How long does it take to crack DES and AES? Suppose that a single evaluation of a block-cipher (DES or AES) takes 10 operations, and the computer can do $10^{15}$ such operations per second.

How long would it take for to recover a DES key, using a brute-force search? How about a 128-bit AES key?
Assume that 1 evaluation of {DES, AES} takes 10 operations, and we can perform $10^{15}$ operations per second. Trivially, that means we can evaluate $10^{14}$, or about $2^{46.5}$ {DES, AES} encryptions per second. This is a simplistic view: we are ignoring here the cost of testing whether we found the correct key, and the key schedule cost.

So on our hypothetical machine, a 56-bit DES key would take, on average, $2^{55}/2^{46.5} = 2^{8.5} \approx 362$  seconds to find. Similarly, a 128-bit AES key would take $2^{127}/2^{46.5} = 2^{80.5}$ seconds $\approx 2^{55}$ (or approximately $36$ quadrillion) years to find.