Generation of primes

In computational number theory, a variety of algorithms make it possible to generate prime numbers efficiently. These are used in various applications, for example hashing, public-key cryptography, and search of prime factors in large numbers.

For relatively small numbers, it is possible to just apply trial division to each successive odd number. Prime sieves are almost always faster.

Prime sieves

A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes, the faster but more complicated sieve of Atkin^[1], and the various wheel sieves^[2] are most common.

A prime sieve works by creating a list of all integers up to a desired limit and progressively removing composite numbers (which it directly generates) until only primes are left. This is the most efficient way to obtain a large range of primes; however, to find individual primes, direct primality tests are more efficient.

Large primes

For the large primes used in cryptography, it is usual to use a modified form of sieving: a randomly chosen range of odd numbers of the desired size is sieved against a number of relatively small odd primes (typically all primes less than 65,000). The remaining candidate primes are tested in random order with a standard probabilistic primality test such as the Baillie-PSW primality test or the Miller-Rabin primality test for probable primes.

Alternatively, a number of techniques exist for efficiently generating provable primes. These include generating prime numbers p for which the prime factorization of p − 1 or p + 1 is known.

Complexity

The sieve of Eratosthenes is generally considered the easiest sieve to implement, but it is not the fastest. It can find all the primes up to N in time O(N), while the sieve of Atkin and most wheel sieves run in sublinear time O(N/log log N). The sieve of Atkin takes space N^1/2+o(1); segmented Eratosthenes' sieve takes slightly less space than O(N^1/2). Sorenson^[3] shows an improvement to the wheel sieve that takes even less space at O(N/((log N)^Llog log N) for any L > 1.

See also

• Formula for primes

References

1. ^ A. Atkin, D.J. Bernstein, Prime sieves using binary quadratic forms, Mathematics of Computation 73 (2004), pp. 1023–1030. [4]
2. ^ Paul Pritchard, "Improved Incremental Prime Number Sieves", Algorithmic Number Theory Symposium 1994, pp. 280–288.
3. Sorenson J. P. (1998). "Trading Time for Space in Prime Number Sieves". Lecture Notes in Computer Science. 1423: 179–195. doi:10.1007/BFb0054861.