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.
Write about what you love and this book will be a success. 
Descriptions of attacks against cryptosystems are strongly encouraged, however full SageMath implementations should not be included, as this has the potential for destroying CryptoHack challenges, or making all attacks known by so many people that CTFs become a total nightmare!! 
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
Gauss' Lemma and its ten thousand corollaries
Euclid's algorithm
Modular Arithmetic
Morphisms et al. 
Frobenius endomorphism
Number Theory
Mainly thinking things like
Prime decomposition and distribution
Primality testing
Euler's theorem
Factoring
Legendre / Jacobi symbol
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
A Differential Approach to Cryptanalysis
Elliptic Curves
Weierstrass
Montgomery
Edwards
Counting points (Schoof's algorithm)
Complex multiplication
​Good reference, thanks Joachim​
Generating Elliptic Curves
​Generating Anomalous curves​
Generating curves of prime order
Generating supersingular curves Wikipedia​
​Generating non-supersinular curves of low embedding degree​
Generating curves of arbitary order (hard)
​Thesis on the topic​
Sage implementation ChiCube's script​
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
Mumford representation of divisors
Computing the order of the Jacobian
​For characteristic 2^n: Example 56​
Hyper Metroid example
Security background
Basic Concepts 
Confidentiality, Integrity etc
Encryption, Key generation
Attacker goals + Attack games
Defining Security - Perfect security, semantic security
Proofs of security + Security Reductions
Asymmetric Cryptography
RSA
Textbook protocol
Padding
Bleichenbacher's Attack
OAEP
Coppersmith
Håstad's Attack
Franklin-Reiter Attack
Wiener's Attack
RSA's Integer fattorization Attacks
​
Fermat Factoring Attack
Quadratic Sieve Attack
Number Fielde Sieve Attack
RSA Digital Signature Scheme
Timing Attacks on RSA
RSA with Chinese Remainder Theorem (CRT)
​Fault Attack on RSA-CRT​
​Bellcore Attack (Low Voltage 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
SHA family
BLAKE Hash family
// TODO: Insert Attacks
Isogeny Based Cryptography
Isogenies
Isogeny graphs
Torsion poins
SIDH
SIKE
BIKE
Cryptographic Protocols
Zero-knowledge proofs
Schnorr proof of knowledge for dlog
Core definitions
Proof of equality of dlog
Proof of knowledge of a group homomorphism preimage
Formal Verification of Security Protocols
Definition of Formal Verification
Uses of Formal Verification
Handshake protocols, flawed protocols
The external threat: Man-In-The-Middle attacks
Attacking the (flawed) Needham-Shroeder public key exchange protocol 
​
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)
RC4 stream Cipher and its variants (H. Rosen, CRC Press, 2013)
Formal Models and Techniques for Analyzing Security Protocols (Cortier, IOS Press, 2011)
Algebraic Shift Register Sequences (Goresky && Klapper, Cambridge Press,  2012)
The Modelling and Analysis of Security Protocols (Schneider, Pearson, 2000)
Secure Transaction Protocol Analysis (Zhang && Chen, Springer, 2008)