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content/Basic Algebra/Basic Algebra.md

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 > [!Info] 
 >  [[Basic Algebra]] provides foundational knowledge essential for understanding cryptographic systems. It covers key algebraic concepts such as groups, fields, and rings, which are important to understand cryptographic algorithms.
 
-[[Binary Operation]]
-[[Monoid]]
-[[Group]]
-[[Subgroup]]
-[[Cyclic Subgroup]]
-[[Ring]]
-[[Binary Relation]]
-[[Equivalence Relation]]
-[[Quotient Ring]]
-[[Ring Homomorphism]]
-[[Field]]
-[[Finite Field]]
-[[Cyclotomic polynomial]]
-[[Discrete Fourier Transform]]
-[[Fast Fourier Transform]]
-[[Number Theoretic Transform]]
-[[Residue Number System]]
-[[Chinese Remainder Theorem]]
-[[Automorphism]]
-[[Elliptic Curves]]
-[[Discrete logarithm]]
-[[ECDLP]]
+- [[Binary Operation]]
+- [[Monoid]]
+- [[Group]]
+- [[Subgroup]]
+- [[Cyclic Subgroup]]
+- [[Ring]]
+- [[Binary Relation]]
+- [[Equivalence Relation]]
+- [[Quotient Ring]]
+- [[Ring Homomorphism]]
+- [[Field]]
+- [[Finite Field]]
+- [[Cyclotomic polynomial]]
+- [[Discrete Fourier Transform]]
+- [[Fast Fourier Transform]]
+- [[Number Theoretic Transform]]
+- [[Residue Number System]]
+- [[Chinese Remainder Theorem]]
+- [[Automorphism]]
+- [[Elliptic Curves]]
+- [[Discrete logarithm]]
+- [[ECDLP]]
 

content/Basic Algebra/Discrete Fourier Transform.md

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+---
+title: Discrete Fourier Transform
+---
+
+# Definition
+The Discrete Fourier Transform (DFT) refers to the Fourier transform applied to discrete values (discrete time to discrete frequency). When computed using the Fast Fourier Transform (FFT), the DFT can be calculated quickly in O(nlogn) time for n values. Therefore, it can be used to efficiently compute polynomial multiplication or vector convolution, which would otherwise require $O(n^2)$ time (where n is the degree of the polynomial).
+
+# n-th root of unity
+
+^1735c9
+
+The **n-th root of unity** is a concept that plays a crucial role in the computation of the DFT. Mathematically, an n-th root of unity is a complex number $\omega$ that satisfies $\omega^n=1$. Therefore, there can be n such $\omega$, each of which can be expressed as $\omega^k = e^{-2\pi ik/n}$, where k is an integer ranging from 0 to n−1, and i is the imaginary unit. These roots are uniformly distributed on the unit circle.
+![[root of unity.png]]
+[Image source](https://homepages.math.uic.edu/~jan/mcs472/discretefourier.pdf)
+
+# Discrete Fourier Transfrom
+Let x = $[x_0, x_1, ..., x_{n-1}]^T$ be an n-dimensional vector, the discrete fourier transform of x is y = $[y_0, y_1, ..., y_{n-1}]^T$ where ^8ba133
+
+$$y_k = \sum_{j=0}^{n-1} x_j \omega^{jk},\quad w=e^{-i2\pi/n}, k=0,1,2, ..., N-1$$ The inverse transform can be defined as
+
+$$x_n= {1\over N} \sum_{j=0}^{n-1} y_j \omega^{-jk}$$

