Computer Science 433
· Take-home final is now available - deadline is January 17th, 2006 12:00pm (noon)
· Deadline for course project - January 13th, 2006 1:30pm.
An introduction to modern cryptography with an emphasis on the fundamental ideas. We will survey both the basic information and complexity theoretic concepts as well as their (often surprising and counter-intuitive) applications.
Click on the lecture number for notes and/or slides, and links to additional reading material:
Professor: Boaz Barak - 405 CS Building. Email: Phone: 258-0255 (I prefer email)
Undergraduate Coordinator: Donna O'Leary - 410 CS Building - 258-1746 firstname.lastname@example.org
Teaching Assistants: David Xiao ( dxiao@cs )
course mailing list.
Grading: 50% homework, 25% project, 25% take-home final. See syllabus for more details.
This course will be an introduction to modern "post-revolutionary" cryptography with an emphasis on the fundamental ideas (as opposed to an emphasis on practical implementations). Among the topics covered will be private key and public key encryption schemes (including DES/AES and RSA), digital signatures, one-way functions, pseudo-random generators, zero-knowledge proofs, and security against active attacks (e.g., chosen ciphertext (CCA) security). As time permits, we may also cover more advanced topics such as the Secure Socket Layer (SSL/TLS) protocol and the attacks on it (Goldberg and Wagner, Bleichenbacher), secret sharing, two-party and multi-party secure computation, and quantum cryptography.
There are no formal prerequisites for the course, but I will assume that students are able to read and write mathematical proofs. In addition, familiarity with algorithms and basic probability theory will be helpful. I recommend that CS majors take this course after COS 226 and COS 341.If you're interested in the course but are not sure you have sufficient background, or you have any other questions about the course, please contact me at
There are several lecture notes for cryptography courses on the web. In particular the notes of Vadhan, Bellare and Rogaway, Goldwasser and Bellare and Malkin will be useful.
Some good sources for the probability and complexity/algorithms backgrounds are:
A good source for computational number theory is A Computational Introduction to Number Theory and Algebra by Victor Shoup. Note that this book freely available on-line under the creative commons license. Another good book on this topic is A Concrete Introduction to Higher Algebra by Lindsay Childs.
Some other more application-oriented crypto books (note that these books take a much less careful approach to definitions and security proofs than we do in the course):
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.
Ross Anderson Security Engineering
Additional reading: You can find more information about historical ciphers on the web page Alex Biryukov's wonderful Course on Cryptanalysis.
Mathematical proofs: Some students asked me for material on reading, writing and coming up with mathematical proofs. Chapters 1 and 3 of the Lehman-Leighton notes of an MIT course can be useful. Some tips on mathematical writing in general and proofs in particular can be found in these few pages from Knuth, Larrabee, and Roberts. On a lighter and more general note, you might be interested to read Keith Devlin's musing about mathematical proofs.
Reading for next class: We'll start to use probability a lot (although only
very basic things). The handout contains some references. In particular
you might want to take a look at this short
handout by Luca Trevisan.
We'll start also thinking about defining security for encryption schemes. Throughout this course the theme of such definitions will be rigor - mathematical precision and being conservative - making very strong demands on the security. In pages 20-25 of Goldreich's book (Volume 1) he gives a nice description of the motivation behind this approach.
Additional reading: Lecture 2 of Bellare's course discusses the issues in defining security for encryption schemes and perfect security. See also Section 6.4 in the Golwasser-Bellare lecture notes. The definition of perfect secrecy was first given by Shannon in this 1949 paper, but our discussion followed more closely the approach of Goldwasser and Micali who, when referring to the indistinguishability definition for encryption schemes, said: "A good disguise should not allow a mother to distinguish between her own children".
