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finitefield.py
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1 """
2 finite fields.
3 """
4 
5 from __future__ import division
6 import logging
7 import nzmath.gcd as gcd
8 import nzmath.bigrandom as bigrandom
9 import nzmath.bigrange as bigrange
10 import nzmath.arith1 as arith1
11 import nzmath.prime as prime
12 import nzmath.ring as ring
13 import nzmath.rational as rational
14 import nzmath.factor.misc as factor_misc
15 import nzmath.intresidue as intresidue
16 import nzmath.poly.uniutil as uniutil
17 import nzmath.matrix as matrix
18 import nzmath.vector as vector
20 
21 _log = logging.getLogger('nzmath.finitefield')
22 
23 
25  """
26  The base class for all finite field element.
27  """
28  def __init__(self):
29  # This class is abstract and can not be instantiated.
30  if type(self) is FiniteFieldElement:
31  raise NotImplementedError("the class FiniteFieldElement is abstract")
32  ring.FieldElement.__init__(self)
33 
34 
36  """
37  The base class for all finite fields.
38  """
39  def __init__(self, characteristic):
40  # This class is abstract and can not be instantiated.
41  if type(self) is FiniteField:
42  raise NotImplementedError("the class FiniteField is abstract")
43  ring.Field.__init__(self)
44  self.char = characteristic
45  self._orderfactor = None # postpone the initialization
46 
47  def card(self):
48  "Cardinality of the field"
49  raise NotImplementedError("card have to be overridden")
50 
51  def getCharacteristic(self):
52  """
53  Return the characteristic of the field.
54  """
55  return self.char
56 
57  def order(self, elem):
58  """
59  Find and return the order of the element in the multiplicative
60  group of the field.
61  """
62  if not elem:
63  raise ValueError("zero is not in the multiplicative group.")
64  if self._orderfactor is None:
65  self._orderfactor = factor_misc.FactoredInteger(card(self) - 1)
66  o = 1
67  for p, e in self._orderfactor:
68  b = elem ** (int(self._orderfactor) // (p**e))
69  while b != self.one:
70  o = o * p
71  b = b ** p
72  return o
73 
74  def random_element(self, *args):
75  """
76  Return a randomly chosen element og the field.
77  """
78  return self.createElement(bigrandom.randrange(*args))
79 
80  def primitive_element(self):
81  """
82  Return a primitive element of the field, i.e., a generator of
83  the multiplicative group.
84  """
85  raise NotImplementedError("primitive_element should be overridden")
86 
87  def Legendre(self, element):
88  """ Return generalize Legendre Symbol for FiniteField.
89  """
90  if not element:
91  return 0
92 
93  if element == self.one or self.char == 2:
94  return 1 # element of cyclic group order 2**n-1 also always 1
95 
96  # generic method:successive squaring
97  # generalized Legendre symbol definition:
98  # (self/_ring) := self ** ((card(_ring)-1)/2)
99  x = element ** ((card(self)-1) >> 1)
100  if x == self.one:
101  return 1
102  elif x == -self.one:
103  return -1
104  raise ValueError("element must be not in field")
105 
106  def TonelliShanks(self, element):
107  """ Return square root of element if exist.
108  assume that characteristic have to be more than three.
109  """
110  if self.char == 2:
111  return self.sqrt(element) # should be error
112 
113  if self.Legendre(element) == -1:
114  raise ValueError("There is no solution")
115 
116  # symbol and code reference from Cohen, CCANT 1.5.1
117  (e, q) = arith1.vp(card(self)-1, 2)
118 
119  a = element
120  n = self.createElement(self.char+1)
121  while self.Legendre(n) != -1:
122  n = self.random_element(2, card(self)) # field maybe large
123  y = z = n ** q
124  r = e
125  x = a ** ((q-1) >> 1)
126  b = a * (x ** 2)
127  x = a * x
128  while True:
129  if b == self.one:
130  return x
131  m = 1
132  while m < r:
133  if b ** (2 ** m) == self.one:
134  break
135  m = m+1
136  if m == r:
137  break
138  t = y ** (2 ** (r-m-1))
139  y = t ** 2
140  r = m
141  x = x * t
142  b = b * y
143  raise ValueError("There is no solution")
144 
145  def sqrt(self, element):
146  """ Return square root if exist.
