## Lecture Summaries, Homeworks, Solutions, Handouts

[+ a tentative schedule for what's to come]

[Lecture 1, week 1, 4/6/04]  Introduction  (lect1.pdf)

Overview of goals of cryptography. Shannon's definition of secrecy.  Classic ciphers and their insecurity.  One-time pad encryption:  Its security and, unfortunately, it’s impracticality.  Hence, the need for computational notions of hardness.

[Lecture 2, week 1, 4/8/04]  Computational Notion of Hardness (+ short review of complexity)  (lect2.pdf)

Probabilistic algorithms, asymptotic analysis of algorithm running time, polynomial time vs. exponential time, notions of negligible adversarial advantage and of computational hardness.  Example: Indistinguishability of Private-Key Encryption.

Homework 1 (due Thursday, 4/15/04):  (hmw1.pdf)

Solutions to Homework 1: (sol1.pdf)

[Lecture 3, week 2, 4/13/04]  Computational Notion of Hardness (cont):  One-way encryption and RSA example  (lect3.pdf)

We define the notion of one-way secure encryption.  We use RSA encryption as an example to illustrate how efficiency/hardness of known attacks on RSA is captured by computational notion of security involving the notion of efficient algorithms and negligible probability.

[Lecture 4, week 2, 4/15/04]  One-way encryption vs. Indistinguishable encryption.  (lect4.pdf)

We compare the two computational notions of security for encryption.  We show that no deterministic cipher, including the textbook RSA can be indistinguishable.  We show other ways in which encryption which is assumed one-way secure can still have security flaws, e.g. it can leak some specific plaintexts, some specific bits of every plaintext, etc.  This shows the gap between one-way security and indistinguishability for encryption, and motivates finding encryption schemes which satisfy the latter, stronger notion.

* Two Handouts:  Number theory facts, collected by prof. Dan Boneh from Stanford:  (h1-primes.pdf)  ,  (h2-composites.pdf)

Homework 2 (due Thursday, 4/22/04):  (hmw2.pdf)

Solutions to Homework 2: (sol2.pdf)

[Lectures 5-6, week 3, 4/20-22/04]  One-Way Functions, Permutations, and Trapdoor Permutations:  Discrete Log, RSA

One Way Functions are a fundamental concept for cryptography.  These are functions which are easy to compute but hard to invert.  We  define a One Way Function (Collection) [OWF], and we show that the long-standing number theoretical assumption of hardness of computing discrete logarithms gives rise to a OWF collection and a One-Way Permutation collection [OWP].  To do that, we review some basic modular arithmetic for primes from handout h1-primes.pdf .  In the second lecture we go through modular arithmetics for composites from handout h2-composites.pdf , and we define Trapdoor Permutation [TDP] collection, and give RSA as an example.

Untill I write up lecture notes from these, read up on this material in lecture notes from other classes:  John Katz's lecture9.pdf, Tal Malkin's lecture6.pdf, Tal Malkin's lecture8.pdf

(Section 1 of John Katz's lecture 9 at UMD explains how RSA and modular exponentiation are examples of OWFs.  Tal Maklin's lecture 6 at Columbia goes over modular arithmetic in a prime group, and also briefly explains the modular exponentiation OWF example.  Tal Maklin's lecture 8 explains RSA as OWF and shows some other examples of OWFs.)

Homework 3 (due Tuesday, 5/03/04):  (hmw3.pdf)

Solutions to Homework 3: (sol3.pdf)

[Lecture 7, week 4, 4/27/04]  Hard-Core Bits

Just because some value is hard to compute (a discrete logarithm, or a factor of a composite number), it doesn't mean that all the bits of that value are hard to guess.  This is crucial because if you want to encrypt something, you need to know not just that your whole message is hard to decipher, but that every bit in your message is hard to guess.  (Note that maybe the attacker cares about only one bit, for example the one which encodes whether you are sending a “buy stock” or a “sell stock” order to your bank.)  In fact we show that even if the modular exponentiation function Exp is a OWF, its value does leak some bits of the preimage, namely the Least Significant Bit of the preimage.  This motivates the definition of Hard Core Bits for OWFs, which are the bits that are unpredictable given the function value.  We show that the Most Significant Bit of the preimage is such a hard core (i.e. unpredictable) bit for the Exp function.

Before I write up the lecture notes here, try reading these:  Yevgeni Dodis's lecture 4.pdf, Tal Malkin's lecture9.pdf

(Yevgeni defines the hard core bit (sec 1), shows that MSB is hard core for the modular exponentiation, just like we did in class (sec 2), he sketches the proof why the <x,r> bit-wise xor is a universal hard core bit for every OWF (sec 3-4), and then explains how to use hard core bits to create indistinguishably secure bit-by-bit encryption (sec 5), something which we sketched at the end of the class.  Tal Maklin's lecture 9 from Columbia gives a quick sketch of this too.  She defines a hard core bit and then sketches a proof why the <x , r> bit-wise xor defines a hard core bit for every one way function.)