Reading for next class: Next class we'll discuss computational models such as
Boolean circuits and Turing machines. You might want to take a look at
pages 351-360 and 368-375 of Sipser's book. Also, I prepared a description of the computational models we use and the relationships between them. See also this picture
Additional reading: Computational complexity is covered in many places and in particular in Sipser's book. If you prefer PowerPoint slides you can look at Muli Safra's complexity course. In particular the first 5 presentations there (Introduction, Preliminaries, Reductions, Cook Theorem, and NP-complete Problems) roughly cover the material we discussed (and of course also some things we did not discuss). As I already mentioned, once we have an impossibility result, the right thing to do is to try to bypass it. This holds also for NP-completeness results where once a problem is NP-complete, and hence is probably not efficiently solvable, people try to approximately solve it (for example, if we can't color a graph in the minimum number of colors, try to color it within a factor of at most K times the minimum for some k.). This web site tracks the current approximation status of many problems. In many cases we can prove that it is NP-hard to even approximate some problems. For a good exposition of this, see Sanjeev Arora's thesis.
For next class: Next class we'll start use computational hardness for cryptography.
There is no particular source to read, but you might want to think whether or not
we can use worst case hardness for cryptography, and if not, what sort of
hardness will we require.
Additional reading: You should look at Goldreich's
Treatment of pseudorandom generators
(Volume I, pages 101-117). He uses somewhat different notations than we do
(in particular working with the asymptotic definition of computational
indistinguishability and pseudorandomness, and using a uniform model
of computation rather than Boolean circuits).
Additional reading & reading for next class: I believe it is very instructive to read
and compare Goldreich's
proof of PRG length extension
(Volume I, pages 114-117) to ours.
Goldreich Volume II (Chapter 5) contains an extensive discussion of the definitions of encryption schemes.
Additional reading: Pseudorandom functions were defined and constructed by Goldreich, Goldwasser, and Micali -
(see this page for the paper, containing
also the full proof).
As mentioned, there are other more efficient candidate constructions,
including HMAC by Bellare, Canetti and Krawczyk
and a factoring-based PRF by Naor, Reingold and Rosen.
Additional reading: You should take a look at the Bellare-Rogaway chapter on pseudorandom functions and permutations. It does not cover exactly the same material we do (which is why it would be especially worth your while to look at it). Pseudorandom permutations, and their construction based on pseudorandom functions is covered in Goldreich Vol I (link is for the older web version, see there section 3.7 page 114).
A lot more information on the AES and other block ciphers can be found on the web page for Eli Biham's modern cryptology course. In particular, this lecture covers block ciphers. See The AES Lounge for more information about the AES, its security and implementations.
If you are interested in the principles behind the design and attack of block ciphers, see this tutorial by Howard Heys and this course by David Wagner.
Finally if the skipjack/clipper story whetted your appetite for crypto-conspiracies you might want to look at this site.
For next week: Next week we'll begin to talk about the goal of integrity. There's nothing in particular you should read but try to think of the following questions:
Additional reading: Lectures 9 and 10 in David Wagner's
cryptography course discuss MACs, including examples of real-world protocols that can be attacked
due to lack of MACs.
Additional reading: The material about the order of encryption vs. authentication is from this CRYPTO 2001 paper by Hugo Krawczyk. See this expository paper by Victor Shoup for more on the motivation behind chosen ciphertext security. You can find here the CRYPTO 98 paper of Daniel Bleichenbacher that attacked the SSL protocol, mainly using the fact that the underlying encryption scheme was not CCA-secure.
Reading for next class: Next week (probably on Thursday) we'll start discussing
number theory, in preparation for public key encryption schemes. Please take a
look at Shoup's book (available freely on the
web). The more you read of this book the better, however, please read at least Chapter 1 (pages 1-10).
Particularly relevant parts of the rest of the book are:
Chapter 2 (up to and including Section 2.5), first 2 pages of Chapter 7, first 5 pages
of Chapter 8, Chapter 10 (up to and including 10.4.1), first two pages
of Chapter 11, Chapter 12 and Chapter 13.
