Here is the worked example I promised you.

We decided to work with:

	p = 11
	q = 7
	n = 77

	Phe(n) = (p-1) * (q-1) = 60

Now, you may recall we picked e=3. Turns out that you can't do this, because e must be relatively prime to Phe(n), and 3 evenly divides 60.

We could have picked e=7.
Now we need to find d. Recall that d = e^-1 (mod n)
To calculate the inverse of e in (mod n), we use the Extended Euclid Algorithm. Like this:

	60 = 1*60 + 0*7
	 7 = 0*60 + 1*7

From these two, we can compute:

	 4 = 1*60 - 8*7		(60 mod 7 = 4)

Combining the last two lines we have:
	 7 = 0*60 + 1*7
	 4 = 1*60 - 8*7		

From which we calculate the next line:

	 3 = -1 * 60 + 9*7	(3 = 7 mod 4)

Now combining the last two lines:

	 4 = 1*60 - 8*7		
	 3 = -1 * 60 + 9*7	

We get the next line:

	1 = 2*60 - 17*7  (1 = 4 mod 3)

From which we can show that the answer is "43" 

	43 = -17 * 7^-1 (mod 60)

We can prove that:

	43 * 3 = 301 = 1 (mod 60)

So d = 43

Let's encrypt the message "12"

To encrypt, we take m^e (mod n)

	12^7 = 35831808 = 48 (mod 60)

Now let's decrypt "48":

	48 ^ 43 = 1965048198399560713177500537391830916254451560885426333004585474449211392 =  12 (mod 60)


Look at that! It works.