quality training to produce the next generation of scholars, researchers, college and university teachers of mathematics, and mathematicians for government and industry. The Department seeks to hire outstanding scholars, whose creation of new knowledge in mathematics will continue the development of the University as a national and international resource. The faculty of the Department is also committed to providing high quality service to the University, the state and federal government, and the mathematics profession.

The Rutgers Department of Mathematics is usually ranked among the top 15 departments nationally for scholarly excellence. Faculty hiring and promotions are primarily based on scholarly promise and achievement, with appropriate attention given to teaching and service. Managing the mathematical enterprise is not simple. About 20,000 students take mathematics courses during the calendar year (even the summer school runs more than 55 math courses!). Almost all students select majors which require some mathematics courses. Also, most undergraduate colleges have quantitative skills requirements which students must satisfy. These requirements are often satisfied by taking mathematics courses.

What is the course’s role in the undergraduate curriculum?

The course was directed at students with minimal college mathematics preparation who wanted to satisfy the quantitative skills graduation requirement of various colleges.

Rutgers is one of many U.S. institutions of higher education which have graduation requirements involving quantitative reasoning. The ability to think quantitatively and to have a certification that such is possible (a passing grade in an appropriate course!) is probably good. Calculus or precalculus courses certainly fulfill that requirement. Therefore the majority of our undergraduates are now covered, since most undergraduate majors require one of those courses. Unfortunately, the learning communities fostered by many of these courses are not ideal.

At many schools the courses intended to satisfy the quantitative reasoning graduation requirement are not pleasant experiences for the instructors and the students. The students emphatically view these courses as an evil and irrelevant hindrance to their academic journey. The instructors may think that courses discuss esoteric and possibly uninteresting topics taught to an unwilling clientele.

At Rutgers-New Brunswick, the academic niche for a mathematics course directed at non-majors with modest mathematical background is Math 103, called Topics in Mathematics for the Liberal Arts. This 3 credit course has two 80-minute meetings each week for 14 weeks.

I realized that a certain portion of these students are frightened of mathematics. Even after I got to know each class, I saw that when I turned to the board and wrote 10100 or (worse) AB that some of those in the room would blink nervously or laugh or look blank. Ingrained in these intelligent students was a fear or distaste for math as an object of intellectual interest. I don’t know whether this was inborn or if it resulted from previous educational experiences, but great care and gentle persistence must be used. The students must be enticed to study math. By contrast, I’ve taught engineering students for much of my instructional career. Many of these students are correctly sure that math is a principal key to success in an engineering curriculum. Some students in Math 103 are almost as mournfully sure that math is their personal fate, with unpleasant connotations.

The math background I requested was good knowledge of Algebra 2, and some knowledge of analytical geometry. I remarked that some involvement with computers (use of the Web for research on papers and elementary use of Maple) would be part of the course.

Funding Source

Partial support for the creation of this course was provided by the National Science Foundation under grant DUE-9850071.

Related Resources


Backgrounder #4 by Etkina & Mestre, “Implications of Learning Research for Teaching Science to Non-Science Majors”

Backgrounder #5 by Ferguson, “Mathematical and Statistical Reasoning in Compelling Contexts: Quantitative Approaches for Building and Interrogating Personal, Disciplinary, Interdisciplinary and Worldviews”

Backgrounder #7 by McGuire, “Reinventing Myself as a Professor: The Catalytic Role of SENCER”


May 2004, “The SENCER Model Series: Seven New Models Introduced for SSI 2004” (p. 4)

Sept. 2004, “Implications of Learning Research ..” (p. 4)

Nov. 2004, “SENCER SALG Update: Math SALG Developed” (p. 4, 6)

Outside resources:


Both the students and the instructor can be helped by choosing a suitable text to support the “technical” part of the course. Amazon today lists more than 3,000 titles about cryptography. Only a few among these might be suitable for the course described here. The book should be written at an appropriate level and in an appropriate style. Below is a list of titles I would inspect. There certainly may be other suitable texts.

  • Cryptology by Albrecht Beutelspacher (Spectrum), 172 pp paperback, MAA, 1996, $38.95
  • Cryptological Mathematics (Classroom Resource Material) by Robert Edward Lewand, 214 pp paperback, MAA, 2000, $38.95. This is supported by a Web page.
  • Cryptography Demystified by John Hersey, 356 pp paperback, McGraw-Hill, 2002, $34.96
  • Invitation to Cryptography by Thomas H. Barr, 396 pp hardcover, Pearson, 2002, $78.67

Here are two books republished from an earlier era, before public key cryptography and the widespread use of computers. The authors were professionals in the field.

  • Cryptanalysis by Helen F. Gaines, 237 pp paperback, Dover, $8.95
  • Elementary Cryptanalysis: A Mathematical Approach (New Mathematical Library) by Abraham Sinkov, 232 pp paperback, MAA, $25.95

Finally, two books on the history of the subject. The first gives a British point of view, especially interesting regarding the discovery of public key cryptography inside the British version of the National Security Agency before this was done by members of the academic community. The second book, a huge text, is the standard reference on the history of cryptography.

  • The Code Book: The Science of Secrecy from Ancient Egypt to Quantum Cryptography by Simon Singh, 411 pp paperback, Anchor Books/Doubleday, 1999, $15.00
  • The Codebreakers: The Story of Secret Writing by David Kahn, 1200 pp hardcover, Scribner, 1966, $70.00

There are many other books, both nonfiction and fiction with cryptographic themes. Modern fiction with some cryptographic content ranges from a Dorothy Sayers mystery where Lord Peter Wimsey decrypts a message written using the Playfair cipher, to the recent novel Cryptonomicon by Neil Stephenson, which essentially has a tutorial on cryptography and its history buried in its 1168 pages!