Class: SyMath::Poly::Galois

Inherits:
SyMath::Poly show all
Defined in:
lib/symath/poly/galois.rb

Overview

Class representing a galois field, i.e finite field

Instance Attribute Summary collapse

Attributes inherited from SyMath::Poly

#arr, #var

Instance Method Summary collapse

Methods inherited from SyMath::Poly

#%, #*, #**, #+, #-, #-@, #/, #==, #[], #degree, #lc, #sort_factors, #sort_factors_multiple, #strip!, #to_m, #zero?

Constructor Details

#initialize(args) ⇒ Galois

Create gl instance from dup or a gl of the same order.



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# File 'lib/symath/poly/galois.rb', line 9

def initialize(args)
  if args.key?(:dup)
    init_from_dup(args[:dup], args[:p])
    return
  end

  if args.key?(:arr)
    init_from_array(args[:arr], args[:p], args[:var])
    return
  end
  
  raise 'Bad arguments for Poly::Galois constructor'
end

Instance Attribute Details

#p ⇒ Object (readonly)

Returns the value of attribute p.



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# File 'lib/symath/poly/galois.rb', line 6

def p
  @p
end

Instance Method Details

#add(g) ⇒ Object

Sum two polynomials



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# File 'lib/symath/poly/galois.rb', line 447

def add(g)
  ret = @arr.clone

  rd = degree
  gd = g.degree
  
  if gd > rd
    (gd - rd).times { ret.unshift(0) }
    rd += gd - rd
  end

  (0..gd).each do |i|
    ret[rd - i] = (ret[rd - i] + g[gd - i]) % @p
  end
  
  return new_gl(ret).strip!
end

#add_ground(a) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 484

def add_ground(a)
  return add(new_gl([a]))
end

#assert_order(g) ⇒ Object

Assert that two fields have the same order



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# File 'lib/symath/poly/galois.rb', line 54

def assert_order(g)
  if @p != g.p
    raise 'Fields do not have the same order'
  end
end

#ddf_zassenhaus ⇒ Object

Deterministic distinct degree factorization



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# File 'lib/symath/poly/galois.rb', line 251

def ddf_zassenhaus()
  x = gl_x
  
  i = 1
  g = x
  factors = []

  f = self
  b = f.frobenius_monomial_base

  while 2*i <= f.degree
    g = g.frobenius_map(f, b)
    h = f.gcd(g - x)

    if h.arr != [1]
      factors << [h, i]

      f = f / h
      g = g % f
      b = f.frobenius_monomial_base
    end

    i += 1
  end

  if f.arr != [1]
    return factors + [[f, f.degree]]
  else
    return factors
  end
end

#diff ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 396

def diff()
  d = degree
  res = @arr.each_with_index.map { |e, i| e*(d - i) % @p }
  res.pop

  return new_gl(res).strip!
end

#div(g) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 572

def div(g)
  assert_order(g)

  df = degree
  dg = g.degree

  if g.zero?
    raise 'Division by zero'
  elsif df < dg
    return [zero, self.clone]
  end

  inv = invert(g[0])

  h = @arr.clone
  dq = df - dg
  dr = dg - 1

  (0..df).each do |i|
    coeff = h[i]

    a = [0, dg - i].max
    b = [df - i, dr].min
    (a..b).each do |j|
      coeff -= h[i + j - dg]*g[dg - j]
    end

    if i <= dq
      coeff *= inv
    end

    h[i] = coeff % @p
  end

  return [new_gl(h[0..dq]), new_gl(h[dq + 1..-1]).strip!]
end

#edf_zassenhaus(n) ⇒ Object

Equal degree factorization



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# File 'lib/symath/poly/galois.rb', line 291

def edf_zassenhaus(n)
  factors = [self.clone]

  if degree <= n
    return factors
  end

  nn = degree / n

  if @p != 2
    b = frobenius_monomial_base
  end

  while factors.size < nn
    r = random_gl(2*n - 1)
  
    if @p == 2
      h = r
      
      (2**(n*nn - 1)).times do
        r = r.pow_mod(2, self)
        h += r
      end

      g = gcd(h)
    else
      h = r.pow_pnm1d2(n, self, b)
      g = gcd(h - 1)
    end
    
    if g.arr != [1] and g != self
      factors = g.edf_zassenhaus(n) + (self / g).edf_zassenhaus(n)
    end
  end

  return sort_factors(factors)
end

#factor ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 170

def factor()
  (lc, f) = monic

  if f.degree < 1
    return [lc, []]
  end

  factors = []

  f.sqf_list[1].each do |g, n|
    g.factor_sqf[1].each do |h|
      factors << [h, n]
    end
  end

  return [lc, sort_factors_multiple(factors)]
end

#factor_sqf ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 91

def factor_sqf()
  (lc, f) = monic

  if f.degree < 1
    return [lc, zero]
  end

  factors = f.zassenhaus

  return [lc, factors]
end

#frobenius_map(g, b) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 228

def frobenius_map(g, b)
  f = self
  m = g.degree

  if f.degree >= m
    f = rem(g)
  end

  if f.zero?
    return zero
  end
  
  n = f.degree
  sf = new_gl([f[-1]])

