Algebra concepts

Classic Algebra: 整数 有理数 实数 复数 加减乘除

Abstract Algebra: 构造新的number system,通常只说 加和乘(减和除用inverse表示),并且加和乘用更通用的 composition来代表,在abstract algebra中,如果不说具体的additive或multiplicative,通常是用 ◦ / + / x 或直接xy不带任何符号来表示一种composition方式,composition可以是加或乘法或模运算甚至是set permutation等

group 闭合,Single composition law: 要么加,要么乘(必须存在inverse)

​ abelian group:满足交换律

​ Cyclic Groups

​ subgroup

ring 闭合,multiple composition law: 既要加(满足additive composition law),又要乘(满足multiplicative composition law,环中乘法不一定有单位元也不一定要满足交换律,不一定存在inverse)

​ 只从满足Additive composition law的加法环ring来说 === abelian group

​ communitative ring ==- ring+满足乘法交换律 例子:Integer Z = {0,1, -1, 2, -2, . . .}

field 闭合,既可以加也可以乘(由加群和乘“群”组合而成)

​ 加群 === abelian group

​ 乘“群” === communitative ring+multiplicative inverse(except additive identity 所以不是群,当然也可以说除掉addtive identity构成群)

​ finite field : field with finite number of elements, the number of elements called order,除去additive identity的乘法结构为循环群

例子:prime finite field:Fp , p = 2, 3, 5, 7 ,11, 13, ….. Fp = p, Z/(pZ)={ 0¯, 1¯, 2¯,(p-1)¯ }

符号:

= ≠

根号 √

∞ 无穷

◦ composition

⊆表示包含于 improper subset,下面有不等号≠的表示真包含于proper subset,但在同济版高等数学中,⊂表示包含于,下面有不等号的表示真包含于。

交集 ∩ 并集 U

∃ 存在

∀ 任意

∈ 属于 ∉

≡ 同余符号 

累加累乘

\[\sum_{i=1}^n \frac{1}{i^2} \quad and \quad \prod_{i=1}^n \frac{1}{i^2} \quad and \quad \bigcup_{i=1}^{2} R\]

https://math.meta.stackexchange.com/questions/5020/mathjax-basic-tutorial-and-quick-reference

希腊字母

uppercase Σ, lowercase σ

https://reference.wolfram.com/language/guide/GreekLetters.html.zh

domain -> codomain bijective surjective injective

https://www.mathsisfun.com/sets/injective-surjective-bijective.html

Cryptography

block ciphers VS stream ciphers

3 problems:

PKI: public key infrastructure (Public-Key Cryptosystems) === Asymmetric Cryptography

DLP: Discrete Logarithm Problem

DHK: Diffie–Hellman key exchange

Encryption

Symmetric

Confusions and Diffusions

Theory: finite field/Galois field/Prime Fields/Extension Fields GF(2m)

DES

AES

Asymmetric

Signature

Theory: hash function(for signing long message)

RSA

Elgamal

DSA

ECDSA

Key Establishment

就是如何发布和管理相关的密钥