Prime numbers are important building blocks of mathematics. They are used in many encryption techniques because it is difficult to find the prime factors of a number but easy to multiply prime numbers. This property makes prime numbers suitable for ensuring secure communication and transactions. However, current primality tests that determine if a number is prime are probabilistic and have a small chance of error. Faster deterministic tests are needed to securely use larger prime numbers for stronger encryption.

1. Why prime numbers are of great importance and their use in everyday life
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By VincS

2. Definition of prime number
A prime number (also prime)is a natural integer, greater than 1, which can
be divided only by 1 or by itself (without generating remainder).
Definition of composited number
Conversely, a composited number is a natural integer, greater than 1, which can be
divided by two or more prime numbers (without generating remainder).

3. Primes:the building blocks of mathematics
By the two definitions, we can understand how prime numbers can be considered as the
building blocks of mathematics, because with them (along with the number 1, which is a
special number) are formed all the natural integers.
• 1 (special number = 1 x 1 x 1 x ……)
• 2 (prime number)
• 3 (prime number)
• 4 (composited number) = 2 x 2
• 5 (prime number)
• 6 (composited number) = 2 x 3
• 7 (prime number)
• 8 (composited number) = 2 x 2 x 2
• 9 (composited number) = 3 x 3
• 10 (composited number) = 2 x 5
• … etcetera …

4. The number 1 is special!
Since thousands of years mathematecians have discussed if the number 1 is a prime
number or not (as in the modern trend of thought).
The number 1 is actually a special number (as the number 0 [zero] is special for other
reasons); in fact it can be at the same time a prime number and a composite number.
Here's why:
• the number 1 falls in the specificity of prime numbers as it can only be divided by 1 or
by itself without generating remainder
• the number 1 can, at the same time, be the result of 1x1x..x1 (staying the same) and then
fall in the specificity of dialed numbers

5. Why prime numbers are currently of great
importance?
Because they are suitable for techniques that, more than others, ensure the confidentiality of
communications, and their reliability; among many others, for example, economic transactions.
Some examples:
https://; no one can understand what is sent via the browser
WEP/WAP keys; no one can use for free your ADSL contract via wi-fi
cryptography; only who knows the password can access documents
electronic signature; no one can sign documents on your behalf
POS; no one can catch your PIN
on-line banking; no one can steal your money
etc.etc..

6. In which way?
Simplifing, for a trivial calculator is very easy to do the multiplication of two prime
numbers ...
52639 x 47353 = 2492614567
... but it is rather difficult (even for large computers) do the reverse as finding the two
prime numbers starting knowing the result of the product.
On the web there are thousands of documents that explain how this property is used to
ensure privacy. If you want to try other examples, there are many network utilities that
generate prime numbers and other running factorization of composite numbers.

7. Suggested tools to perform test
To generate the nth prime number (and others stuffs) I can recommend ...
http://primes.utm.edu/nthprime/
To perform the decomposition I can recommend ...
http://www.spiega.com/rez/scomposizione_fattori.php?

8. Is it a prime number?
Unfortunately, a particular characteristic of prime numbers is that is not easy, even for
larger computers, determining whether they are effectively prime numbers or if they are
not.
To do this, in a short time, probabilistic methods are used (eg. the Rabin-Miller). These
methods, therefore, have a margin of error that is inversely proportional to the time the
test is performed.
Other methods, such as deterministic methods (eg. Wilson or AKS), have no margin of
error, but they are almost impossible to use because it would take a long time to run a
single test, making it useless for any application.

9. Where is the problem?
The limitation of not being able to generate and verify big numbers certainly primes
in a short time, makes unsecure the encryption of information because the
decomposition of generated products would be relatively simple. Do you want to
test it?
Oversimplifying, try to imagine that a certain prime number is the secret information
you want to transmit to a friend of yours through the network. Previously you both
friends shared a "probably" prime being your key of encryption and decryption. Let
be their product the way you think to safely send it through the network. You are
sure that no one will be able to do the decomposition of the product even if it would
be intercepted. But are you really sure?

10. Here is the problem!
Let’s get the prime number secret information 47353 (16 bits wide) and multiply it by the "probably" prime number encryption
key (16 bits wide) 34387 (someone sold it to you as to be a prime number but indeed it is = 137 x 251 ) .
Now let’s run on the calculator 34387 x 47353 = 1628327611 and let’s insert the result in an on-line decomposition tool (click
below) ....
http://www.spiega.com/rez/scomposizione_fattori.php?q=1628327611
You'll see that in a few moments the result of the breakdown will appear!!!
Your secret number has been discovered!
Just to double check, let’s insert the product of a encryption key “certainly” prime number
52639 x 47353 = 2492614567 (click below) ...
http://www.spiega.com/rez/scomposizione_fattori.php?q=2492614567
The tool is unable to calculate the factors and mistakes it for a prime number!
Your secret number is safely at your friend’s home that can so divide …
2492614567/52639 = 47353 … to get so the secret information you would let him knowing!

11. Prime numbers bigger and bigger!
Is it clear now why the inability to determine if a number is really prime endangers the secrecy of any kind of communications?
It 'easy to understand that greater are the two prime numbers and how greater will be the difficult to make the
decomposition of the product. Surely you have heard of 128-bit encryption, 256-bit but also over (512 and 1024 bits). Here
below are a few ...
52639 ..................................................................................... it’s a 16 bits wide prime number
3732374119............................................................................. it’s a 32 bits wide prime number
9431969683375056557 ......................................................... it’s a 64 bits wide prime number
234162490561541866917184511796682725377 .................. it’s a 128 bits wide prime number
Increasing the availability of the computing power, we must continually increase the number of encryption bits (few years ago it
was 128 and now has growth up to 1024).
It’s not our goal to explore these issues but what above reported is just enough to understand the importance of finding a
primality test that uses a deterministic method and fast.
Is it really impossible?

12. The importance of execution time
However, if you still have doubts about the importance that may have the speed in
calculation you could search on any search engine (eg. Google) the following sentences
"execution time primality test“ (without the quotes). A few thousand have been
found? Please give a look.

13. Has everything been said?
Has everything been written?
It's amazing how, in the third millennium, we can still discover
interesting things about prime numbers. An example is the website ...
www.VincS.it
... where is stated and proved a theorem so simple, yet original, that it
could have been written in the year 300 BC by Greek mathematicians.