(Draft) Lattice-based crypto - introduction

During the Corona situation, I decided to learn something new, and it was lattice-based cryptography. Intro to LWE from knapsack problem From this slide 1D (modular) knapsack problem Given $a_1, a_2,\cdots, a_n$ and $t$, $q$, where all variables are integers. Find $\{x_i\}_{i=1}^{n}\in\{0, 1\}^{n}$ s.t. , $$ t=\sum_{i=1}^{n}x_1a_i \bmod q. $$ Vector modular knapsack problem Given $\pmb{a}_1, \pmb{a}_2,\cdots, \pmb{a}_n$ and $\pmb{t}$, $q$, where $\pmb{a},\pmb{t}\in\mathbb{Z}^{m\times1}$ and $q \in \mathbb{Z}$. Find $\pmb{x}=\{x_i\}_{i=1}^{n}\in\{0,1\}^{m\times1}$ s.t. , $$ \pmb{t}=\sum_{i=1}^{n}x_i\pmb{a}_i \bmod q.

Galois Theory (Updating continuously)

Good material Visual Group Theory - Lecture 6 on YouTube is the best Galois theroy lecture. You can fined slides of the lecture at below. Kelin four group Hasse diagrams complex number is algebraically closed. fundamental theorem of algebra Any plynomial with complex coefficients, having degree d, has at least one complex root.(Gauss) How to listen it I listened 6.1 - 6.3 at the first day.

Category Theory (Apr.2021. updating continuously)

Materials Category Theory by Steve Awodey Very kind “Category Theory For Beginners” by Richard Southwell. My memo: Introduction done My memo: Functor done Definition of a category Objects and arrows exist. There should be identity arrows for all objects. Arrows should be composited. Associativity holds. Terminology: domain and codomain $\mathrm{dom}(f)\xrightarrow{f} \mathrm{cod}(f)$ Directed graphs and categories If $f:A\rightarrow B$ and $g:B\rightarrow C$ exist but $h := g\cdot f$ doesn’t, it violates the rule of composition.