Princeton University
Computer Science Department

Computer Science 433

Boaz Barak

Fall 2007

Lecture notes | Reading | Admin

Course Summary

Cryptography or "secret writing" has been around for about 4000 years, but was revolutionized in the last few decades. The first aspect of this revolution involved placing cryptography on more solid mathematical grounds, thus transforming it from an art to a science and showing a way to break out of the "invent-break-tweak" cycle that characterized crypto throughout history. The second aspect was extending cryptography to applications far beyond simple codes, including some paradoxical impossible-looking creatures such as public key cryptography , zero knowledge proofs, and playing poker over the phone.

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.

The main prerequisite for this course is ability to read, write (and perhaps enjoy!) mathematical proofs. In addition, familiarity with algorithms and basic probability theory will be helpful. No programming knowledge is needed. 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

Lecture Notes and Handouts.

  1. Overview of course, crypto history
  2. Perfect secrecy and its limitations
  3. Computational model, computational security
  4. Computational indistinguishability, pseudorandom generators
  5. CPA security, Pseudorandom functions
  6. Pseudorandom permutations, block-ciphers, AES
  7. The authentication problem, CCA security
  8. Message authentication codes
  9. One way permutations and pseudorandom generators, Goldreich-Levin Theorem
  10. One way permutations, pseudorandom generators, commitment schemes (cont)
  11. Some basic number theory (guest lecturer Benny Applebaum)
  12. Public key cryptography and Rabin encryption
  13. Signature schemes
  14. CCA security for public key encryption
  15. Zero knowledge proofs
  16. Zero knowledge proofs (cont)
  17. Forward security, identity-based encryption
  18. Secret sharing, visual cryptography, threshold signatures
  19. Oblivious transfer, Private information retrieval
  20. Secure 2-party and multiparty computation
  21. Electronic voting
  22. Cramer-Shoup Cryptosystem
  23. Quantum computing and cryptography
  24. Course summary

Lecture 1: Tuesday, September 18, 2007
Overview of course, crypto history

Overview of crypto goals and history. Some classical ciphers and how they were broken. Outline of crypto 1970's "revolution". Admin info about the course.
Powerpoint Slides
Handout 1 - Mathematical Background
Excerpt from Katz-Lindell Book on Principles of Modern Cryptography (pages 18-27)

KL Book: Chapter 1 - introduction.

Additional reading: You can find more information about historical ciphers on the web page Alex Biryukov's wonderful Course on Cryptanalysis.

Mathematical proofs: Some links 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.
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. The Katz-Lindell excerpt explains some of the motivation behind this. See also pages 20-25 of Goldreich's book (Volume 1).

Lecture 2: Thursday, September 20, 2007
Perfect secrecy and its limitations

Perfect Secrecy, the one-time pad. Limitations of perfect secrecy.
Lecture Outline
Homework 1 - due September 27th (LaTeX source).
I prefer you use LaTeX to write your solution. Here is a short guide for LaTex by Dave Xiao (you might want to look at the source files for the guide: latex-guide.tex and macros.tex).

KL Book: Chapter 2 - Perfectly secret encryption

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 the Turing machine. You might want to take a look at this chapter from my upcoming book with Sanjeev Arora.

Lecture 3: Tuesday, September 25, 2007
Computational model, computational security

Computational models and computational security.
Lecture Outline

Additional reading: You might want to look at the Boolean circuits and NP completeness chapter from my upcoming book with Sanjeev Arora. Some handouts from the last offer of this course: Powerpoint Slides (did not use these in class) computational models handout and figure. 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.

Lecture 4: Thursday, September 27, 2007
Computational indistinguishability, pseudorandom generators

Computational indistinguishability, Pseudorandom generators, length extension.
Lecture Outline
Handout: Pages 215-220 from KL book

KL Book: Chapter 3 - Private key encryption and pseudorandomness

Additional reading: Please take a look at Section 6.4.2 of the Katz-Lindell book. You might also want to look at Goldreich's Treatment of pseudorandom generators (Volume I, pages 101-117).

