class Crypto::Prime

• Crypto::Prime
• Reference
• Object

Overview

The set of all prime numbers. Adapted from the Ruby implementation at https://github.com/ruby/prime.

Example

Prime.each(100) do |prime|
  p prime # => 2, 3, 5, 7, 11, ...., 97
end

Prime is Enumerable:

Prime.first 5 # => [2, 3, 5, 7, 11]

Generators

A "generator" provides an implementation of enumerating pseudo-prime numbers and it remembers the position of enumeration and upper bound. Furthermore, it is an external iterator of prime enumeration which is compatible with an Enumerator.

Prime::PseudoPrimeGenerator is the base class for generators. There are few implementations of generator.

[Prime::EratosthenesGenerator] Uses Eratosthenes' sieve. [Prime::TrialDivisionGenerator] Uses the trial division method. [Prime::Generator23] Generates all positive integers which are not divisible by either 2 or 3. This sequence is very bad as a pseudo-prime sequence. But this is faster and uses much less memory than the other generators. So, it is suitable for factorizing an integer which is not large but has many prime factors. e.g. for Prime#prime?.

Included Modules

• Enumerable(Int128 | Int16 | Int32 | Int64 | Int8)

Defined in:

Class Method Summary

• .coprime?(a, b)

  Returns true if the integer a is coprime (relatively prime) to integer b.

• .each(ubound = nil, generator = EratosthenesGenerator.new, &block : Int::Signed -> )

  Iterates the given block over all prime numbers.

• .factorize(value, generator = Generator23.new)

  Returns the factorization of value.

• .int_from_factorization(pd : Indexable(Indexable(Int)))

  Re-composes a prime factorization and returns the product.

• .prime?(n, k = 10)

  Returns true if value is a prime number, else returns false.

• .random(start, stop, count, generator = Generator23.new, random = Random::DEFAULT)

  Return count random primes in the given range.

• .random(bits, random = Random::DEFAULT)

Instance Method Summary

• #each(ubound = nil, generator = EratosthenesGenerator.new, &block : Int::Signed -> )

  see .each

• #factorize(value, generator = Generator23.new)

  see .factorize

• #int_from_factorization(pd : Indexable(Indexable(Int)))

  see .int_from_factorization

• #prime?(n, k = 10)

  see .prime?

• #random(start, stop, count, generator = Generator23.new, random = Random::DEFAULT)

  see .random

• #random(bits, random = Random::DEFAULT)

Class Method Detail

RSA::Math.coprime?(6, 35) # => true
RSA::Math.coprime?(6, 27) # => false

p_1**e_1 * p_2**e_2 * ... * p_n**e_n,
``

factorize returns an array of pairs of integers:

Each pair consists of a prime number -- a prime factor --
and a natural number -- its exponent (multiplicity).

## Parameters
`value`: An arbitrary integer.
`generator`: Optional. A pseudo-prime generator.
             `generator`.succ must return the next
             pseudo-prime number in ascending order.
             It must generate all prime numbers,
             but may also generate non-prime numbers, too.

### Exceptions
`DivisionByZeroError`: when `value` is zero.

## Example

  [[p_1, e_1], [p_2, e_2], ..., [p_n, e_n]],

  p_1**e_1 * p_2**e_2 * ... * p_n**e_n.

Prime.int_from_factorization([[3, 2], [5, 1]]) # => 45
3**2 * 5                                       # => 45

Instance Method Detail