COMS W4261: Introduction to
Lecture Summaries |
Required and Suggested Reading |
Grading and Policies |
- 12/14/16 The exam location tomorrow is in the usual classroom: 1024 Mudd, 10am-12noon. Good luck!
- 12/6/16 Office hours for next week:
- Mon 12/12:
- Tue 12/13:
- 1:00-3:00 Luke
- 5:00-7:00 Edo
- Wed 12/14:
- 10:00-12:00 Prof Malkin
- 2:00-4:00 Edo
- 12/6/16 Homework 5 is on the Homework page. It is due next Tue by 11pm
electronically on courseworks. No late days will be accepted. This
homework is shorter and counts as half a homework for grading
- 12/6/16 Reminder: the exam will take place Thursday
December 15 10am-12noon (location to be determined). The exam is
open book open notes, but no electronic equipment and no sharing of
materials during the exam. Students are responsible for everything
covered in class starting with lecture 15 and on the homework
starting with HW 3.
- 12/3/16 Office hours change this week: Prof Malkin's office hours on Wednesday 12/7 will be 8:50am-9:50am (instead of 1-2). Other office hours will be held as usual.
- 11/30/16 Edo will lead a recitation on Friday 12/2
from 12-1:15 in our usual classroom, focusing on the number
theoretic material that we are using.
This is a three-credit graduate level course. It can be
credited to all degree programs, subject to advisor approval. It is also
a theory elective for the PhD program in computer science, a suitable elective
for the MS foundations or security tracks, and a suitable class for undergrads
We meet Tuesday and Thursday, 10:10-11:25am at 1024 Mudd.
Questions? Email the instructors and/or the TAs.
(tal at cs)
Office hours: Wednesday 1:00pm-2:00pm, Thursday 11:30am-12:30pm, 514 CS Building
Ghada Almashaqbeh (ghada at cs)
Office hours: Monday 3:30-5:00pm, DSI Mudd 4th floor
Luke Kowalczyk (luke at cs)
Office hours: Tuesday 1:00-3:00pm, DSI Mudd 4th floor
Edo Roth (enr2116 at columbia)
Office hours: Wednesday 2:00-4:00pm, TA room Mudd 1st floor
Class Description and
Lectures and Readings
This course is an introduction
to modern cryptography. In general, cryptography aims to construct
efficient schemes achieving some desired functionality, even in an adversarial
environment. For example, the most basic question in cryptography is that
of secure communication across an insecure channel: Can Alice send a message to
Bob so that Bob understands the message, but no eavesdropper does? How
can Bob be sure that the message received was sent by Alice? Another question is that of
secure computation in an insecure environment: Can a group of parties
perform some distributed computation (e.g., coordinate an attack, or tally a
vote), so that an adversary controlling the communication channels and some of
the parties cannot disrupt the computation or learn extra information?
While cryptography is an ancient field, the emergence of modern cryptography
in the last few decades is characterized by several important features
distinguishing it from classical cryptography. For one thing, the
availability of computers and the wide spread of networked information systems
and the Web, has dramatically increased both the need for good cryptography,
and the possibilities that it can offer. In addition to the classical
military and national security applications, a wide scope of financial, legal,
and social cryptographic applications has emerged, from using a credit card
on-line or sending an encrypted email, to more ambitious goals of electronic
commerce, electronic voting, contract-signing, database privacy, and so on.
The most important characteristic of modern cryptography is its rigorous,
scientific approach, based on firm complexity-theoretical foundations.
In contrast to the classical approach based on ad-hoc solutions (design a
scheme that seems very hard to break, and hope for the best), modern
cryptography aims for specific, rigorously quantifiable security guarantees,
based on precise mathematical definitions and provably secure protocols.
What You Will Learn in This Class (Hopefully!)
- Definitions: Why
and how to identify, conceptualize and rigorously formalize goals (e.g.,
what does it mean for communication to be secure?)
- Protocol design and
analysis: Constructions and proofs of security (according to
- Foundations: Limits of
what is possible to achieve, computational assumptions and their
implications in cryptography.