[Lecture 8, week 4, 4/29/04]  Computational Indistinguishability, Pseudorandomness, and Pseudorandom Number Generators

We define what it means for two random variables to be "computationally indistinguishable".  We illustrate this notion with indistinguishably secure encryption:  This is an encryption s.t. the distribution of ciphertext of message m1 is computationally indistinguishable from the distribution of ciphertexts of message m2, for any m1,m2 of the same length.  We then define a Pseudorandom Number Generator [PRG].  It is an algorithm whose outputs are longer than inputs, and whose outputs are distributed in a computationally indistinguishable way from random numbers.

Here are other people's lectures on this material:  Yevgeni Dodis's lecture 5.ps, Tal Malkin's lecture10.pdf, John Katz's lecture10.pdf,

[Lecture 9, week 5, 5/04/04]  PKI Infrastructure, guest lecture by Einar Mykletun.

This lecture was on public key infrastructures, combining public key and symmetric key encryptions, and on using signatures and revocation to implement certificates.

Here are the notes from that lecture:  Einar's lecture.pdf

[Lectures 10-11, week 6, 5/11-13/04]  Pseudorandom Generators and Private-Key Encryption

Construction of a pseudorandom generator [PRG] from hard-core bits of a one-way permutation.  Using PRG to implement a public-key encryption for one-bit messages.  Using PRG to create a private-key encryption (the stream cipher).

Notes for these lectures are in Dodis's lecture 4 (se notes to lecture 7 above) and Dodis's lecture 8 (see notes to lecture 8 above).  These notes contain a construction of a PRG from a one-way permutation and its hardcore bit, construction of one-bit public-key encryption from trapdoor permutation, and a construction of a stream cipher from a PRG.

Homework 4 (due Thursday, 5/20/04):  (hmw4.pdf)

Solutions to Homework 4: (sol4.pdf)

[Lectures 12-13, week 7, 5/18-20/04]  Trapdoor Permutations and Secure Public Key Encryption, Diffie-Hellman and ElGamal Encryption

With a simple extension of the OWP-based PRG construction we get a TDP-based public key encryption with short ciphertexts.  Similarly to this general construction (general because assuming any TDP), we can get a practical public key encryption based on the specific assumption of Decisional Diffie Hellman [DDH] assumption, namely the ElGamal encryption.

Notes from Yevgeni Dodis's lecture covering this material: Dodis's lecture6.pdf (for the TDP-based public-key encryption and related material), Dodis's lecture7.pdf (for the DDH assumption, Diffie-Hellman key exchange, and the ElGamal encryption)

[Lecture 14, week 8, 5/25/04]  Stronger Notion of Encryption Security:  Chosen Ciphertext Attack.

We motivate stronger security notions for encryption: security against Chosen Ciphertext Attack (and its restricted version, called Lunchtime Attack), and security in the sense of Non-Malleability of encryption.  We show that our public-key encryption schemes, either the general TDP-based one or the DDH-based ElGamal encryption, are insecure against such attacks.

Notes on Chosen Ciphertext Attack: notes from Mihir's lectures on PK encryption (pages 5-6), paper by Victor Shoup on "why chosen ciphertext security matters", Shoup's paper

[Lecture 15, week 8, 5/27/04]  Pseudorandom Functions (part 1)

We define PRF, a pseudorandom function family, and we give an application to a stateless authentication scheme.

Notes from Yevgeni Dodis's lecture covering material for this lecture and the next: Dodis's lecture9.pdf

Homework 5 (due Thursday, 6/03/04):  (hmw5.pdf)

Solutions to Homework 5:  (sol5.pdf)

[Lecture 16, week 9, 6/1/04]  Pseudorandom Functions (part 2)

We construct a PRF (albeit not a very efficient one) from any PRG.  We then show that based on the DDH assumption a similar construction leads to a very efficient PRF.

Notes for this lecture are Dodis's lecture9, same as above.

[Lecture 17, week 9, 6/3/04]  Ciphers from Pseudorandom Functions

A (CPA,CCA1)-secure encryption scheme from a PRF.  Other modes of using PRFs and PRPs to encrypt.

The notes for this lecture are here: handout2.pdf , also read Dodis's lecture10.pdf, section 3, on this and other constructions of ciphers from PRFs and PRPs.

Homework 6 (due Thursday, 6/10/04):  (hmw6.pdf)

Solutions to Homework 6:  (sol6.pdf)

[Lecture 18-19, last week]  Pseudorandom Permutations, Feistel Permutations

Luby-Rackoff construction of a PRP from a PRF

Homework 7 (due last day of the final week, Friday, 6/18/04):  (hmw7.pdf)

[Tentative Schedule for Upcoming Lectures:]

• Message Authentication

Definition of Message Authentication Codes.  Constructions from PRFs and PRPs.

• Identification Schemes, Zero Knowledge

Security models for identification schemes.  Passive vs. active attacks.  Constructions and proofs.  An ID scheme based on the discrete log problem.  Related concept of Zero-Knowledge Proofs.

• Signature Schemes

Definition of secure signatures.  Collision-Resistant Hashing.  Claw-Free Permutations.

• Random Oracle Model, Signatures, Encryption

Random Oracle Model [ROM] for hash functions.  Chaining hash functions for arbitrary input length. Secure signature scheme from full-domain hashing and one-way functions (in ROM).  Fiat-Shamir construction for secure signatures from zero-knowledge identification schemes (in ROM).  Secure Encryption from one-way permutations (in ROM).