Additional reading: One-way permutations and the hard-core bits are covered extensively in Goldreich Volume 1 (although it is perhaps too extensive for our purposes). Vadhan's lecture notes cover one-way functions, Commitment schemes and hardcore bits (see also lecture 10 and 11 there). Luca Trevisan's lecture notes on pseudorandomness have a nice presentation of the proof of the Goldreich-Levin theorem. Note: You might want to look at these sources after you tried to tackle Exercise 1 on your own.
In many places there is an emphasis not so much on one way permutations but on one way functions. One-way functions are a generalization of one-way permutations in the sense that every one-way permutation is a one-way function but not necessarily vice versa. The definition is the natural way you'd generalize the definition one-way permutations to the case where the function f() may not be one-to-one: since for a given y there may be many x's such that y=f(x), adversary is successful if it manages to find one of them.
Reading for Thursday: In addition to Chapter 1 (pages 1-10)
please also read the first 5 pages of Chapter 8
in Shoup's book (pages 180-184, not including
Additional reading: Please take a look at the notes for
this lecture by Shafi Goldwasser.
As mentioned above, Shoup's book is an excellent resource
for computational number theory.
Additional reading: There are many sources for public key encryption.
I particularly recommend the Goldwasser and Bellare.
Dan Boneh has an excellent survey on the
Decisional Diffie Hellman (DDH) assumption we mentioned in class.
Additional readings: One of the best non-technical explanation of zero knowledge appears in this paper by Naor, Naor and Reingold published in the prestigious Journal of Craptology.
The most extensive treatment of Zero Knowledge is in Goldreich, Vol I, Chapter 3 (see also fragments on the web). For a shorter version, see chapter 4 of foundations of cryptography -- a primer). Goldreich also has tutorial on zero knowledge including the basic notions and more recent developments as well. protocol QR I described in class is also described in these UCB computer security lecture notes.
You can find on line the
original paper of Goldwasser, Micali and Rackoff presenting zero knowledge. Zero knowledge is also the topic
of many dissertations (including my own), I particularly recommended
Uri Feige's thesis
Additional reading: I strongly recommend you look at the following lecture notes of an MIT course by Silvio Micali (one of the inventors of zero knowledge). Lectures 1 to 5 cover the material we talked about in class. If you are interested in more about zero knowledge then the rest of the lectures are a good place to start. A good overview of the material is in pages 1 to 17 of Goldreich's tutorial on zero knowledge. For full proofs see Goldreich's book or the fragments on the web. In particular, reduction of error by sequential composition is covered in section 4.3.4 of the fragments, and the protocol for 3-coloring is covered in section 4.4.
If you like slides, you can see some
of this material in PowerPoint format from Muli Safra's course (see also
these slides from Ely Porat).
Additional reading: Fuller exposition and proofs for this material can be
found in Chapter 6 of Goldreich's book (Vol II) or in the fragments on the web.The
Goldwasser-Micali-Rivest factoring based hash function is obtained through the intermediate notion
of claw-free permutations. This construction is described somewhat tersely
in the Golswasser-Bellare notes and with a bit more detail in Goldreich's sections
2.4.5 (Vol I) and 220.127.116.11 (Vol II),
Additional reading: The paper of Bellare and Rogaway suggesting the random oracle model can be found here. One of the most powerful critiques of this model is in this paper by Canetti, Goldreich and Halevi (be sure to look at the authors' opinions at the end). Helger Lipmaa collected some links related to the random oracle model.
As mentioned in class, Bellare and Rogaway built on
an earlier work of
Fiat and Shamir that gave a different construction for signature
and identification schemes using a "crazy" hash function based on zero knowledge proofs.
