Summary

Cryptography is the science of designing algorithms and protocols that guarantee privacy, authenticity, and integrity of data when parties are communicating or computing in an insecure environment. The recent explosion of electronic communication and commerce has expanded the significance of cryptography far beyond its historical military role into all of our daily lives. For example, cryptography provides the technology that allows you to use your credit card to make on-line purchases without allowing other people on the internet to learn your credit card number.

The past 25 years have also seen cryptography transformed from an ad hoc collection of mysterious tricks into a rigorous science based on firm complexity-theoretic foundations. It is this modern, complexity-theoretic approach to cryptography that will be the focus of this course. Specifically, we will see how cryptographic problems can be given precise mathematical definitions. Then we will construct algorithms which provably satisfy these definitions, under precisely stated and widely believed assumptions. For example, we will see how to prove statements of the flavor "Encryption algorithm X hides all information about the message being transmitted, under the assumption that factoring integers is computationally infeasible." (Of course, this kind of statement will be given a precise meaning.)

What can you hope to learn from this course?

Definitions: Why it is important to precisely define cryptographic problems, and how to do so for several important problems (encryption, authentication, digital signatures, ...). The kinds of subtleties that arise in such definitions, and how to critically evaluate and interpret cryptographic definitions.

Constructions & Proofs of Security: Examples of general & concrete solutions to various cryptographic problems, and how to prove that they satisfy the definitions mentioned above (based on precisely stated assumptions).

Foundations: The assumptions on which modern cryptography is based, and their implications.

Theory vs. Practice: This course will focus on theory, but we will discuss how the theory relates to what is actually done in practice.

Applications: If time permits, we will see one or two examples of how to address cryptographic issues in higher-level protocol problems, such as auctions, voting, or electronic cash.

Security: This is not a course on security, but if time permits, we will discuss how cryptography fits into the broader contexts of network and systems security.

What this course will NOT teach you:

Acronyms: There are many different cryptographic algorithms, protocols, and standards out there, each their own acronym. It is not the aim of this course to cover these specific systems, which may come and go, but rather the general principles on which good cryptography is based. Understanding these principles will enable you to evaluate the specific systems you encounter outside this course, on your own. (This is not to say that the course will be without examples, but the examples will be selectively chosen mainly for illustrative purposes.)

Hacking: We will not teach you how to "break" or "hack" systems.

Security: We will not teach you "how to secure your system". Cryptography is only one part of security, albeit an important one.

Everything there is to know about cryptography: Cryptography is a vast subject, and we will not attempt to be comprehensive here. Instead, we aim to convey the main principles, philosophy, and techniques which guide the subject, focusing on the most basic primitives, such as encryption and digital signatures. This should put you in a good position to read about other topics on your own or take more advanced courses on cryptography.

Tentative List of Topics

Introduction
Review of Algorithms and Probability
Private-Key Encryption: Defining Security
Computational Number Theory
One-Way Functions
Pseudorandom Generators & Pseudorandom Functions
Private-Key Encryption: Constructions
Private-Key Encryption in Practice: Block Ciphers
Trapdoor Functions & Public-Key Encryption
Message Authentication, Digital Signatures, and Hashing
Zero-Knowledge Proofs
Protocols
Network & Systems Security
Policy Issues
Conclusions & what we didn't cover

Prerequisites

The formal prerequisite for the course is one prior course in theoretical computer science, such as CSCI E-207 or E-124. (Students with strong math backgrounds may be able to manage with extra background reading and/or taking E-124 concurrently; come to my office hours to discuss.) The main skills that will be assumed from these courses are:

The ability to understand and write formal mathematical definitions and proofs.
Comfort with reasoning about algorithms, such as proving their correctness and analyzing their running times.

It is also important that you are familiar with basic probability . Additional background that will be helpful:

Complexity Theory: NP-completeness, reductions
Randomized Algorithms, such as a primality testing algorithm.
Basic Number Theory: modular arithmetic, Chinese Remainder Theorem.
Probability Theory: independence, conditional probabilities, expectation, Bayes' Law.

While it is not necessary to have had exposure to all of these topics prior to CSCI E-177, familiarity with none will probably make it quite difficult to keep up.

