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a and b have the same remainders when divided by c
Cryptography
Tools that underline most modern security protocols
If p is prime, then for any integer n coprime with n,
Used by Egyptians 4000 years ago
Sieves Primality Test
Played a crucial role in both world wars
Two main stages : Classic era and Computer era
-It is the fastest method to use for a 100% accuracy in finding prime numbers.
-It's main Drawback is when computing prime numbers above 10 million. Since it takes unreasonably long time.
- Sieves method is quicker and more efficient than the Trial method
- It has a much lower Complexity than the Trial method
-Most efficient method for small primes (below 10 million)
Example: User passwords databases in websites
1-Confidentiality: To prevent unauthorized acces to data
Symmetric key cryptography
Example: Online banking
2-Data Integrity: To keep unauthorized parties from manipulating data
3-Authentication: 1-Entity(user) authentication 2-Data origin authentication
Public key cryptography
Example: Purchase orders for big companies
4-Non-repudiation: Prevents an entity from denying previous commitments
And we iterate the process
Analysis of an algorithm
How efficient is an algorithm or piece of code?
Probalistic primality test
The probability to be wrong stating p is prime is 0.25^k, k being the number of iteration.
Complexity of Miller-Rabin method in function of the input (n) the number of iterations (x)
Prime numbers and Cryptography