https://zcc31415926.github.io/Contents/blogs/diffie_hellman/diffie_hellman.html

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67<h1 id="discussion-the-diffie-hellman-key-exchange-algorithm">Discussion: The Diffie-Hellman Key-Exchange Algorithm</h1>
68
69<p><a href="/">Return to main</a></p>
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71<p><a href="/Contents/blogs.html">Return</a></p>
72
73<p>The Diffie-Hellman algorithm is a key-exchange algorithm for safe communications in networks.
74The algorithm is as follows:</p>
75<ol>
76  <li>the sender sets a prime number $p$, a base number $g$ and a random number $a$</li>
77  <li>the sender calculates $A=g^a\%p$</li>
78  <li>the sender sends $p$, $g$ and $A$ to the receiver</li>
79  <li>the receiver sets a random number $b$</li>
80  <li>the receiver calculates $B=g^b\%p$ and $s1=A^b\%p$</li>
81  <li>the receiver sends $B$ to the sender</li>
82  <li>the sender calculates $s2=B^a\%p$</li>
83  <li>$s1 \equiv s2 = s$, and the shared key is set $s$</li>
84</ol>
85
86<p>In the algorithm, $A$ and $B$ are named public keys; $a$ and $b$ are named private keys.
87In communications, private keys are hidden and unknown to other users;
88public keys are transmitted through the network.
89However, with only public keys, $p$ and $g$, $s$ cannot be determined.
90The sender and the receiver can cooperate to determine a shared key without knowledge of each other’s private key.</p>
91
92<p>Proof of the algorithm:</p>
93
94<p>let $x$ and $y$ be two positive integers that satisfy:</p>
95
96\[g^a = px + A \\
97g^b = py + B\]
98
99<p>then,</p>
100
101\[A^b = (g^a-px)^b \\
102A^b\%p = (g^{ab}-C_b^1\times g^{a(b-1)}px+C_b^2\times g^{a(b-2)}(px)^2+\cdots)\%p = g^{ab}\%p\]
103
104<p>Similarly,</p>
105
106\[B^a\%p = g^{ab}\%p\]
107
108<p>Therefore, $s1 \equiv s2$.</p>
109
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