147  """
148  if not element or element == self.one:
149  return element # trivial case
150 
151  # element of characteristic 2 always exist square root
152  if self.char == 2:
153  return element ** ((card(self)) >> 1)
154 
155  # otherwise,
156  return self.TonelliShanks(element)
157 
158 
160  """
161  The class for finite prime field element.
162  """
163  def __init__(self, representative, modulus, modulus_is_prime=True):
164  if not modulus_is_prime and not prime.primeq(abs(modulus)):
165  raise ValueError("modulus must be a prime.")
166 
167  FiniteFieldElement.__init__(self)
168  intresidue.IntegerResidueClass.__init__(self, representative, modulus)
169 
170  # ring
171  self.ring = None
172 
173  def __repr__(self):
174  return "FinitePrimeFieldElement(%d, %d)" % (self.n, self.m)
175 
176  def __str__(self):
177  return "%d in F_%d" % (self.n, self.m)
178 
179  def getRing(self):
180  """
181  Return the finite prime field to which the element belongs.
182  """
183  if self.ring is None:
184  self.ring = FinitePrimeField.getInstance(self.m)
185  return self.ring
186 
187  def order(self):
188  """
189  Find and return the order of the element in the multiplicative
190  group of F_p.
191  """
192  if self.n == 0:
193  raise ValueError("zero is not in the group.")
194  if not hasattr(self, "orderfactor"):
195  self.orderfactor = factor_misc.FactoredInteger(self.m - 1)
196  o = 1
197  for p, e in self.orderfactor:
198  b = self ** (int(self.orderfactor) // (p**e))
199  while b.n != 1:
200  o = o*p
201  b = b**p
202  return o
203 
204 
206  """
207  FinitePrimeField is also known as F_p or GF(p).
208  """
209 
210  # class variable
211  _instances = {}
212 
213  def __init__(self, characteristic):
214  FiniteField.__init__(self, characteristic)
215  self.registerModuleAction(rational.theIntegerRing, self._int_times)
216  # mathematically Q_p = Q \ {r/s; gcd(r, s) == 1, gcd(s, p) > 1}
217  # is more appropriate.
218  self.registerModuleAction(rational.theRationalField, self._rat_times)
219 
220  @staticmethod
221  def _int_times(integer, fpelem):
222  """
223  Return k * FinitePrimeFieldElement(n, p)
224  """
225  return FinitePrimeFieldElement(integer * fpelem.n, fpelem.m)
226 
227  @staticmethod
228  def _rat_times(rat, fpelem):
229  """
230  Return Rational(a, b) * FinitePrimeFieldElement(n, p)
231  """
232  return FinitePrimeFieldElement(rat * fpelem.n, fpelem.m)
233 
234  def __eq__(self, other):
235  if self is other:
236  return True
237  if isinstance(other, FinitePrimeField):
238  return self.char == other.char
239  return False
240 
241  def __hash__(self):
242  return self.char
243 
244  def __ne__(self, other):
245  return not (self == other)
246 
247  def __str__(self):
248  return "F_%d" % self.char
249 
250  def __repr__(self):
251  return "%s(%d)" % (self.__class__.__name__, self.char)
252 
253  def __hash__(self):
254  return self.char & 0xFFFFFFFF
255 
256  def issubring(self, other):
257  """
258  Report whether another ring contains the field as a subring.
259  """
260  if self == other:
261  return True
262  if isinstance(other, FiniteField) and other.getCharacteristic() == self.char:
263  return True
264  try:
265  return other.issuperring(self)
266  except:
267  return False
268 
269  def issuperring(self, other):
270  """
271  Report whether the field is a superring of another ring.
272  Since the field is a prime field, it can be a superring of
273  itself only.
274  """
275  return self == other
276 
277  def __contains__(self, elem):
278  if isinstance(elem, FinitePrimeFieldElement) and elem.getModulus() == self.char:
279  return True
280  return False
281 
282  def createElement(self, seed):
283  """
284  Create an element of the field.
285 
286  'seed' should be an integer.