Grading

Weekly problem sets: 50%
Two midterm quizzes: 10% each
Final exam: 25%
Class participation: 5%

Your class participation grade is based on participation in sections, but can also be boosted by participation in section, emailing the course staff, and/or coming to office hours or section with "good" questions or comments. A "good" question is one which is not just aimed to help you answer questions on the problem set or exam. It is one that shows genuine interest in the material and that you have been thinking about the course material on your own. Do not be afraid of asking "stupid" questions! Class participation also includes viewing the online lectures, participating in sections either in person or online, and participating in discussions on the course website. We will account for the different nature of class participation for distance students when computing their class participation grades.

Problem Sets & Collaboration Policy

The course will have weekly problem sets, due as posted on each assignment and the course website. They will be due either in the course box marked CS120 in the basement of Maxwell Dworkin or electronically by submission to the Assignments dropbox on the course website. If you prefer to handwrite your assignments, you may scan them and submit them electronically instead of submitting them in the course box. Please remember to write your name on the assignment and indicate it is for CSCI E-177.

Assignments are due on the date specified on the assignment. It is important that students keep up with the lecture material by completing the assignments on time; in addition, we cannot return graded assignments or solutions until all students have handed in their assignments. Late work will only be accepted in case of exceptional unforeseen circumstances by prior application to the teaching assistant: if something comes up that may prevent you from turning your work in on time, ask for more time immediately, not when you run out of time the day before the assignment is due. Project deadlines at work and obligations for other courses are not exceptional, and you should budget your time accordingly.

Collaboration on homework is permitted with small groups of other students, provided that you limit your collaboration to verbal discussion of solutions in general terms and not specific language of the solutions to be handed in. Because email tends to result in specific, crafted language, students are expressly forbidden from exchanging email about solutions to problem sets. You may collaborate via in-person meetings, telephone, instant messenger, or web conference; please clearly indicate the names of any collaborators on your assignment.

Sections

There will be weekly sections, which will be used to clarify difficult points from lecture, review background material, go over previous homework solutions, and sometimes provide interesting supplementary material. We will attempt to schedule sections at a time when distance students can participate over online chat. Section times will be arranged in the first week or two of class.

Readings

There is no required text for the course other than the lecture notes, but you may find the following to be useful references (but beware that some of the notation, conventions, and definitions may differ slightly from lecture):

Jonathan Katz and Yehuda Lindell. An Introduction to Modern Cryptography. This is a preliminary version of a textbook in-writing that the authors have graciously allowed us to use. Its level and contents seem to fit this course very well, so copies of the relevant chapters will be handed out in section and available for printing via the course website. The preliminary state of the book means, however, that some chapters are not yet written (particularly the ones relevant to the beginning of the course) and that there may be some errors. In return for the authors' sharing this book with us, we should compile a list of errors and constructive suggestions to send the authors at the end of the term. We will set up a discussion tool on the course website for this purpose.
Oded Goldreich. Foundations of Cryptography. This two-volume set is a very comprehensive and definitive treatment of the theoretical foundations of cryptography. Volumes I and II cover most of what we'll be doing in this course far greater depth, though the treatment is more abstract than ours. Volume I contains most of the still-unwritten material from the Katz-Lindell text. If you plan to continue on in cryptography (particularly as a researcher), I highly recommend purchasing these books.

Other texts on cryptography take a much less careful approach to definitions and proofs of security than we do. Still, they can serve as good references for more examples of concrete cryptosystems used in practice and some high-level ideas. After this course, you should understand how to critically evaluate the merits or deficiencies of the cryptosystems described in the books below (and indeed we urge you to have a critical eye when reading them):

Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone. Handbook of Applied Cryptography.
Douglas R. Stinson. Cryptography: Theory and Practice.
Bruce Schneier. Applied Cryptography.

For background reading on probability, algorithms, complexity theory, and number theory, I recommend:

Thomas Cormen, Charles Leiserson, Ron Rivest, and Cliff Stein. Introduction to Algorithms.
Michael Sipser. Introduction to the Theory of Computation.

Related Courses at Harvard Extension School

CSCI E-207 - Introduction to Formal Systems and Computation, CSCI E-124 - Data Structures and Algorithms. The two possible prerequisites. Both are highly recommended and offer material not covered in this course.
MATH E-313 - Classical Mathematics II: Number Theory and Analysis. Number Theory is the main source of the hard computational problems on which cryptography is built. An excellent complement if taken before or after this course.