# 0x042 Cryptography

This page will cover classic cryptography, quantum cryptography and block chains. Main reference for classic cyptography will be Handbook of Applied Cryptography.

Contents Index

## Classic Cipher

Personally I think the code book by Simon Singh is a very good reference for old ciphers.

### Substitution Ciphers

Substitution Ciphers can be easily attacked with frequency analysis (e.g: character e appears about 13% in a standard English text)

• scytale: a transposition cipher used in ancient Greece
• Caesar Cipher
• Vigenere Ciper
• ROT13: $(a+13)%26$

### enigma

#### encryption

• three roters
• one plugboard
• Alice message -> plugboard -> three roters -> plugboard -> encryption

#### complexity

• roter combination: (5x4x3)
• roter initialization: (26x26x26)
• plugboard combination: (factorial(26)/(factorial(6)xfactorial(10)xpower(2,10)))
• total complexity: 10^21

#### decryption

• Bob set up the roter and plugboard with correct combination and feed the encrypted message, and the output will be the original message

#### how to break (bombe)

• basiclly, brute force with several tricks to reduce complexity
##### Tricks
• use ‘weather report’ as a key word
• same char would not be mapped to the same char

## Symmetric-key Cipher

Alice and Bob use the same key: A cipher defined over $(\mathcal{K}, \mathcal{M}, \mathcal{C})$ is a pair of efficient algorithms $(E,D)$ where $C: \mathcal{K} \times \mathcal{M} \rightarrow \mathcal{C}$ and $D: \mathcal{K} \times \mathcal{C} \rightarrow \mathcal{M}$

Definitions

• Consistency: $D(k, E(k, m))==m$
• Perfect secrecy: A cipher $(E,D)$ over $(\mathcal{K}, \mathcal{M}, \mathcal{C})$ is said to have perfect secrecy if $P(E(k, m_0)==c) == P(E(k, m_1)==c)$ for all $m_0, m_1 \in \mathcal{M}, c \in \mathcal{C}$ where $k$ is uniformly random sampled over $\mathcal{K}$ (Rought idea is cipher text give no info about plain text). Hard to use in practice

Vernam 1917. One Time Pad is defined by taking XOR between text and key of same length: $\mathcal{M}, \mathcal{E} \in \{0, 1\}^n, \mathcal{K} \in \in \{0, 1\}^n$.

### Stream Cipher

Idea: Expand a short random seed (seed is key) into a long pseudorandom bit string, and then apply One Time Pad

## Asymmetric-key Cipher

Alice and Bob use different keys

### ECC (Elliptic Curve Crypto)

Elliptic curve is a set of $(x,y)$ over Galois Field whose points are satisfying $y^2=x^3+ax+b$ along with inf points

Definition

• scalar multiplication: $Q = dP = P + P … + P$. this operation will generate cyclic group

#### ECC Discrete logarithm

Given a elliptic curve $<G>$ where G is the base pointer, and a pointer E, find a which satisfying $X=aG$

Properties

• No efficient solution for discrete logarithm (EC is hard to break)
• efficient solution for reverse exp operation (EC is hard to encode and decode)

#### Curve

• NIST curve: 192, 224, 256, 384, 521 bit (521 is selected for computation efficiency with a Mersenne prime)
• Brainpool curve: 160, 192, 224, 256, 320, 384, 512 bit (slow than NIST curve)

## Cryptographic Hash

Crypto hash takes any string as input, produces fixed size output, and should be efficiently computable. It should meet the following security properties.

• collision-free: Nobody can find a pair $(x, y)$ such that $x!=y; H(x) == H(y)$
• hiding: If $r$ is chosen from distribution with high min entropy. then given $H(r|x)$, it is infeasible to find $x$
• puzzle-friendly: For every possible output $y$ and $k$ chosen from distribution with high-min entropy, it is infeasible to find $H(k|x)=y$.
• avalanche effect: a small change in input space would cause a drastic change in output space

## Block Chain

### Data Structures

Hash Pointer: a pair of (pointer to an info, hash of the info). It can verify that the info has not been changed.

Merkle tree: binary tree using hash pointer, it can be used to prove membership in $log(n)$. Sorted merkle tree can prove non membership