287  """
288  return FinitePrimeFieldElement(seed, self.char, modulus_is_prime=True)
289 
290  def primitive_element(self):
291  """
292  Return a primitive element of the field, i.e., a generator of
293  the multiplicative group.
294  """
295  if self.char == 2:
296  return self.one
297  fullorder = card(self) - 1
298  if self._orderfactor is None:
299  self._orderfactor = factor_misc.FactoredInteger(fullorder)
300  for i in bigrange.range(2, self.char):
301  g = self.createElement(i)
302  for p in self._orderfactor.prime_divisors():
303  if g ** (fullorder // p) == self.one:
304  break
305  else:
306  return g
307 
308  def Legendre(self, element):
309  """ Return generalize Legendre Symbol for FinitePrimeField.
310  """
311  if not element:
312  return 0
313  if element.n == 1 or element.m == 2:
314  return 1 # trivial
315 
316  return arith1.legendre(element.n, element.m)
317 
318  def SquareRoot(self, element):
319  """ Return square root if exist.
320  """
321  if not element or element.n == 1:
322  return element # trivial case
323  if element.m == 2:
324  return element.getRing().one
325  return arith1.modsqrt(element.n, element.m)
326 
327  def card(self):
328  "Cardinality of the field"
329  return self.char
330 
331  # properties
332  def _getOne(self):
333  "getter for one"
334  if self._one is None:
336  return self._one
337 
338  one = property(_getOne, None, None, "multiplicative unit.")
339 
340  def _getZero(self):
341  "getter for zero"
342  if self._zero is None:
344  return self._zero
345 
346  zero = property(_getZero, None, None, "additive unit.")
347 
348  @classmethod
349  def getInstance(cls, characteristic):
350  """
351  Return an instance of the class with specified characteristic.
352  """
353  if characteristic not in cls._instances:
354  cls._instances[characteristic] = cls(characteristic)
355  return cls._instances[characteristic]
356 
357 
358 FinitePrimeFieldPolynomial = uniutil.FinitePrimeFieldPolynomial
359 uniutil.special_ring_table[FinitePrimeField] = FinitePrimeFieldPolynomial
360 
361 
363  """
364  ExtendedFieldElement is a class for an element of F_q.
365  """
366  def __init__(self, representative, field):
367  """
368  FiniteExtendedFieldElement(representative, field) creates
369  an element of the finite extended field F_q.
370 
371  The argument representative has to be a polynomial with
372  base field coefficients, i.e., if F_q is over F_{p^k}
373  the representative has to F_{p^k} polynomial.
374 
375  Another argument field mut be an instance of
376  ExtendedField.
377  """
378  if isinstance(field, ExtendedField):
379  self.field = field
380  else:
381  raise TypeError("wrong type argument for field.")
382  if (isinstance(representative, uniutil.FiniteFieldPolynomial) and
383  representative.getCoefficientRing() == self.field.basefield):
384  self.rep = self.field.modulus.mod(representative)
385  else:
386  _log.error(representative.__class__.__name__)
387  raise TypeError("wrong type argument for representative.")
388 
389  def getRing(self):
390  """
391  Return the field to which the element belongs.
392  """
393  return self.field
394 
395  def _op(self, other, op):
396  """
397  Do `self (op) other'.
398  op must be a name of the special method for binary operation.
399  """
400  if isinstance(other, ExtendedFieldElement):
401  if other.field is self.field:
402  result = self.field.modulus.mod(getattr(self.rep, op)(other.rep))
403  return self.__class__(result, self.field)
404  else:
405  fq1, fq2 = self.field, other.field
406  emb1, emb2 = double_embeddings(fq1, fq2)
407  return getattr(emb1(self), op)(emb2(other))
408  if self.field.hasaction(ring.getRing(other)):
409  # cases for action ring elements
410  embedded = self.field.getaction(ring.getRing(other))(other, self.field.one)
411  result = self.field.modulus.mod(getattr(self.rep, op)(embedded.rep))
412  return self.__class__(result, self.field)
413  else:
414  return NotImplemented
415 
416  def __add__(self, other):
417  """
418  self + other
419 
420  other can be an element of either F_q, F_p or Z.
