General Information

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 as well.  
We meet Tuesday and Thursday, 10:35-11:50am at 1024 Mudd.

Use the links above to navigate the class pages.

Questions?  Email the instructors and/or the TAs.  

Instructor

Teaching Assistant

Class Description and Syllabus 

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 three 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 established definitions). 
• 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.

• Information-theoretic (perfect) secure encryption: one-time pad, Shannon's impossibility result 
• Pseudorandom generators, functions, and permutations, one-way functions and permutations, hard-core predicates 
• Number theory and computational hardness:  factoring, RSA, discrete-log, DH, DDH
• Private-key encryption: definitions of security and constructions,  block-ciphers and (a little) cryptanalysis
• Trapdoor functions and permutations,  key exchange 
• Public-key encryption:  definitions of security and constructions 
• Message authentication codes, digital signatures, hash functions
• Zero  knowledge proofs, identification protocols
• 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), and COMS E6184 (Anonymity and Privacy) classes.  

• Details of specific protocols and 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 addressed here.
• 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.

Required Text

We will use the book “Introduction to Modern Cryptography” by Jonathan Katz and Yehuda Lindell, Chapman and Hall/CRC Press. 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. 

Prerequisites

• 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 following areas:

• 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, but if you do not have any background in any of them, you are likely to find it hard to keep up.

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 (40%), midterm (30%), and final (30%). Dates and format of the midterm and final will be announced.
See the Homeowrk page for homework policies.

All students are presumed to be aware of the departmental policy regarding academic honesty.