In this post we're going to look at several methods for generating various kinds of pseudoprimes and we're also listing some open-problems that motivate the generation of such numbers.
In this post we're going to look at some algorithms for checking if a given number is either prime or composite. Édouard Lucas , Leonhard Euler , Carl Pomerance and Pierre de Fermat Prime numbers play an important role in public-key cryptography (e.g. RSA algorithm ) which require fast generation of large random prime numbers that have around `600` digits (or more). The simplest way to generate a prime number, is to pick a random odd integer and check its primality, repeating this process until a prime is found. This approach requires a fast algorithm for checking the primality of a number. In 2002, Manindra Agrawal , Neeraj Kayal , and Nitin Saxena published the AKS primality test , unconditionally proving that the primality of a number can be checked in polynomial time, with an asymptotic time complexity of `Õ(log(n)^12)`. Later on, in 2005, Carl Pomerance and Hendrik Lenstra improved the complexity of the AKS algorithm to run in just `Õ(log(n)^6)` steps.
The Fibonacci sequence is, without doubt, one of the most popular sequences in mathematics and in popular culture, named after Italian mathematician Leonardo of Pisa (also known as Fibonacci , Leonardo Bonacci, Leonardo of Pisa, Leonardo Pisano Bigollo, or Leonardo Fibonacci), who first introduced the numbers in Western European with his book Liber Abaci , in 1202.
RSA is a practical public-key cryptographic algorithm, which is widely used on modern computers to communicate securely over large distances. The acronym of the algorithm stands for Ron Rivest , Adi Shamir and Leonard Adleman , which first published the algorithm in 1978. # Algorithm overview Choose `p` and `q` as distinct prime numbers Compute `n` as `n = p*q` Compute `\phi(n)` as `\phi(n) = (p-1) * (q-1)` Choose `e` such that `1 < e < \phi(n)` and `e` and `\phi(n)` are coprime Compute the value of `d` as `d ≡ e^(-1) mod \phi(n)` Public key is `(e, n)` Private key is `(d, n)` The encryption of `m` as `c`, is `c ≡ m^e mod n` The decryption of `c` as `m`, is `m ≡ c^d mod n` # Generating `p` and `q` In order to generate a public and a private key, the algorithm requires two distinct prime numbers `p` and `q`, which are randomly chosen and should have, roughly, the same number of bits. By today standards, it is recommended that each prime number to have a