The Bellare-Rogaway signature scheme with a tighter security proof can be found
Additional reading: Chosen ciphertext security was defined by Rackoff and Simon. As mentioned in the notes for lecture 9, Victor Shoup has a very nice expository paper about this concept. The first construction that was proven to be chosen-ciphertext secure was given in this paper of Dolev, Dwork and Naor. However, this construction is quite complicated and inefficient. Perhaps the simplest (although rather inefficient) construction and analysis of a "random-oracle free" CCA-secure encryption is given in this paper by Lindell. Both constructions use an ingredient that we did not talk about in class (non-interactive zero knowledge proofs) but is described in Goldreich's book (Vol I).
The scheme we presented in class was taken from the original random-oracle paper by Bellare and Rogaway. An efficient encryption scheme with a proof of "almost CCA security" in the random oracle model is the OAEP scheme of Bellare and Rogaway. However, in this paper by Shoup he shows some "holes" in that proof and gives a different random-oracle based construction. Perhaps the simplest and most efficient encryption that has a proof of CCA security in the random oracle model is this one by Dan Boneh.
In this wonderful paper of Cramer and Shoup they present an efficient encryption scheme that has a "real" (no random oracles) proof of CCA security based on the DDH assumption.
Even if a scheme is proven to be CCA secure, this only implies real world security
if the real world adversary does not have access to additional information
from observing say the time it takes servers
to answer queries or other such things - see this
paper for a demonstration of this issue.
Additional reading: See Bleichenbacher's paper for more information about his attack on RSA as used in the SSL protocol. Some related attacks can be found here. As mentioned in the reading for the previous lecture, even when using assumed CCA secure schemes, an adversary may use timing and/or error message information to attack a scheme, as demonstrated in this CRYPTO 2001 paper by Manger.
The SSL protocol is also described in
these notes by Lindell. Some attacks on SSL V3.0 are described in
this paper by Schneier and Wagner (although it is pre-Bleichenbacher, and so does not
include many strong attacks, to quote from a summary of a talk by Wagner on this work:
"In general, this analysis was informal, not formal, meaning that it can only illustrate
flaws in the protocol, not prove that it's correct."). The attack on the
pseudorandom generator used by Netscape
was given in this
paper by Goldberg and Wagner.
Additional reading: You can find more about PGP key reconstruction on the PGP user guide.
Here's a nice puzzle
about visual cryptography. You can find the relevant papers from Moni
Naor's web page. See also Doug Stinson's page on
You can find a nice and relatively simple threshold RSA signature scheme in
this paper by Shoup. See this paper of Tal Rabin
on how to convert the scheme we saw in class to a general threshold, robust, and proactive scheme.
I was actually once involved in writing
Java implementation of
proactively secure protocols.
Additional reading: Dan Boneh's group has a
web page on Identity Based Encryption where you can
find the original Boneh-Franklin paper and also download encrypted email software based on IBE.
The forward-secure encryption scheme given in class is from
this note by Canetti and Halevi which is a simplification of the construction of
this paper by Canetti, Halevi and Katz (the latter however is
better in the sense that it does not need a large storage by the sender and also does not
use the random oracle model). See this paper by Bellare
and Yee for a treatment of forward security for private key primitives.
Additional reading: Some web resources on oblivious
transfer are this page by
Benn Lynn and this
page by Helger Lipmaa. A full exposition with constructions and
proofs of oblivious transfer and secure function evaluation can be
found in Goldreich's book (Vol II). This paper of
Kushilevitz and Ostrovsky gave the first computationally secure
PIR with single server and sublinear communication. It also
discusses some possible applications for PIR. Amos Beimel has a webpage on
private information retrieval. This
project is about obtaining practical PIR protocols. See this
paper and the announcement for this workshop
for some information on the connections between PIR protocols and
other objects. See also this survey
on private information retrieval by Gasarch.
Collaborating with your classmates on assignments is OK and even encouraged. You must, however, list your collaborators for each problem. The assignment questions have been carefully selected for their pedagogical value and may be similar to questions on problem sets from past offerings of this course or courses at other universities. Using any preexisting solutions from these other sources is strictly prohibited.