From d8c4aad7c099604b2b2c7ac9ab55201dac94f616 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?KevinZ=C3=B8nda?=
 <33132228+KevinZonda@users.noreply.github.com>
Date: Tue, 9 May 2023 01:16:34 +0100
Subject: [PATCH] rsa

---
 Week3/0x02 RSA.md | 41 +++++++++++++++++++++++++++++++++++++++++
 1 file changed, 41 insertions(+)
 create mode 100644 Week3/0x02 RSA.md

diff --git a/Week3/0x02 RSA.md b/Week3/0x02 RSA.md
new file mode 100644
index 0000000..984c401
--- /dev/null
+++ b/Week3/0x02 RSA.md	
@@ -0,0 +1,41 @@
+# 0x02 RSA
+
+## 算法
+
+包含3个算法，Gen，Enc 和 Dec
+
+### Gen
+
+输入安全参数（security parameter）$\lambda$
+- 生成拥有同样 $\lambda$ bits 的两个不同素数，$p, q$
+- 计算 $N=pq$, $\phi(N)=(p-1)(q-1)$
+- 选择一个随机整数 $e\in (1, \phi{(N)})$ 满足 $\text{gcd}(e, \phi{(N)})=1$
+- 定义 $\mathbb{Z}^+_N=\{x \mid x\in(0, N)\quad \text{and}\quad \text{gcd}(x,N)=1\}$
+- 计算 $d$ 满足 $ed\equiv 1 \quad({\text{mod }(\phi(N))})$
+- 公钥 $PK=(e, N)$, 私钥 $SK=e, d, N$
+
+### Enc
+$\text{INPUT: } PK, m\in\mathbb{Z}^+_N$
+$\text{OUTPUT: } c = m^e (\text{mod }(N))$
+
+### Dec
+$\text{INPUT: } SK, c\in\mathbb{Z}^+_N$
+$\text{OUTPUT: } m = c^d (\text{mod }(N))$
+
+### Example
+
+1. INPUT: $p = 3$, $q = 11$
+2. $N=pq=33$, $\phi(N)=(p-1)(q-1)=2\times 10=20$
+4. Let $e=7$
+	- Note: $\text{gcd}(e,\phi(N))=\text{gcd}(7, 20)=1$
+5. $d=3$, ➡️  $e\times d=7\times 3 = 21=20+1=\phi{(N)}+1$
+6. $PK=(e, N)=(7, 33)$
+7. $SK=(e, d, N)=(7, 3, 33)$
+
+$\mathbb{Z}^+_N=\{1,2,4,5,7,8,10,13,14,16,17,19,20,23,25,26,28,29,31,32\}$
+Enc
+1. $m=4$
+2. $c=m^e (\text{mod }(N))=4^7 \mod{33}=16$
+Dec
+1. $c=16$
+2. $m= c^d (\text{mod }(N))=16^3\mod{33}=4$