「Cryptography-MIT6875」: Lecture 16

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Topics Covered:

  • IP for Quadratic Non-Residuosity
  • Non-interactive ZK
    • NIZK in The Common Random String(CRS) Model
    • Construction in CRS Model: Blum-Feldman-Micali’88 (quadratic residuosity)
    • NIZK for QNR
    • NIZK for 3SAT
  • Proofs vs. Argument

「Cryptography-MIT6875」: Lecture 15

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Topics Covered:

  • Sequential vs Parallel Repetition: reduce soundness error
  • Proof of Knowledge
    • PoK of DLOG
  • Non-Interactive ZK(NIZK)
    • NIZK in The Random Oracle Model
      • NIZK for 3COL
    • NIZK in The Common Random String Model (Lecture 16)

「Cryptography-MIT6875」: Lecture 12

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Topics Covered:

  • Construction of CRHF from Discrete Log

  • Digital Signatures only from OWF

  • Direct Constructions:Trapdoor Permutation and the Hash-and-Sign Paradigm.

  • Random Oracles.


「Cryptography-MIT6875」: Lecture 11

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Today’s topic is Many-time Digital Signatures.

Topics Covered:

  • Many-time, stateful, signature schemes.
  • Naor-Yung construction: stateless EUF-CMA-secure signature schemes.

「Cryptography-MIT6875」: Lecture 10

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Today’s topic is Digital Signatures.

Topics Covered:

  • Motivation for Digital Signatures.
  • Definition: EUF-CMA Security
  • One-time signatures: Lamport’s Scheme.
    • how to sign a single bit, once
    • how to sign $n$ bits, once
  • Collision-resistant hashing
    • how to sign polynomially many bits, once.

「Cryptography-MIT6875」: Lecture 9

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Topics Covered:

  • Quadratic Residue and Quadratic Residuosity Assumption
  • Goldwasser-Micali Encryption and Homomorphism
  • Diffie-Hellman Key Exchange
  • El Gamal Encryption

「Cryptography-MIT6875」: Lecture 7

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

It is indeed possible for $F(x)$ to leak a lot of information about $x$ even if $F$ is one-way.

The hardcore predicate $B(x)$ represent the specific piece of information about $x$ which is hard to compute given $F(x)$.

Topics Covered:

  • Definition of one-way functions (OWF)
  • Definition of hardcore bit/predicate (HCB)
  • One-way permutations → PRG.
    (In fact, one-way functions → PRG, but that’s a much harder theorem.)
  • Goldreich-Levin Theorem: every OWF has a HCB.
    (Proof for an important special case.)

「Cryptography-MIT6875」: Lecture 6 - Number Theory

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^
The motif of this blog is Number Theory, including the second half of Lecture 5 and the Lecture 6.

It’s an excellent opportunity to learn Number Theory in manner of English.

An important point in this blog is that we focus more on the statements, which is useful in later lectures, rather than the proof.

About 70% of the content in this blog is originally and literally from the lecture notes. I just organize and refine it according to the logic of the professor’s narration since the lecture note is awesome.

The rest is my own understanding and derivation of some theorems. And I will be learning Number Theory and completing the omitted proof.

So this blog will be updated continuously.

Topics covered:

  • Groups, Order of a group and the Order of an element, Cyclic Groups.
  • The Multiplicative Group $\mathbb{Z}_N^*$ and $\mathbb{Z}_P^*$ for a prime $P$.
  • Generators of $\mathbb{Z}_P^*$.
  • Primes, Primality Testing.
  • The Discrete Logarithm (DLOG) problem and a candidate OWF.
  • Diffie-Hellman assumptions: DDH and CDH.

「Cryptography-MIT6875」: Lecture 5
「Cryptography-MIT6875」: Lecture 4
「Cryptography-MIT6875」: Lecture 3

「Cryptography-MIT6875」: Lecture 3

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Topics:

  • The Hybrid Argument.
  • An application: PRG length extension.
  • The notion of pseudorandom functions: Definition, motivation, discussion and comparison with PRGs.
  • PRG implies (stateful) secret-key encryption.
  • PRFs imply (stateless) secret-key encryption.

「Cryptography-MIT6875」: Lecture 2

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Science wins either way.

Topics:

  • How to circumvent Shannon’s lower bound: the computational adversary
  • Definition of computational security
  • The definition of pseudorandom generators(PRG)

「Cryptography-MIT6875」: Lecture 1

In this series, I will learn MIT 6.875, Foundations of Cryptography, lectured by Vinod Vaikuntanathan.
Any corrections and advice are welcome. ^ - ^

Everything you’ve ever wanted is on the other side of fear.

Topics:

  • Introduction to cryptography
  • Secure Communication and Shannon’s definitions of perfect secrecy
  • Perfect Indistinguishability definitions.
  • The One-time pad construction
  • Shannon’s lower bound.