content/Basic Algebra/Fast Fourier Transform.md

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+---
+title: Fast Fourier Transfrom
+---
+# Definition
+Fast Fourier Transform (FFT) is a technique that utilizes the periodicity of the root of unity to perform the [[Discrete Fourier Transform]] in \(O(n \log n)\) time. [Root of Unity]([[Discrete Fourier Transform#^1735c9]]) has the following property: $$\omega^{n/2+j}=-\omega^j$$ Using this property, we can divide the DFT calculation of length \(N\) into two DFT calculations of length \(N/2\). This is known as the Danielson-Lanczos Lemma and forms the basis of the FFT.
+
+# Danielson-Lanczos Lemma
+The DFT of length \(N\) (where \(N\) is even) can be expressed as the sum of two DFTs of length \(N/2\). One is formed from the even-indexed terms, and the other from the odd-indexed terms. [The DFT formula]([[Discrete Fourier Transform#^8ba133]]) can be expressed, according to the Danielson-Lanczos Lemma, as follows: 
+
+$$ 
+\begin{align} 
+y_k &= \sum_{j=0}^{n-1} x_j \omega^{jk} \\ & = x_0\omega^0 + x_2\omega^{2k} + x_4\omega^{4k} + \ldots + x_{n-2}\omega^{k(n-2)} \\ & \quad + \omega^k (x_1\omega^0 + x_3\omega^{2k} + x_5\omega^{4k} + \ldots + x_{n-1}\omega^{k(n-2)}) \\ &= \sum_{j=0}^{n/2-1} x_{2j} \omega^{2kj} + \omega^k \sum_{j=0}^{n/2-1} x_{2j+1} \omega^{2kj} \\ &= y^E_k + \omega^k y_k^O 
+\end{align}
+$$
+
+# Algorithm: Cooley-Tukey Algorithm
+Cooley-Tukey algorithm is designed for Fast Fourier Transform (FFT) algorithm. This algorithm significantly reduces the time complexity of computing the DFT from $O(N^2)$ to $O(N \log N)$.
+
+The Cooley-Tukey algorithm uses a divide-and-conquer method to break down the input signal into smaller parts. Cooley-Tukey algorithm then performs the Fourier transform on each part, and combines the results to obtain the final transform. It is particularly effective for input lengths that are powers of two.
+
+**Pseudo code**
+```
+function FFT(A, ω)
+  Input: Coefficient representation of a polynomial A(x) of degree ≤ n − 1, where n is a power of 2
+  Output: Value representation A(ω^0), . . . , A(ω^n−1)
+
+  if ω = 1: return A(1)
+  express A(x) in the form Ae(x^2) + xAo(x^2)
+  call FFT(Ae, ω^2) to evaluate Ae at even powers of ω
+  call FFT(Ao, ω^2) to evaluate Ao at odd powers of ω
+  for j = 0 to n − 1:
+    compute A(ω^j) = Ae(ω^2j) + ω^jAo(ω^2j)
+  return A(ω^0), . . . , A(ω^n−1)
+
+```

