Preface
Introduction
1. Kerckhoff's Principle
2. Symmetric encryption
3. Secret sharing
4. Asymmetric encryption
5. P versus NP problem
6. Trapdoor and one way functions
7. Randomness
8. Hashes and Random Oracle
9. Zero Knowledge Proofs
10. Fully Homomorphic Encryption
11. Secure Multi-Party Computation
Building Blocks
1. XOR and one-time pads
2. Natural Numbers
3. Integers
4. Divisibility and greatest common divisors
5. Modular arithmetic and congruences
6. Groups
7. Rings
8. Fields
9. Fermat's little theorem and Euler, Cauchy and Lagrange theorems
10. Discrete logarithm
11. Primitive roots
12. Polynomials
13. Polynomial interpolation
14. The Arithmetic of Elliptic Curves
15. Pairings and field extensions
16. Lattices
17. Ring learning with errors
Secret sharing
1. Diffie–Hellman key exchange
2. Elliptic Curves Diffie Helman
3. ElGamal
Symmetric encryption
1. Information vs Computational Security
2. Bit Operations
3. Stream and block ciphers
4. AES
5. ChaCha20 and XChaCha20
6. MiMC
7. Rescue and Vision
Asymmetric encryption
1. Primality Testing
2. RSA
3. Fast Multiplication Algorithms
4. Elliptic Curves
5. The discrete logarithm problem
Hash functions
1. Merkle Tree
2. BLAKE2b
3. Pedersen
4. Poseidon
Pseudorandom number generators
Signatures
1. Schnorr
2. MuSig
Message Authentication Codes
1. Poly1305
SNARKs
1. Overview
2. Arithmetization
3. Probabilistically Checkable Proof
4. Interactive oracle proof
5. Polynomial commitments
6. Multiscalar Multiplication
7. Groth16
8. Plonk
STARKs
1. AIR
Fully Homomorphic Encryption
Secure Multi-Party Computation

Fields

We know examples of fields from elementary math. The rational, real and complex numbers with the usual notions of sum and multiplication are examples of fields (these are not finite though).

A finite field is a set equipped with two operations, which we will call \( + \) and \( * \) . These operations need to have certain properties in order for this to be a field:

If \( a \) and \( b \) are in the set, then \( c=a+b \) and \( d=a*b \) should also be in the set. This is what is mathematically called a closed set under the operations \( + \), \( * \).
There is a zero element, \( 0 \), such that \( a+0=a \) for any \( a \) in the set. This element is called the additive identity.
There is an element, \( 1 \), such that \( 1*a=a \) for any \( a \) in the set. This element is the multiplicative identity.
If \( a \) is in the set, there is an element \( b \), such that \( a+b=0 \). We call this element the additive inverse and we usually write it as \( -a \).
If \( a \) is in the set, there is an element \( c \) such that \( a*c=1 \). This element is called the multiplicative inverse and we write is as \( a^{-1} \).