One of the ways in which I function, is to write down thoughts in this blog, that may seem clear to me at first, but which, once written down, require further thought and refinement.
I’ve written numerous times about Public Key Cryptography, in which the task needs to be solved, to generate 1024-bit prime numbers – or maybe even, larger prime numbers – And I had not paid much attention to the question, of how exactly to do that efficiently. Well only yesterday, I read a posting of another blogger, that inspired me. This blogger explained in common-sense language, that a probabilistic method exists to verify whether a large number is prime, that method being called “The Miller-Rabin Test”. And the blogger in question was named Antoine Prudhomme.
This blogger left out an important part in his exercise, in which he suggested some working Python code, but that would be needed if actual production grade-code was to generate large prime numbers for practical cryptography. He left out the eventual need, to perform more than just one type of test, because this blogger’s main goal was to explain the one method of testing, that was his posting subject.
I decided to modify his code, and to add a simple Fermat Test, simply because (in general,) to have two different probabilistic tests, reduces the chances of false success-stories, even further than Miller-Rabin would reduce those chances by itself. But Mr. Prudhomme already mentioned that the Fermat Test exists, which is much simpler than the Miller-Rabin Test. And, I added the step of just using a Seive, with the known prime numbers up to 65535, which is known not to be prime itself. The combined effect of added tests, which my code performs prior to applying Miller-Rabin, will also speed the execution of code, because I am applying the fastest tests first, to reduce the total number of times that the slower test needs to be applied, in case the candidate-number could in fact be prime, as not having been eliminated by the earlier, simpler tests. Further, I tested my code thoroughly last night, to make sure I’ve uploaded code that works.
Here is my initial, academic code:
(Corrected 10/03/2018, 23h20 … )
(Updated 10/08/2018, 9h25 … )