[Cryptography] PGP -- Can someone help me understand something?

[Stephan Neuhaus]

On 09.08.18 08:45, Matt Maxson wrote:
> The question was, basically, if someone has access to both a PGP encrypted email and a plain text version of the same email, can an attacker determine the key.  The answer given was "no".
> 
> I don't understand.  Why can't that happen?  For example, if I have 10 + x = 50  (this can be replaced with any formula that has exactly one unknown), I can solve for X.  In my thinking, isn't the unknown in the equation simply the key?  Sure, the maths are more complex, but it should be a trivial issue to work backwards and solve for the key.

The short answer is that there are functions that are invertibe, but 
which are hard to invert. For example, I'm telling you that I know some 
x for which

   3^x mod 127 = 68.

Now I ask you, what is x? You can solve this problem easily, but it will 
probably be by trial and error, not by "working backwards". The point is 
that you can't just "work backwards", or if you can, fame and fortune 
await. So it's not always "a trivial issue to work backwards".

As always, the details are complicated, but I believe that this covers 
the essentials.

Fun,

Stephan