Lecture 5: Tuesday, October 2, 2007
CPA, Pseudorandom functions

Encryption schemes secure against Chosen-Plaintext Attack (CPA), Pseudorandom functions.
Lecture outline

KL book: Pages 82-93 and 221-225. (Sections 3.5, 3.6.1, 3.6.2 and 6.5). See also the following excerpt from Goldreich's book on the construction of PRF's from PRG's. (This excerpt is from a draft - see Goldreich Vol I for the updated version.)

Additional reading:Goldreich Volume II (Chapter 5) contains an extensive discussion of the definitions of encryption schemes.

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.

Lecture 6: Thursday, October 4, 2007
Pseudorandom permutations, block-ciphers, AES

Pseudorandom permutations, block ciphers and the AES.
Powerpoint slides.

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:

Lecture 7: Tuesday, October 9, 2007
The authentication problem, CCA security

Short recap of past lectures (powerpoint)
Lecture outline

KL book: Section 3.7 (pages 103-104),

Additional reading: 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. (These sources talk about public key encryption, but the underlying message is the same.)

Lecture 8: Thursday, October 11, 2007
Message authentication codes

lecture outline
Exercise 4

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. The material about the order of encryption vs. authentication is from this CRYPTO 2001 paper by Hugo Krawczyk.

Lecture 9: Tuesday, October 16, 2007
One way permutations and pseudorandom generators

One-way permutations and hard-core bits, with applications to pseudorandom generators and commitment schemes.
Lecture outline

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.

Lecture 10: Thursday, October 18, 2007
One way permutations, pseudorandom generators, commitment schemes (cont)

Reading for next time: Next week we'll start discussing number theory, in preparation for public key encryption schemes. Some resources for number theory include Chapter 7 and Appendix B of the KL book. There's also an excellent book of Victor Shoup (available freely on the web). The more you read of this book the better, however, I recommend you look in Chapter 1 (pages 1-10) and the first 5 pages of Chapter 8 (pages 180-184, not including exercise 8.1). Other 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, Chapter 10 (up to and including 10.4.1), first two pages of Chapter 11, Chapter 12 and Chapter 13. See also the mathematical background appendix from my upcoming book with Sanjeev Arora.

Lecture 11: Tuesday, October 23, 2007
Some basic number theory - Benny Applebaum

Some basic number theory. Guest lecturer: Benny Applebaum

slides (taken from Eli Biham's Technion course)

Additional reading: As mentioned above, Shoup's book is an excellent resource for computational number theory.

Lecture 12: Thursday, October 25, 2007
Public key cryptography and Rabin encryption

Public Key Cryptography and the Rabin Encryption Scheme
Lecture outline
Homework 6 (LaTeX source)

Additional reading: KL Book Sections 10.1 , 10.2, 10.4, 10.7, 11.2

Lecture 13: Tuesday, November 6, 2007
Signature schemes

Signature schemes
Lecture outline

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 (Vol II). 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 Limpaa 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 here.

Lecture 14: Thursday, November 8, 2007
CCA security for public key encryption

Security against Chosen Ciphertext Attack (CCA) for public key encryptions, random-oracle based constructions.
Lecture outline
Homework 7 (LaTeX source) -- due November 15th, 2007.

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 first 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.

Lecture 15: Tuesday, November 13, 2007
Zero knowledge proofs

Zero Knowledge Proofs.
Lecture outline
HW8 (LaTeX source) November 29

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

Lecture 16: Thursday, November 15, 2007
Zero knowledge proofs (cont)

Zero Knowledge Proofs (continued).
Lecture outline

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).

Lecture 17: Tuesday, November 20, 2007
Forward security, identity-based encryption

Forward security, identity based encryption.
Lecture outline

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.

Lecture 18: Tuesday, November 27, 2007
Secret sharing, visual cryptography, threshold signatures

Secret sharing, visual cryptography, threshold signatures.
Lecture outline

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 visual cryptogeaphy. 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.

