Editing Diffie–Hellman key exchange (section)

=== Secrecy chart ===
The chart below depicts who knows what, again with non-secret values in {{blue|blue}}, and secret values in '''{{red|red}}'''. Here [[Alice and Bob#Cast of characters|Eve]] is an [[Eavesdropping#Network attacks|eavesdropper]] – she watches what is sent between Alice and Bob, but she does not alter the contents of their communications.
* ''<span style="color:blue">g</span>'', public (primitive root) base, known to Alice, Bob, and Eve. ''<span style="color:blue">g</span>'' = <span style="color:blue">5</span>
* ''<span style="color:blue">p</span>'', public (prime) modulus, known to Alice, Bob, and Eve. ''<span style="color:blue">p</span>'' = <span style="color:blue">23</span>
* '''''<span style="color:red">a</span>''''', Alice's private key, known only to Alice. '''''<span style="color:red">a</span>''''' = '''<span style="color:red">6</span>'''
* '''''<span style="color:red">b</span>''''', Bob's private key known only to Bob. '''''<span style="color:red">b</span>''''' = '''<span style="color:red">15</span>'''
* ''<span style="color:blue">A</span>'', Alice's public key, known to Alice, Bob, and Eve. ''<span style="color:blue">A</span>'' = ''<span style="color:blue">g</span><sup>'''<span style="color:red">a</span>'''</sup>'' mod ''<span style="color:blue">p</span>'' = <span style="color:blue">8</span>
* ''<span style="color:blue">B</span>'', Bob's public key, known to Alice, Bob, and Eve. ''<span style="color:blue">B</span>'' = ''<span style="color:blue">g</span><sup>'''<span style="color:red">b</span>'''</sup>'' mod ''<span style="color:blue">p</span>'' = <span style="color:blue">19</span>

{| border="0" cellspacing="0" cellpadding="2"
|-
| valign="top" |
{| class="wikitable"
|+ Alice
|- style="text-align:center;" 
! Known
! Unknown
|-
| ''<span style="color:blue">p</span>'' = <span style="color:blue">23</span>
|
|-
| ''<span style="color:blue">g</span>'' = <span style="color:blue">5</span>
|
|-
| '''''<span style="color:red">a</span>''''' = '''<span style="color:red">6</span>'''
| '''''<span style="color:red">b</span>'''''
|-
| ''<span style="color:blue">A</span>'' = <span style="color:blue">5</span><sup>'''''<span style="color:red">a</span>'''''</sup> mod <span style="color:blue">23</span>
|
|-
| ''<span style="color:blue">A</span>'' = <span style="color:blue">5</span><sup>'''<span style="color:red">6</span>'''</sup> mod <span style="color:blue">23</span> = <span style="color:blue">8</span>
|
|-
| ''<span style="color:blue">B</span>'' = <span style="color:blue">19</span>
| 
|-
| '''''<span style="color:red">s</span>''''' = <span style="color:blue">B</span><sup>'''''<span style="color:red">a</span>'''''</sup> mod <span style="color:blue">23</span>
|
|-
| '''''<span style="color:red">s</span>''''' = <span style="color:blue">19</span><sup>'''<span style="color:red">6</span>'''</sup> mod <span style="color:blue">23</span> = '''<span style="color:red">2</span>'''
|
|}
| valign="top" | 
{| class="wikitable"
|+ Bob
|- style="text-align:center;" 
! Known
! Unknown
|-
| ''<span style="color:blue">p</span>'' = <span style="color:blue">23</span>
|
|-
| ''<span style="color:blue">g</span>'' = <span style="color:blue">5</span>
|
|-
| '''''<span style="color:red">b</span>''''' = '''<span style="color:red">15</span>'''
| '''''<span style="color:red">a</span>'''''
|-
| ''<span style="color:blue">B</span>'' = <span style="color:blue">5</span><sup>'''''<span style="color:red">b</span>'''''</sup> mod <span style="color:blue">23</span>
|
|-
| ''<span style="color:blue">B</span>'' = <span style="color:blue">5</span><sup>'''<span style="color:red">15</span>'''</sup> mod <span style="color:blue">23</span> = <span style="color:blue">19</span>
|
|-
| ''<span style="color:blue">A</span>'' = <span style="color:blue">8</span>
| 
|-
| '''''<span style="color:red">s</span>''''' = <span style="color:blue">A</span><sup>'''''<span style="color:red">b</span>'''''</sup> mod <span style="color:blue">23</span>
|
|-
| '''''<span style="color:red">s</span>''''' = <span style="color:blue">8</span><sup>'''<span style="color:red">15</span>'''</sup> mod <span style="color:blue">23</span> = '''<span style="color:red">2</span>'''
|
|}
| valign="top" | 
{| class="wikitable"
|+ Eve
|- style="text-align:center;" 
! Known
! Unknown
|-
| ''<span style="color:blue">p</span>'' = <span style="color:blue">23</span>
|
|-
| ''<span style="color:blue">g</span>'' = <span style="color:blue">5</span>
|
|-
|
| '''''<span style="color:red">a</span>''''', '''''<span style="color:red">b</span>'''''
|-
| &nbsp;
| &nbsp;
|-
| &nbsp;
| &nbsp;
|-
| ''<span style="color:blue">A</span>'' = <span style="color:blue">8</span>, ''<span style="color:blue">B</span>'' = <span style="color:blue">19</span>
| 
|-
| &nbsp;
| &nbsp;
|-
|
| '''''<span style="color:red">s</span>'''''
|}
|}

Now '''''<span style="color:red">s</span>''''' is the shared secret key and it is known to both Alice and Bob, but ''not'' to Eve. Note that it is not helpful for Eve to compute ''<span style="color:blue">AB</span>'', which equals ''<span style="color:blue">g</span>''<sup>'''''<span style="color:red">a</span>'''''+'''''<span style="color:red">b</span>'''''</sup> mod <span style="color:blue">p</span>.

Note: It should be difficult for Alice to solve for Bob's private key or for Bob to solve for Alice's private key. If it is not difficult for Alice to solve for Bob's private key (or vice versa), then an eavesdropper, [[Alice and Bob#Cast of characters|Eve]], may simply substitute her own private / public key pair, plug Bob's public key into her private key, produce a fake shared secret key, and solve for Bob's private key (and use that to solve for the shared secret key). [[Alice and Bob#Cast of characters|Eve]] may attempt to choose a public / private key pair that will make it easy for her to solve for Bob's private key.