The principles and techniques
underlying the above will be illustrated through specific examples drawing from
the basic cryptographic primitives. Through these examples, which are
very important on their own, you will also learn to critically evaluate and
interpret cryptographic definitions and security proofs (i.e., what is
and what is not guaranteed?).
While the class will focus on the theoretical foundations, we will discuss the relation
to how things are actually done in practice.
The material covered in the class should prepare you to make sense of some
current research papers in cryptography, and to study further on your own (or
take an advanced class). Opportunities for research under my supervision
may be available for interested students who do well in the class.
Tentative List of Topics
The following is an ambitious list of topics to be covered.
Depending on time, some of the topics may be omitted.
(perfect) secure encryption: one-time pad, Shannon's
- Pseudorandom generators,
functions, and permutations, one-way functions and permutations, hard-core
- Number theory and
computational hardness: factoring, RSA, discrete-log, DH, DDH
- Private-key encryption:
definitions of security and constructions, block-ciphers and (a
- Trapdoor functions and
permutations, key exchange
- Public-key encryption:
definitions of security and constructions
- Message authentication codes,
digital signatures, hash functions
- Zero knowledge proofs,
- Protocols: secret-sharing,
oblivious transfer, secure multi-party computation, and higher level
protocols and applications as time permits
What You Will Not Learn in This Class
The following topics are outside of the scope of this class.
Some aspects of these topics are taught in COMS W4180 (Network Security),
COMS 4187 (Security Architecture and Engineering), COMS E6184 (Anonymity
and Privacy), and COMS W3995 Computers and Society classes.
- Details of specific
standards currently used: We focus on the general underlying
principles in cryptography (with examples to illustrate them), rather than
describing the many specific schemes currently in use.
- How to implement the
studied protocols: We focus on the theoretical aspects,
including asymptotic efficiency and provable security.
Implementation of the protocols involves additional issues, both in
terms of security and efficiency, depending on the architecture, operating
system, hardware, and so on.
- How to secure your
computer: Keep in mind that cryptography is only one (quite
important) part of security. Having good cryptography is necessary,
but not sufficient, to assure the security of your system.
- Social and legal issues: We
focus on cryptographic technology. How this technology could and
should be used, and towards which goals, is a fascinating subject, not
- Everything worth knowing
about cryptography: We focus on the main underlying ideas and
the basic primitives. This should prepare you to study further on
your own (or take an advanced cryptography class), in order to explore the many exciting cryptographic topics that we will not get to here.
We will use the book “Introduction to Modern
Cryptography” by Jonathan Katz and Yehuda Lindell, Chapman and Hall/CRC
Press, 2nd edition. This book will be on reserve in the engineering library, and
available from the Columbia bookstore.
Additional papers and handouts
may occasionally be distributed in class.
Recommendations for some other textbooks (not required) appear here.
- Mathematical maturity,
comfort with abstract reasoning, ability to understand and write formal
definitions and proofs.
- Strongly recommended:
one of COMS W4231 (analysis of algorithms) or COMS W3261
(computability and models of computation) or COMS W4236 (introduction to
computational complexity), or equivalent.
The following skills will be assumed:
- Comfort with basic discrete
math and probability (COMS W3203 or equivalent)
- Comfort with Big-O notation,
ability to reason about algorithms (correctness, running time)
It will also help to have background in at least some of the
- Complexity theory (polynomial
time algorithms, NP-completeness, reductions)
- Randomized algorithms (what
is a randomized polynomial time algorithm?)
- Probability theory (what is
conditional probability? how to compute expectation?)
- Basic number theory (modular
arithmetic, Chinese Remainder Theorem, ...)
These topics will be briefly covered in class as needed, but if you do
not have any background in any of them, you are likely to find it hard to keep
The appendix of the textbook reviews some background, and
additional references for background reading can be found here.
Grading and Policies
The grading will be based on homework (50%) a
midterm (25%) and a final (25%).
The midterm will take place November 3rd during class, and
the tentative date for the final is during finals week (as scheduled
Students are expected to attend class.
No laptops or other electronic equipment are to be used during class,
unless you have obtained prior approval from the instructor.
See the Homework page for homework
All students are presumed to be aware of the departmental policy
regarding academic honesty.