Lecture 19: Thursday, November 29, 2007
Oblivious transfer, Private information retrieval

Oblivious transfer, private information retrieval.
Lecture outline
HW9 (LaTeX source) December 6

Additional reading: Some web resources on oblivious transfer are this page by Benn Lynn and this page by Helger Limpaa. 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.

Lecture 20: Tuesday, December 4, 2007
Secure 2-party and multiparty computation

The slides shown in class were taken from Joe Kilian's tutorial and Vitaly Shmatikov's lecture.

For a proof of security of Yao's grabled circuit protocol in the honest-but-curious model see this paper by Lindell and Pinkas.

Lecture 21: Thursday, December 6, 2007
Electronic voting

Lecture outline
Slides (mostly taken from Tal Moran).
HW10 (LaTeX source) December 17

Additional reading: I took much of the material from the excellent MIT course by Canetti and Rivet. I also highly recommend the following slides by Tal Moran. Here are some additional links (curtesy of Tal Moran). The main contenders for practical voter-verifiable e-voting schemes are Punchscan and Pret-a-voter. Scantegrity is another, slightly newer, idea of Chaum's. All these systems are based on the notion of a 2-part receipt that the voter makes some checks on in the booth and then gets to keep only one part. The voting system presented in class was based on the secrecy of the order of events in the booth, this idea was suggested in this paper by Neff. Another interesting suggestion is Rivest's "Three Ballot" scheme that does not use cryptography in the voter verification step at all, but it has some serious problems such as reliance on very strong assumptions about the voting hardware (the paper mentions a vote buying attack also but this seems to be less serious if the number of candidates is very small compared to the number of voters). Finally, there is a lot of literature about "internet" voting protocols (that ignore the human component). Until 2004, all the cryptographic voting schemes were of this type. Helger Lipmaa's list ( has many of the notable ones.

Lecture 22: Tuesday, December 11, 2007
Cramer-Shoup Cryptosystem

Lecture outline

Lecture 23: Thursday, December 13, 2007
Quantum computing and cryptography

Quantum Computing and Cryptography
Slides (PowerPoint format)

For a good starting point on quantum computing, you can do far worst than explore Scott Aaronson's home page

Lecture 24: Monday, December 17, 2007
Course summary

Summary of the course.

Some reading: An excellent discussion of provable security appears in this survey/position paper of Ivan Damgard. It refers also to issues raised in these 2004 and 2006 papers of Koblitz and Menezes, where they criticize proofs of security. In my view, some of the issues they raise are valid, though not novel, but the entire discussion is incomplete and rather misleading. See also this AMS Notices article by Koblitz and these responses (including one by me). Perhaps what bothers me most about these discussion is that they tend to focus too much on proofs of security, as opposed to precise definitions of security, whereas in my opinion the latter are even more fundamental to both the theory and practice of cryptography (although proofs are of course necessary to show that "crazy-sounding" definitions such as chosen-message secure signatures or chosen-ciphertext secure encryption can actually be satisfied).

Additional reading:

Administrative Information

Lectures: Tuesday and Thursday 1:30pm-2:50pm, Room: Friend 108

Professor: Boaz Barak - 405 CS Building. Email: Phone: 609-981-4982 ?> (I prefer email)

Undergraduate Coordinator: Donna O'Leary - 410 CS Building - 258-1746

Teaching Assistant: Rajsekar Manokaran ( rajsekar@cs )

course mailing list.

Grading: 50% homework, 50% take-home final. See syllabus for more details.

Homework assignment Due
HW1 (LaTeX source) September 27
HW2 (LaTeX source) October 4
HW3 (LaTeX source) October 11
HW4 (LaTeX source) October 18
HW5 (LaTeX source) October 25
HW6 (LaTeX source) November 8
HW7 (LaTeX source) November 15
HW8 (LaTeX source) November 29
HW9 (LaTeX source) December 6
HW10 (LaTeX source) December 13


The textbook for this book is Introduction to Modern Cryptography by Katz and Lindell. However, we will not completely follow the book in this course. Some of the lectures will follow Oded Goldreich's book Foundations of Cryptography Volumes 1 and 2. Note that some of this material is online in the form of "fragments" of the book.

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

Honor Code for this class

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.