GLWE (#86)

* GLWE

* Update i18n/ko/Basic Cryptography/General LWE.md

Co-authored-by: hwam <137725515+hwamzzx@users.noreply.github.com>

* Update i18n/ko/Basic Cryptography/General LWE.md

Co-authored-by: hwam <137725515+hwamzzx@users.noreply.github.com>

* Update i18n/ko/Basic Cryptography/General LWE.md

Co-authored-by: hwam <137725515+hwamzzx@users.noreply.github.com>

* glwe

* chore: copy General LWE.md to Basic Cryptography directory

---------

Co-authored-by: user <user@MacBook-Pro-4.local>
Co-authored-by: hwam <137725515+hwamzzx@users.noreply.github.com>
Co-authored-by: yijun-lee <yijunlee.000@gmail.com>

Author: 4 people

Cryptography

@@ -1 +1 @@
-Subproject commit f60836de4bba01e0e33e053997551e88e79fc331
+Subproject commit e58ebf555964007b0f82b882aa11232eb6269030

content/Basic Cryptography/Basic Cryptography.md

@@ -20,4 +20,5 @@
 - [[Hash function]]
 - [[Merkle Tree]]
 - [[Digital Signature]]
-- [[Schnorr Signature]]
+- [[Schnorr Signature]]
+- [[General LWE]]

content/Basic Cryptography/General LWE.md

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+## GLWE: general LWE - Secret Key Encryption
+
+LWE와 RLWE을 일반화한 종류의 암호문이다. 
+
+1. secret key generation :
+    $\vec{S}=(S_0,\dots, S_{k-1})\in \mathcal{R}^k \text{ where R: uniform binary distribution}$
+<br>
+
+2. encryption :
+    $q,p: \text{positive integers}\\ q: \text{ciphertext modulus}\\p: \text{plaintext modulus}\\ \Delta(=q/p): \text{scaling factor}$ 
+    <br>
+
+    - $(A_0, \dots, A_{k-1},B)\in GLWE_{S,\sigma}(\Delta M)\subseteq \mathcal{R}_q^{k+1}\\A_i: \text{sampled uniformly random from Rq}$
+    - $B = \sum_{i=0}^{k-1} A_i \cdot S_i + \Delta M + E \in \mathcal{R}_q \\(R_q: \text{coefficients sampled from gaussian distribution } \chi_\sigma)$ <br>
+    A 부분은 주로 mask, B 부분은 body라고 불린다. 다항식 $\Delta M$은 message M의 encoding으로도 불린다.
+    최종적으로 secret key $\vec{S}=(S_0,\dots, S_{k-1})\in \mathcal{R}^k$ 하에서 encrypt된 cipher text  $(A_0, \dots, A_{k-1},B)\in GLWE_{S,\sigma}(\Delta M)\subseteq \mathcal{R}_q^{k+1}$를 얻을 수 있다. 
+<br>
+
+3. Decryption :
+    - $B - \sum_{i=0}^{k-1} A_i \cdot S_i = \Delta M + E \in \mathcal{R}_q$
+    - $M = \left\lfloor \frac{\Delta M + E}{\Delta} \right\rfloor.$<br>
+    message M은 $\Delta$와 곱해지며 $\Delta M + E$의 MSB부분에, E는 LSB 부분에 생성됨을 확인할 수 있으며, $\left|E\right|<\Delta/2 $라면 decryption을 통한 원본 message의 복원이 가능하다. 

i18n/ko/Basic Cryptography/General LWE.md

@@ -0,0 +1,22 @@
+## GLWE: general LWE - Secret Key Encryption
+
+LWE와 RLWE을 일반화한 종류의 암호문이다. 
+
+1. secret key generation :
+    $\vec{S}=(S_0,\dots, S_{k-1})\in \mathcal{R}^k \text{ where R: uniform binary distribution}$
+<br>
+
+2. encryption :
+    $q,p: \text{positive integers}\\ q: \text{ciphertext modulus}\\p: \text{plaintext modulus}\\ \Delta(=q/p): \text{scaling factor}$ 
+    <br>
+
+    - $(A_0, \dots, A_{k-1},B)\in GLWE_{S,\sigma}(\Delta M)\subseteq \mathcal{R}_q^{k+1}\\A_i: \text{sampled uniformly random from Rq}$
+    - $B = \sum_{i=0}^{k-1} A_i \cdot S_i + \Delta M + E \in \mathcal{R}_q \\(R_q: \text{coefficients sampled from gaussian distribution } \chi_\sigma)$ <br>
+    A 부분은 주로 mask, B 부분은 body라고 불린다. 다항식 $\Delta M$은 message M의 encoding으로도 불린다.
+    최종적으로 secret key $\vec{S}=(S_0,\dots, S_{k-1})\in \mathcal{R}^k$ 하에서 encrypt된 cipher text  $(A_0, \dots, A_{k-1},B)\in GLWE_{S,\sigma}(\Delta M)\subseteq \mathcal{R}_q^{k+1}$를 얻을 수 있다. 
+<br>
+
+3. Decryption :
+    - $B - \sum_{i=0}^{k-1} A_i \cdot S_i = \Delta M + E \in \mathcal{R}_q$
+    - $M = \left\lfloor \frac{\Delta M + E}{\Delta} \right\rfloor.$<br>
+    message M은 $\Delta$와 곱해지며 $\Delta M + E$의 MSB부분에, E는 LSB 부분에 생성됨을 확인할 수 있으며, $\left|E\right|<\Delta/2 $라면 decryption을 통한 원본 message의 복원이 가능하다.