  (1..n).each do |i|
    sf += b[i]*f[n - i]
  end

  return sf
end

#frobenius_monomial_base ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 203

def frobenius_monomial_base()
  n = degree

  if n == 0
    return []
  end

  b = []
  b << one

  if @p < n
    (1..n - 1).each do |i|
      mon = b[i - 1].lshift(@p)
      b << mon % self
    end
  elsif n > 1
    b << new_gl([1, 0]).pow_mod(@p, self)
    (2..n - 1).each do |i|
      b << (b[i - 1]*b[1]) % self
    end
  end

  return b
end

#gcd(g) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 404

def gcd(g)
  assert_order(g)

  f = self
  # Euclidian algorithm for gcd      
  while !g.zero?
    (f, g) = [g, f.rem(g)]
  end

  return f.monic[1]
end

#gcdex(g) ⇒ Object

Extended gcd for two polynomials



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# File 'lib/symath/poly/galois.rb', line 340

def gcdex(g)
  assert_order(g)

  if self.zero? and g.zero?
    return [one, zero, zero]
  end

  (p0, r0) = monic
  (p1, r1) = g.monic

  if zero?
    return [zero, new_gl([invert(p1)]), r1]
  end

  if g.zero?
    return [new_gl([invert(p0)]), zero, r0]
  end

  s0, s1 = new_gl([invert(p0)]), zero
  t0, t1 = zero, new_gl([invert(p1)])

  while true
    (qq, rr) = r0.div(r1)

    if rr.zero?
      break
    end

    r0 = r1
    (c, r1) = rr.monic

    inv = invert(c)

    s = s0 - s1*qq
    t = t0 - t1*qq

    s0 = s1
    s1 = s*inv

    t0 = t1
    t1 = t*inv
  end

  return [s1, t1, r1]
end

#gl_x ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 49

def gl_x()
  new_gl([1, 0])
end

#init_from_array(arr, p, var) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 29

def init_from_array(arr, p, var)
  @arr = arr
  @p = p
  @var = var
end

#init_from_dup(dup, p) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 23

def init_from_dup(dup, p)
  @arr = dup.arr.map { |t| t % p }
  @p = p
  @var = dup.var
end

#invert(a) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 72

def invert(a)
  p = @p
  raise "No inverse - #{a} and #{p} not coprime" unless a.gcd(p) == 1

  return p if p == 1

  m0, inv, x0 = p, 1, 0

  while a > 1
    inv -= (a / p) * x0
    a, p = p, a % p
    inv, x0 = x0, inv
  end
  
  inv += m0 if inv < 0

  return inv
end

#lshift(n) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 609

def lshift(n)
  if zero?
    return zero
  else
    return new_gl(@arr + [0]*n)
  end
end

#monic ⇒ Object

Divide coefficients by lc



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# File 'lib/symath/poly/galois.rb', line 387

def monic()
  if zero?
    return self.clone
  end

  c = lc
  return [c, mul_ground(invert(c))]
end

#mul(g) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 497

def mul(g)
  df = degree
  dg = g.degree

  dh = df + dg
  if dh > 0
    h = [0]*(dh + 1)
  else
    h = []
  end

  (0..dh).each do |i|
    coeff = 0

    a = [0, i - dg].max
    b = [i, df].min
    (a..b).each do |j|
      coeff += self[j]*g[i - j]
    end

    h[i] = coeff % p
  end

  ret = new_gl(h)
  ret.strip!
  return ret
end

#mul_ground(a) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 525

def mul_ground(a)
  if a == 0
    return zero
  else
    return new_gl(@arr.map { |e| a*e % @p})
  end
end

