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CS 120: Introduction to Cryptography
| Problem Sets
& Collaboration Policy | Sections | Readings
Lecturer: Prof. Salil Vadhan
and Dr. Alon Rosen
Teaching Fellow: Florin Ciocan
Course website: http://people.seas.harvard.edu/~salil/cs120
Staff e-mail: firstname.lastname@example.org
Past CUE evalutions: Fall 01, Spring 03
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
- 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
- Review of Algorithms and
- Private-Key Encryption:
- Computational Number Theory
- One-Way Functions
- Pseudorandom Generators
& Pseudorandom Functions
- Private-Key Encryption:
- Private-Key Encryption in
Practice: Block Ciphers
- Trapdoor Functions &
- Message Authentication,
Digital Signatures, and Hashing
- Zero-Knowledge Proofs
- Network & Systems
- Policy Issues
- Conclusions & what we
The formal prerequisite for the course is one prior course
in theoretical computer science, such as CS 121 or 124. (Students
with strong math backgrounds may be able to manage with extra background reading
and/or taking CS 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
It is also important that you are familiar with basic probability . Additional background that will be
- Complexity Theory:
- Randomized Algorithms, such
as a primality testing algorithm.
- Basic Number Theory: modular
arithmetic, Chinese Remainder Theorem.
- Probability Theory:
independence, conditional probabilities, expectation, Bayes'
While it is not necessary to have had exposure to all
of these topics prior to CS 120, familiarity with none will probably make it
quite difficult to keep up.
- Weekly problem sets: 50%
- Two in-class quizzes: 10%
- Final exam: 25%
- Class participation: 5%
Your class participation grade is based on participation in
lecture, but can also be boosted by participation in section 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"
Problem Sets & Collaboration Policy
The course will have weekly problem sets, due TBA (in the
box labelled CS 120 in the basement of Maxwell Dworkin.) You are allowed 6 late days for the semester, of
which at most 2 can be used on any individual problem set. (1 late day = 24
hours exactly). In case of an emergency which requires an exception to
these rules, please have your senior tutor call me.
Students are encouraged to discuss the course material and the homework
problems with each other in small groups (2-3 people).
Discussion of homework problems may include brainstorming and verbally walking
through possible solutions, but should not include one person telling the
others how to solve the problem. In addition, each person must write up
their solutions independently, and these write-ups should not be checked
against each other or passed around.
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.
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 CS
120 very well, so copies of the relevant chapters will be handed out in
class. 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):
For background reading on probability, algorithms,
complexity theory, and number theory, I recommend:
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