Book Plan

Philosophy

The aim of CryptoBook is to have a consolidated space for all of the mathematics required to properly learn and enjoy cryptography. The focus of any topic should be to introduce a reader to a subject in a way that is fun, engaging and with an attempt to frame it as an applied resource.

The second focus should be to cleanly implement the various topics using SageMath, so that there is a clear resource for a new reader to gain insight on how SageMath might be used to create the objects needed.

Proposed topics

This list is not complete so please add to it as you see fit.

Mathematical Background

Fundamentals

• Congruences

• GCD, LCM

  • Bézout's Theorem

Number Theory

Mainly thinking things like

• Prime decomposition and distribution

• Primality testing

• Euler's theorem

Abstract Algebra

Mainly thinking things like:

• Groups, Rings, Fields, etc.

• Abelian groups and their relationship to key-exchange

• Lagrange's theorem and small subgroup attacks

Basilar Cryptanalysis forms

• Introduction to Cryptanalysis

• A linear Approach to Cryptanalysis

• Matsui's Best biases algorithm

Elliptic Curves

• Weierstrass

• Montgomery

• Edwards

Generating Elliptic Curves

• Generating curves of prime order

• Generating supersingular curves

Hyperelliptic curves

• Generalization of elliptic curves

• Recovering a group structure using the Jacobian

• Example: genus one curves, jacobian is isomorphic to the set of points

Security background

• Basic Concepts

  • Confidentiality, Integrity etc

  • Encryption, Key generation

Asymmetric Cryptography

RSA

• Textbook protocol

• Padding

  • Bleichenbacher's Attack

Paillier Cryptosystem

• Textbook protocol

ElGamal Encryption System

• Textbook protocol

• ElGamal Digital Signature Scheme

Diffie-Hellman

• Textbook protocol

• Strong primes, and why

Elliptic Curve Cryptography

• ECDSA

• EdDSA

Symmetric Cryptography

One Time Pad

• XOR and its properties

• XOR as One Time Pad

• Generalized One Time Pad

Block Ciphers

• AES

Stream Ciphers

• Affine

• RC4

Hashes

• Introduction

• Trapdoor Functions

• MD family

Isogeny Based Cryptography

• Isogenies

• Isogeny graphs

• Torsion poins

Cryptographic Protocols

Zero-knowledge proofs

• Schnorr proof of knowledge for dlog

• Core definitions

• Proof of equality of dlog

Formal Verification of Security Protocols

• Definition of Formal Verification

• Uses of Formal Verification

• Handshake protocols, flawed protocols

Usefull Resources ( Books, articles ..) // based on my material

• Cryptanalytic Attacks on RSA (Yan, Springer, 2008)

• Algorithmic Cryptanalysis (Antoine Joux, CRC Press, 2009)

• Algebraic Cryptanalysis (Brad, Springer, 2009)