421  """
422  if other is 0 or other is self.field.zero:
423  return self
424  return self._op(other, "__add__")
425 
426  __radd__ = __add__
427 
428  def __sub__(self, other):
429  """
430  self - other
431 
432  other can be an element of either F_q, F_p or Z.
433  """
434  if other is 0 or other is self.field.zero:
435  return self
436  return self._op(other, "__sub__")
437 
438  def __rsub__(self, other):
439  """
440  other - self
441 
442  other can be an element of either F_q, F_p or Z.
443  """
444  return self._op(other, "__rsub__")
445 
446  def __mul__(self, other):
447  """
448  self * other
449 
450  other can be an element of either F_q, F_p or Z.
451  """
452  if other is 1 or other is self.field.one:
453  return self
454  return self._op(other, "__mul__")
455 
456  __rmul__ = __mul__
457 
458  def __truediv__(self, other):
459  return self * other.inverse()
460 
461  __div__ = __truediv__
462 
463  def inverse(self):
464  """
465  Return the inverse of the element.
466  """
467  if not self:
468  raise ZeroDivisionError("There is no inverse of zero.")
469  return self.__class__(self.rep.extgcd(self.field.modulus)[0], self.field)
470 
471  def __pow__(self, index):
472  """
473  self ** index
474 
475  pow() with three arguments is not supported.
476  """
477  if not self:
478  return self # trivial
479  if index < 0:
480  index %= self.field.getCharacteristic()
481  power = pow(self.rep, index, self.field.modulus)
482  return self.__class__(power, self.field)
483 
484  def __neg__(self):
485  return self.field.zero - self
486 
487  def __pos__(self):
488  return self
489 
490  def __eq__(self, other):
491  try:
492  if self.field == other.field:
493  if self.rep == other.rep:
494  return True
495  except AttributeError:
496  pass
497  return False
498 
499  def __hash__(self):
500  return hash(self.rep)
501 
502  def __ne__(self, other):
503  return not (self == other)
504 
505  def __nonzero__(self):
506  return bool(self.rep)
507 
508  def __repr__(self):
509  return "%s(%s, %s)" % (self.__class__.__name__, repr(self.rep), repr(self.field))
510 
511  def trace(self):
512  """
513  Return the absolute trace.
514  """
515  p = self.field.char
516  q = p
517  alpha = self
518  tr = alpha.rep[0]
519  while q < card(self.field):
520  q *= p
521  alpha **= p
522  tr += alpha.rep[0]
523  if tr not in FinitePrimeField.getInstance(p):
524  tr = FinitePrimeField.getInstance(p).createElement(tr)
525  return tr
526 
527  def norm(self):
528  """
529  Return the absolute norm.
530  """
531  p = self.field.char
532  nrm = (self ** ((card(self.field) - 1) // (p - 1))).rep[0]
533  if nrm not in FinitePrimeField.getInstance(p):
534  nrm = FinitePrimeField.getInstance(p).createElement(nrm)
535  return nrm
536 
537 
539  """
540  ExtendedField is a class for finite field, whose cardinality
541  q = p**n with a prime p and n>1. It is usually called F_q or GF(q).
542  """
543  def __init__(self, basefield, modulus):
544  """
545  ExtendedField(basefield, modulus)
546 
547  Create a field extension basefield[X]/(modulus(X)).
548 
549  The modulus has to be an irreducible polynomial with
550  coefficients in the basefield.
551  """
552  FiniteField.__init__(self, basefield.char)
553  self.basefield = basefield
554  self.modulus = modulus
555  if isinstance(self.basefield, FinitePrimeField):
556  self.degree = self.modulus.degree()
557  else:
558  self.degree = self.basefield.degree * self.modulus.degree()
559  bf = self.basefield
560  while hasattr(bf, "basefield"):
561  # all basefields can be used as a scalar field.
562  self.registerModuleAction(bf.basefield, self._scalar_mul)
563  bf = bf.basefield
565  # integer is always a good scalar
566  self.registerModuleAction(rational.theIntegerRing, self._scalar_mul)
567 
568  @classmethod
569  def prime_extension(cls, characteristic, n_or_modulus):
570  """
571  ExtendedField.prime_extension(p, n_or_modulus) creates a
572  finite field extended over prime field.
