I would like to start my question with a quote:
If an encrypted document and its source can be obtained, it is possible to attempt to find the 56-bit key. An exhaustive search by conventional means would make it necessary to search 2 to the power 55 keys before hitting the correct one. This would take more than a year even if one billion keys were tried every second, by comparison, Grover's algorithm could find the key after only 185 searches.
This quote is from A Brief History of Quantum Computing By Simon Bone and Matias Castro
As a reader, I wonder how the authors got the magic number of 185. Unfortunately, that is not justified.
I thought about it myself. To calculate the number of iterations, the Grover algorithm uses this formula:
$$k=\frac{\pi}{4\cdot \sin^{-1}\left(\frac{1}{\sqrt{2^n}}\right)}-0.5$$
If I just do that for the number $2^{56}$ (DES keysize), then it follows that you need k iterations:
$$k=\frac{\pi}{4\cdot \sin^{-1}\left(\frac{1}{\sqrt{2^{56}}}\right)}-0.5=210828713$$
That's still not the number the authors suggest. Therefore I ask here, if anyone can imagine, how the authors came to the number. Is my consideration correct?
Assuming it were $2^{16}$, then you would need about 200 iterations, which are still not 185. I am not aware of a cryptographic system with a key length of $2^{16} $ ...