content/Basic Algebra/Residue Number System.md

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+---
+title: Residue Number System
+---
+Residue Number System(RNS)은 정수를 표현하고 연산하는 데에 사용되는 number system의 하나로, 병렬 연산에 유리하다는 특징이 있다. RNS에서는 정수를 여러 개의 서로 다른 서로소인 정수들(이후 기수 집합이라고 부르게 되는 것)으로 표현하며, 기반이 되는 원리에는 [[Chinese remainder theorem]]가 있다. 즉 서로소인 정수들이 주어졌을 때, 이들에 대한 연립 합동식의 해가 유일하다는 것에 기반하여 여러 서로소 정수들로 하나의 정수를 나타낼 수 있는 것이다.
+
+## RNS Representation
+RNS에서 정수 X는 다음과 같이 k개의 정수로 나타낼 수 있다.
+$$X \equiv \{x_1, x_2, x_3, ..., x_k\} \quad (mod \space \{m_1, m_2, m_3, m_4, ..., m_k\})$$
+여기서 ${m_1, m_2, m_3, …, m_k}$ 는 RNS의 기수(base) 집합이다.
+
+예를 들어 기수 집합 {3, 5, 7}을 사용하는 RNS에서 정수 23은 (2, 3, 2)로 표현된다.
+$$
+
+\begin{align} 23 \equiv 2 \quad& (mod \space 3) \\ 23 \equiv 3 \quad& (mod \space 5) \\ 23 \equiv 2 \quad& (mod \space 7) \\ \end{align}
+
+$$
+
+## Arithmetic operations
+### Add
+RNS에서의 덧셈은 단순한 residue끼리의 덧셈을 수행하여 이루어진다. 뺄셈 및 곱셈도 마찬가지로 수행된다.
+$$ \begin{align} \{x_1, x_2, x_3, ..., x_k\} +\{y_1, y_2, y_3, ..., y_k\} =\{& x_1 +y_1 (mod \space m_1), \\ & x_2 +y_2 (mod \space m_2), \\ &..., \\ &x_k +y_k (mod \space m_k)\} \end{align} $$
+
+### Subtract
+$$ \begin{align} \{x_1, x_2, x_3, ..., x_k\} +\{y_1, y_2, y_3, ..., y_k\} =\{& x_1 -y_1 (mod \space m_1), \\ & x_2 -y_2 (mod \space m_2), \\ &..., \\ &x_k -y_k (mod \space m_k)\} \end{align} $$
+
+### Multiply
+$$ \begin{align} \{x_1, x_2, x_3, ..., x_k\} +\{y_1, y_2, y_3, ..., y_k\} =\{& x_1 \cdot y_1 (mod \space m_1), \\ & x_2 \cdot y_2 (mod \space m_2), \\ &..., \\ &x_k \cdot y_k (mod \space m_k)\} \end{align} $$
+
+### Divide
+나눗셈은 나누는 수의 모듈로 역원을 이용해 계산한다. 따라서 다음과 같이 수행된다.
+
+$$ \begin{align} \{x_1, x_2, x_3, ..., x_k\} +\{y_1, y_2, y_3, ..., y_k\} =\{& x_1 \cdot y_1^{-1} (mod \space m_1), \\ & x_2 \cdot y_2^{-1} (mod \space m_2), \\ &..., \\ &x_k \cdot y_k^{-1} (mod \space m_k)\} \end{align} $$
+
+### Comparison
+동일성의 경우 두 수의 residue들이 동일하면 두 수도 동일하다고 판단한다.
+대소비교의 경우 직접적으로 수행하기 어려우며 일반적으로는 전체 값을 복원하여 연산한다.

content/Basic Cryptography/Basic Cryptography.md

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 > [[Basic Cryptography]] introduces the fundamental principles and techniques of cryptography. This includes basic encryption and decryption methods, and the principles behind key cryptographic algorithms such as symmetric and asymmetric encryption.
 
 
-[[Public key cryptography]]
-[[Symmetric key cryptography]]
-[[Diffie-Hellman]]
-[[RSA]]
-[[ElGamal]]
-[[ECDSA]]
-[[Post quantum cryptography]]
-[[Learning with Errors(LWE)]]
-[[Ring Learning with Errors(RLWE)]]
-[[Module Learning with Errors(MLWE)]]
+- [[Public key cryptography]]
+- [[Symmetric key cryptography]]
+- [[Diffie-Hellman]]
+- [[RSA]]
+- [[ElGamal]]
+- [[ECDSA]]
+- [[Post quantum cryptography]]
+- [[Learning with Errors(LWE)]]
+- [[Ring Learning with Errors(RLWE)]]
+- [[Module Learning with Errors(MLWE)]]

content/Privacy Enhancing Technologies (PET)/Privacy Enhancing Technologies (PET).md

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 > [[Privacy Enhancing Technologies (PET)]] explains advanced cryptographic methods designed to protect user privacy. It covers technologies such as zero-knowledge proofs, secure multi-party computation, and homomorphic encryption.
 
 
-[[Homomorphic Encryption]]
-[[Secure Multiparty Computation]]
-[[Differential Privacy]]
-[[Federated Learning]]
-[[Trusted Execution Environment]]
-[[Zero Knowledge Proofs]]
+- [[Homomorphic Encryption]]
+- [[Secure Multiparty Computation]]
+- [[Differential Privacy]]
+- [[Federated Learning]]
+- [[Trusted Execution Environment]]
+- [[Zero Knowledge Proofs]]