how do you calculate the private exponent in asymmetric key encryption Here is an excellent paper on the math of asymmetric key encryption: http://www.mathaware.org/mam/06/Kaliski.pdf‎  

See the example on Page 6.  

The public key = $55$
Primes used to calculate public key are $5$ and $11$.

$e = 3$

Now see the appendix:
$L = \mathrm{LCM}(p-1, q-1) = 20$

The paper states $de = 1 \mod L$

I can't figure out how he gets the value of $d = 7$
You compute the modular inverse of $e \pmod {20}$ with the Extended Euclidean Algorithm, but in this simple case with $e=3$ you can guess $d=7$ because $3\times 7 = 21$.