# Why are Prime Numbers so frequently used in Security Systems?

Is it related to the fact that we do not know any rule regarding their distribution to date?

For example we have generated lists of several billion Consecutive Prime Numbers and it turns out that there does not seem to be a rule regarding the gaps between each number or at least not on an acceptable human scale .. ... .....

Here you can download 2 of our small generated lists, the first one containing the first 10'000'000 and the second one containing the first 100'000'000 consecutive Prime Numbers all listed in the correct order with their exact positions, as well as the gap between each number.

Are there any available very long and detailled lists of Large Prime Numbers that allow for larger-scale analysis of the gaps between numbers ?

Or is there a mathematical rule to quantify those number gap or to know their emerging sequence?

Regards
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Senior DeveloperCommented:
If you're interested in this kind of Mathematics, then you need some studying about Riemann's work and his zeta function.

But this really hard stuff. The conjecture based on the zeta function, called Riemann hypothesisis part of the Millenium Problems list of the Clay Mathematics Institute. You'll get 1 million \$, when you either proof right or wrong.

And for the question in your subject: Cause the factorization problem is a hard to calculate problem
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Commented:
Why are Prime Numbers so frequently used in Security Systems?

A very good question, which leads to another good question:
If you aren't sure, why did you use them in your security system?

The distribution of primes has been studied extensively:
https://en.wikipedia.org/wiki/Prime_number_theorem

As have the gaps between primes:
https://en.wikipedia.org/wiki/Prime_gap

There is also recent work, looking for patterns in the first 400 billion primes:
http://www.independent.co.uk/news/science/maths-experts-stunned-as-they-crack-a-pattern-for-prime-numbers-a6933156.html

400 billion is only  4 x 10^11.  But people have found all the primes up to 10^18
https://primes.utm.edu/notes/faq/LongestList.html

The last link also indicates why lists of primes are not that useful:
Small primes are too easy to find. The can be found far faster than they can be read from a hard disk.
Long lists just waste storage, and if placed on the Internet, they just waste bandwidth.
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Commented:
Why are Prime Numbers so frequently used in Security Systems?

Actually, I believe systems based on prime numbers are out of favor because the large key sizes required to insure security are very inefficient.
RSA does use primes, but it also allows for public and private keys, and so it is still the standard for key exchange.
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CreationAuthor Commented:

Of course, we know that Mr. Riemann's work is one of the key elements of this issue.

If we use Prime Numbers as part of our security system (very small one), it is only because we have found a way to make the calculation of prime numbers faster than the current known systems.

We did not create a new algorithm for the number primality test, but a preprocessing algorithm that allows for much faster results in combination with the best known algorithms.

For example, to calculate the list of 100'000'000 consecutive prime numbers (this is a very small list) and this in Mono-CPU-Thread mode, we obtained a time saving of 6% compared to the use only the Miller-Rabin algorithm contained in the famous GMP library.

We are in the process of completing the development of the Multi-Threading part of our pre-processing algorithm, we will post the performance results once the first lists are calculated.

Best regards,
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RetiredCommented:
The prime numbers used in encryption systems are much larger than you've imagined. Ron Rivest (the R in RSA) asked this question to academia in about 1995, "The Factoring Challenge". Can you factor this number into two primes:
103476330920053572286873676368850992545116734031879550656996221305168759307650257059

The factors  are:
342526708406385189575946388957261768583317   and
302097116459852171130520711256363590397527

Google either prime number or ("The Factoring Challenge") or go to:
https://www2.cs.arizona.edu/~collberg/Teaching/466-566/2012/.../Handout-7.pdf

BTW: Ron Rivest is the creator of the Message Digest series ending with MD5.
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Commented:
I think this has been resolved.
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