From 7c412050570ef229dd78cbcffbf80f23728a630d Mon Sep 17 00:00:00 2001
From: Prefetch
Date: Sat, 13 May 2023 15:42:47 +0200
Subject: Improve knowledge base

---
 .../concept/diffie-hellman-key-exchange/index.md   | 26 +++++++++++++---------
 1 file changed, 16 insertions(+), 10 deletions(-)

(limited to 'source/know/concept/diffie-hellman-key-exchange')

diff --git a/source/know/concept/diffie-hellman-key-exchange/index.md b/source/know/concept/diffie-hellman-key-exchange/index.md
index 4735209..3525881 100644
--- a/source/know/concept/diffie-hellman-key-exchange/index.md
+++ b/source/know/concept/diffie-hellman-key-exchange/index.md
@@ -7,18 +7,19 @@ categories:
 layout: "concept"
 ---
 
-In cryptography, the **Diffie-Hellman key exchange** is a method
-for two parties to securely agree on an encryption key,
-when they can only communicate over an insecure channel.
+In cryptography, the **Diffie-Hellman key exchange** is a method for two parties,
+who can only communicate over an insecure channel,
+to securely agree on an encryption key.
 
 The fundamental assumption of the Diffie-Hellman scheme,
 upon which its security rests,
 is that the following function $$f(n)$$ is a **trapdoor function**,
 which means that calculating $$f$$ is easy,
-but its inverse $$f^{-1}$$ is extremely hard to find:
+but its inverse $$f^{-1}$$ is extremely hard:
 
 $$\begin{aligned}
-    f(n) = g^n \bmod p
+    f(n)
+    \equiv g^n \bmod p
 \end{aligned}$$
 
 Where $$n$$ is a natural number, and $$p$$ is a prime.
@@ -39,9 +40,11 @@ Alice and Bob each choose a secret number from $$\{0, ..., p \!-\! 2\}$$, respec
 and then privately calculate $$A$$ and $$B$$ as follows:
 
 $$\begin{aligned}
-    A = g^a \bmod p
-    \qquad \quad
-    B = g^b \bmod p
+    A
+    \equiv g^a \bmod p
+    \qquad \qquad
+    B
+    \equiv g^b \bmod p
 \end{aligned}$$
 
 Finally, they transmit these numbers $$A$$ and $$B$$
@@ -50,13 +53,16 @@ and then each side calculates $$k$$, which is the desired secret key:
 
 $$\begin{aligned}
     \boxed{
-        k = A^b \bmod p = B^a \bmod p = g^{ab} \bmod p
+        k
+        \equiv A^b \bmod p
+        = B^a \bmod p
+        = g^{ab} \bmod p
     }
 \end{aligned}$$
 
 The point is that $$k$$ includes both $$a$$ *and* $$b$$,
 but each side only needs to know *either* $$a$$ *or* $$b$$.
-And, due to the trapdoor assumption,
+Thanks to the trapdoor assumption,
 the eavesdropper knows $$A$$ and $$B$$,
 but cannot recover $$a$$ or $$b$$.
 
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