#neg ⇒ Object

Return the negative of the polynomial



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# File 'lib/symath/poly/galois.rb', line 493

def neg()
  return new_dup(@arr.map { |t| -t % @p })
end

#new_gl(arr) ⇒ Object

Convenience method for creating a new gl of the same order and the same variable



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# File 'lib/symath/poly/galois.rb', line 37

def new_gl(arr)
  return SyMath::Poly::Galois.new({ :arr => arr, :p => @p, :var => @var })
end

#one ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 45

def one()
  return new_gl([1])
end

#pow_mod(n, g) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 416

def pow_mod(n, g)
  if n == 0
    return one
  elsif n == 1
    return self % g
  elsif n == 2
    return self**2 % g
  end

  f = self
  h = one

  while true
    if n.odd?
      h = (h*f) % g
      n -= 1
    end

    n >>= 1

    if n == 0
      break
    end

    f = f**2 % g
  end

  return h
end

#pow_pnm1d2(n, g, b) ⇒ Object

Compute f**(p**n - 1) / 2 in GF(p)/(g) (Utility function for edf_zassenhaus)



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# File 'lib/symath/poly/galois.rb', line 190

def pow_pnm1d2(n, g, b)
  f = rem(g)
  h = f
  r = f

  (n - 1).times do
    h = h.frobenius_map(g, b)
    r = (r*h) % g
  end

  return r.pow_mod((@p - 1)/2, g)
end

#quo(g) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 564

def quo(g)
  return div(g)[0]
end

#random_gl(n) ⇒ Object

Generate random polynomial of degree n



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# File 'lib/symath/poly/galois.rb', line 284

def random_gl(n)
  ret = [1]
  (n).times { ret << rand(@p) }
  return new_gl(ret)
end

#rem(f) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 568

def rem(f)
  return div(f)[1]
end

#sqf_list ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 114

def sqf_list()
  n = 1
  sqf = false
  factors = []
  r = @p

  (lc, f) = monic

  if degree < 1
    return [lc, []]
  end

  f = self
  
  while true
    ff = f.diff

    if !ff.zero?
      g = f.gcd(ff)
      h = f / g
    
      i = 1
    
      while h.arr != [1]
        gg = g.gcd(h)
        hh = h / gg

        if hh.degree > 0
          factors << [hh, i*n]
        end

        g = g / gg
        h = gg
        i += 1
      end

      if g.arr == [1]
        sqf = true
      else
        f = g
      end
    end

    if !sqf
      d = f.degree/r

      f = new_gl((0..d).map { |i| f[i*r] })
      n = n*r
    else
      break
    end
  end

  return [lc, factors]
end

#sqf_p ⇒ Object

Return true if polynomial is square free



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# File 'lib/symath/poly/galois.rb', line 104

def sqf_p()
  (lc, f) = monic

  if f.zero?
    return true
  else
    return f.gcd(f.diff).arr == [1]
  end
end

#sqr ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 533

def sqr()
  d = degree
  dh = 2*d
  h = []

  (0..dh).each do |i|
    coeff = 0

    jmin = [0, i - d].max
    jmax = [i, d].min

    n = jmax - jmin + 1
    jmax = jmin + n/2 - 1

    (jmin..jmax).each do |j|
      coeff += self[j]*self[i - j]
    end

    coeff += coeff

    if n.odd?
      elem = self[jmax + 1]
      coeff += elem**2
    end

    h << coeff % @p
  end

  return new_gl(h).strip!
end

#sub(g) ⇒ Object

Subtract a polynomial from this one



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# File 'lib/symath/poly/galois.rb', line 466

def sub(g)
  ret = @arr.clone

  rd = degree
  gd = g.degree

  if gd > rd
    (gd - rd).times { ret.unshift(0) }
    rd += gd - rd
  end

  (0..gd).each do |i|
    ret[rd - i] = (ret[rd - i] - g[gd - i]) % @p
  end
  
  return new_gl(ret).strip!
end

#sub_ground(a) ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 488

def sub_ground(a)
  return sub(new_gl([a]))
end

#to_dup ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 60

def to_dup()
  ret = @arr.map do |e|
    if e <= @p / 2
      e
    else
      e - @p
    end
  end
  
  return SyMath::Poly::DUP.new({ :arr => ret, :var => @var })
end

#to_s ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 617

def to_s()
  return @arr.to_s + '/' + @p.to_s
end

#zassenhaus ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 329

def zassenhaus()
  factors = []

  ddf_zassenhaus.each do |f|
    factors += f[0].edf_zassenhaus(f[1])
  end
  
  return sort_factors(factors)
end

#zero ⇒ Object 



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# File 'lib/symath/poly/galois.rb', line 41

def zero()
  return new_gl([])
end