Class: FasterPrime::PrimalityTest::APRCL

Inherits:
Object
  • Object
show all
Defined in:
lib/faster_prime/primality_test.rb

Overview

APRCL Primarity Test

Henry Cohen. A course in computational algebraic number theory. 9.1 The Jacobi Sum Test

Defined Under Namespace

Classes: Composite, Failed, JacobiSumTableClass, ZZeta

Class Method Summary collapse

Instance Method Summary collapse

Constructor Details

#initialize(n, jacobi_sum_table: DefaultJacobiSumTable, t: nil) ⇒ APRCL

Creates an APRCL instance to test an integer that is less than bound



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# File 'lib/faster_prime/primality_test.rb', line 107

def initialize(n, jacobi_sum_table: DefaultJacobiSumTable, t: nil)
  # an integer in question
  @n = n

  # precompute n-1 and (n-1)/2 because n is normally big
  @n1 = n - 1
  @n1_2 = @n1 / 2

  # @eta[pk]: a set of p^k-th roots of unity
  @eta = {}

  # a cache of Jacobi sums
  @js = jacobi_sum_table

  # compute e(t)
  if t
    raise "t must be even" if t.odd?
    @t = t
    @et = compute_e(@t)
    raise "t is too small to test n" if @n >= @et ** 2
  else
    @t = find_t(@n)
    @et = compute_e(@t)
  end
end

Class Method Details

.prime?(n) ⇒ Boolean

Returns true if n is prime, false otherwise.

Returns:



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# File 'lib/faster_prime/primality_test.rb', line 102

def self.prime?(n)
  new(n).prime?
end

Instance Method Details

#prime? ⇒ Boolean

Algorithm 9.1.28 (Jacobi Sum Primality test)

Returns:



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# File 'lib/faster_prime/primality_test.rb', line 134

def prime?
  return false if @n <= 1

  # 1. [Check GCD]
  g = @n.gcd(@t * @et)
  return g == @n && PrimalityTest.prime?(g) if g != 1

  # 2. [Initialize]
  lp = {}
  PrimeFactorization.prime_factorization(@t) do |p, |
    lp[p] = false if p == 2 || Utils.mod_pow(@n, p - 1, p ** 2) == 1
  end

  # 3. [Loop on characters]
  Utils.each_divisor(@t) do |d|
    q = d + 1
    next unless PrimalityTest.prime?(q)
    PrimeFactorization.prime_factorization(d) do |p, k|
      # 4. [Check (*beta)]
      lp.delete(p) if step4(q, p, k)
    end
  end

  # 5. [Check conditions Lp]
  lp.keys.each {|p| step5(p) }

  # 6. [Final trial division]
  r = 1
  1.upto(@t - 1) do
    r = r * @n % @et
    return false if @n % r == 0 && 1 < r && r < @n
  end

  return true

rescue Composite
  return false
end