## Lecture Summaries, Homeworks, Solutions, Handouts

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

Lectures 1-2  (lect1.pdf)

Lectures 3-4  (h1-primes.pdf)  ,  (h2-composites.pdf)  ,  (Dana Angluin's notes on computation and number theory.pdf).

We covered some basic modular arithmetic in the "primes" handout, and the extended Euclidean algorithm for computing gcd and modular inverses from chapter 4 of Dana's notes.

Lecture 5

We showed that modular exponentation can be done efficiently (polynomial time), but we posed the inverse of the exponentiation, namely the discrete logarithm problem, as a problem for which no known efficient algorithm is known.  We looked at two trivial attacks against discrete logarithm: exhaustive search and guessing, and concluded that the first runs in exponential time while the second one has a negligible probability of success.  We saw Shank's discrete logarithm running in time O(\sqrt(q)) and the index calculus methods which run in time about O(2^{|p|^{1/3}), and we translated these two algorithms into bounds on the size of p and q needed to achieve security for the discrete logarithm in practice.  Finally, we stated the discrete logarihtm assumption.

Reading:  Most of this material is in Stinson, chapter 6, sections 6.1, 6.2 (esp 6.2.1, the other attacks are an optional reading), and 6.6.

In the next lecture we'll abstract the assumption that discrete logarithm is hard into an assumption that "exponentiation is a one-way function".  The best lecture notes which introduce one-way functions is Yevgeni Dodis's lecture notes #2.pdf.  For now read up sections 1-7.

[If you are curious why we are skipping Stinson 6.3-5, here is a quick overview of that material: Chapter 6.3 gives another type of evidence that the discrete logarithm is hard.  Namely, it shows that any attack logarithm which is "generic", must run in time at least \Omega(\sqrt(q)).  This shows that a type of a DL computing algorithm like the Shank's algorithm in fact cannot be improved.  This is an optional but very recommended reading.  Sections 6.4 and 6.5 show that the discrete logarithm problem can be posed in other groups than the Z_p^*.  Section 6.4 describes the DL problem in the extension field of Z_p, and section 6.5 descibes the DL problem over elliptic curves.  Both are important, but not essential to the progress of ideas in this class, so they can be left as an optional reading.]

Lecture 6

We have defined a (weak) secure authentication and shown how to use one-way permutation (for example based on modular exponentiation) to build an authentication protocol that satisfies this security property.  One of the weaknesses of this authentication scheme considered in the previous lecture is that it is stateful and that every verifier which might want to authenticate the client needs a separate verification key.

This material is in section 11 of the above lecture notes.

Lectures 7-8

We define the strong notion of security for the authentication scheme (a.k.a. "identification scheme").  We show a public-key (and stateless) authentication scheme which is based on modular exponentiation.  We introduce the concept of zero-knowledge proof and of the simulation proof technique, and we show that this identification scheme is based on a zero-knowledge proof of knowledge of discrete logarithm.  This scheme is similar to the Fiat-Shamir identification scheme, and it forms a basis of the Schnorr Signature scheme [Schnorr, Crypto'89].  We show that this scheme is secure under the discrete logarithm assumption, i.e. the assumption that modular exponentiation is a one-way function.  In its general form, this scheme actually works using any one-way function with certain "homomorphic" properties.

Lectures 9-10

We introduce preimage resistant, second-preimage resistant, and collision resistant hash functions.  We introduce the random oracle model for hash functions, which implies all these properites.  We show Merkle tree and Merkle-Damgard construction of hash function on infinite domain from a hash function that takes element from a small domain (such hash functions are called "compression functions"), and we argue that both constructions preserve all the needed properties of the hash function.  We define message authentication schemes (MACs), and we define a strong notion of security for MACs as resistance against chosen message attack (CMA).  We show a secure MAC scheme based on a hash function, and argue that this scheme is CMA-secure as long as the hash function can be modeled as a random oracle.  We show also a number-theoretical hash scheme, by Pedersen [Crypto'91], which is collision resistant under the discrete logarithm assumption.

The material for these lectures is Stinson's chapter 4, sections 4.1 to 4.4, with section 4.5 as a recommended optional reading.  For a different take, see Tal Malkin's lecture 22.pdf  and Yevgeni Dodis's lecture 11.pdf.

Lecture 11

We define signature schemes and various notions of their security.  We show the Schnorr Signature scheme based on the authentiaction protocol from lectures 7-8 above.  We argue that it is existentially unforgeable under the CMA attack under the discrete logarithm assumption and assuming the random oracle model for a hash function used in this scheme.  We show the standard Digital Signature Standard (DSS) signature, which can be looked at as a variant of the Schnorr signature scheme (historically DSS was first, but Schnorr signature has better understood security).

See Stinson's 7.1, 7.2, 7.4.1, 7.4.2.  For the background reading with optional subjects like one-time signatues (which are also covered in Stinson's chapter 7.5.1), see Yevgeni Dodis's lecture 12.pdf or Tal Malkin's lecture 18.pdf,   lecture 19.pdf, and lecture 21.pdf,

Lectures 12-13

We introduce the RSA permutation, examine its relation to factoring, and argue why it is assumed to be a trapdoor permuation.  We show Rabin's permutation and argue that Rabin is a Trapdoor permutation if and only if factoring RSA moduli is hard.  We show several signature schemes based on RSA permutation, with varying degrees of security: plain RSA, padded RSA, and full-domain hash RSA signature.  Only the last scheme is known to be existentially unforgeable under the CMA attack assuming RSA is a trapdoor permutation and assuming the random oracle model for the hash function.

Stinson covers the RSA cryptosystem in chapter 5.  So far we covered material in Sections 5.1-5.3 and 5.7-5.8.  Sections 5.4-5.6 are optional reading.  We will come back Section 5.9 when discussing RSA encryption.

Homework 1 (due Tuesday, 10/5) (hmw1.pdf)(sol1.pdf)

Homework 2 (due Thursday, 10/14) (hmw2.pdf)(sol2.pdf)

Homework 3 (due Thursday, 10/21) (hmw3.pdf)(sol3.pdf)

Homework 4 (due Thursday, 11/04) (hmw4.pdf)(sol4.pdf)

Homework 5 (due after the final week) (hmw5.pdf)