Generating a Germain Prime efficiently, using Python and an 8-core CPU.

I have spent extensive time, as well as previous blog postings, exploring the subject of how to generate a Germain prime (number), in the Python (3.5) programming language, but taking advantage of parallel computing to some slight degree.

Before I go any further with this subject, I’d like to point out that generally, for production-ready applications, Python is not the best language to use. The main reason is the fact that Python is an interpreted language, even though many modern interpreted languages are compiled into bytecode before being interpreted. This makes a Python script slower by nature, than very well-written C or even C++. But what I aim to do is to use Lego-Blocks to explore this exercise, yet, to use tools which would be used professionally.

The main way I am generating prime numbers is, to start with a pseudo-random, 512-bit number (just as a default, the user can specify different bit-lengths), and then to attempt to divide this number by a list of known, definite primes, that by now only extend to 4096 (exclusively, of course), in an attempt to disprove that number prime. In about 1/8 of all cases, the number survives this test, after which I undertake a more robust, Miller-Rabin approach to try disproving it prime 192 times, probabilistically. If the number has survived these tests, my program assumes it to be prime.

Even though I’ve never been told this, the probability of a non-prime Candidate surviving even a single Miller-Rabin Test, or Witness, is very small, smaller than 1/8. This could be due to the fact that the Witness in question is raised to a high, odd exponent, in its 512-bit modulus etc., after which it would be squared some number of times. Because the first Candidate in the modulus of 4 is 3, that one actually gets squared a subsequent total of zero times. And after each exponentiation, the result could be any number in the modulus, with the possible exception of zero. It needs to become either (1) or (n-1) in the modulus of (n), for the Candidate to survive the test. (:1)

Further, there is no need for the numbers that get used as witnesses, which are pseudo-random, to be of the same, high, cryptographic quality of pseudo-random, as the candidate is, which is being tested.

But there is a sub-category of prime numbers which have recently been of greater interest to me, which is known as the Germain prime numbers, such that the Totient of that main candidate, divided by two, should also be prime. And so, if the density of prime numbers roughly equal to (n) is (1 / log(n)), and if we can assume a uniform distribution of random numbers, then the probability of finding a Germain prime is roughly (1 / (log (n))2), assuming that our code was inefficient enough actually to test all numbers. The efficiency can be boosted by making sure that the random number modulo 4 equals 3.

But one early difficulty I had with this project was, that if I needed to start with a new pseudo-random number for each candidate, on my Linux computers, I’d actually break ‘/dev/urandom’ ! Therefore, the slightly better approach which I needed to take next was, to make my initial random number the pseudo-random one, and then just to keep incrementing it by 4, until the code plodded into a Germain prime.

Even when all this is incorporated into the solution I find that with Python, I need the added advantage of parallel computing. Thus, I next learned about the GIL – The Global Interpreter Lock – and the resulting pitfalls of multi-threaded Python, which is not concurrent. Multi-threading under Python tells the O/S to allocate CPU cores as usual, but then only allows 1 core to be running at any one time! But, even under those constraints, I found that I was benefiting from the fact that my early code was able to test at least 2 candidates for primality simultaneously, those being the actual candidate, as well as its Totient divided by 2. And as soon as either candidate was disproved prime, testing on the other could be stopped quickly. This reduced the necessary number of computations dramatically, to make up for the slowness of multi-threaded Python, and I felt that I was on the right track.

The only hurdle which remained was, how to convert my code into multi-processing code, no longer merely multi-threaded, while keeping the ability for two processes, then, to send each other a shutdown-command, as soon as the present process disproved its number to be prime.

(Updated 6/01/2019, 17h35 … )

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Refining my Python, for generating strong prime numbers…

According to an earlier posting, I had applied the (probabilistic) Miller-Rabin test, after testing whether large prime number candidates are divisible by any in a list of smaller, definite prime numbers, to generate pseudo-random numbers first, and then to continue until one was found to be prime.

That earlier effort had numerous issues, including the fact that it did not generate the kind of strong prime numbers needed for Diffie-Hellman key exchange. I have broadened my horizons slightly, and written more-up-to-date Python 3, which will generate such strong primes, as well as computing the resulting, Primitive Root, which would be used as the generator (g), if the results were ever actually used in cryptography.

I’d like to thank my friend, François Dionne, for giving me help with this project.

The resulting Python script can be downloaded from this URL:

http://dirkmittler.homeip.net/text/Generate_Prime_DMFD_ITC.py

(Updated 5/23/2019, 18h40 : )

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What I’ve learned about RSA Encryption and Large Prime Numbers – How To Generate

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:

http://dirkmittler.homeip.net/text/Generate_Prime_3.py

 

(Corrected 10/03/2018, 23h20 … )

(Updated 10/08/2018, 9h25 … )

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