573 
574  characteristic must be prime. n_or_modulus can be:
575  1) an integer greater than 1, or
576  2) a polynomial in a polynomial ring of F_p with degree
577  greater than 1.
578  """
579  if isinstance(n_or_modulus, (int, long)):
580  n = n_or_modulus
581  if n <= 1:
582  raise ValueError("degree of extension must be > 1.")
583  # choose a method among three variants:
584  #modulus = cls._random_irriducible(characteristic, n)
585  #modulus = cls._small_irriducible(characteristic, n)
586  modulus = cls._primitive_polynomial(characteristic, n)
587  elif isinstance(n_or_modulus, FinitePrimeFieldPolynomial):
588  modulus = n_or_modulus
589  if not isinstance(modulus.getCoefficientRing(), FinitePrimeField):
590  raise TypeError("modulus must be F_p polynomial.")
591  if modulus.degree() <= 1 or not modulus.isirreducible():
592  raise ValueError("modulus must be of degree greater than 1.")
593  else:
594  raise TypeError("degree or modulus must be supplied.")
595  return cls(FinitePrimeField.getInstance(characteristic), modulus)
596 
597  @staticmethod
598  def _random_irriducible(char, degree):
599  """
600  Return randomly chosen irreducible polynomial of self.degree.
601  """
602  cardinality = char ** degree
603  basefield = FinitePrimeField.getInstance(char)
604  seed = bigrandom.randrange(1, char) + cardinality
605  cand = uniutil.FiniteFieldPolynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
606  while cand.degree() < degree or not cand.isirreducible():
607  seed = bigrandom.randrange(1, cardinality) + cardinality
608  cand = uniutil.FiniteFieldPolynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
609  _log.debug(cand.order.format(cand))
610  return cand
611 
612  @staticmethod
613  def _small_irriducible(char, degree):
614  """
615  Return an irreducible polynomial of self.degree with a small
616  number of non-zero coefficients.
617  """
618  cardinality = char ** degree
619  basefield = FinitePrimeField.getInstance(char)
620  top = uniutil.polynomial({degree: 1}, coeffring=basefield)
621  for seed in range(degree - 1):
622  for const in range(1, char):
623  coeffs = [const] + arith1.expand(seed, 2)
624  cand = uniutil.polynomial(enumerate(coeffs), coeffring=basefield) + top
625  if cand.isirreducible():
626  _log.debug(cand.order.format(cand))
627  return cand
628  for subdeg in range(degree):
629  subseedbound = char ** subdeg
630  for subseed in range(subseedbound + 1, char * subseedbound):
631  if not subseed % char:
632  continue
633  seed = subseed + cardinality
634  cand = uniutil.polynomial(enumerate(arith1.expand(seed, cardinality)), coeffring=basefield)
635  if cand.isirreducible():
636  return cand
637 
638  @staticmethod
639  def _primitive_polynomial(char, degree):
640  """
641  Return a primitive polynomial of self.degree.
642 
643  REF: Lidl & Niederreiter, Introduction to finite fields and
644  their applications.
645  """
646  cardinality = char ** degree
647  basefield = FinitePrimeField.getInstance(char)
648  const = basefield.primitive_element()
649  if degree & 1:
650  const = -const
651  cand = uniutil.polynomial({0:const, degree:basefield.one}, basefield)
652  maxorder = factor_misc.FactoredInteger((cardinality - 1) // (char - 1))
653  var = uniutil.polynomial({1:basefield.one}, basefield)
654  while not (cand.isirreducible() and
655  all(pow(var, int(maxorder) // p, cand).degree() > 0 for p in maxorder.prime_divisors())):
656  # randomly modify the polynomial
657  deg = bigrandom.randrange(1, degree)
658  coeff = basefield.random_element(1, char)
659  cand += uniutil.polynomial({deg:coeff}, basefield)
660  _log.debug(cand.order.format(cand))
661  return cand
662 
663  @staticmethod
664  def _scalar_mul(integer, fqelem):
665  """
666  Return integer * (Fq element).
667  """
668  return fqelem.__class__(fqelem.rep * integer, fqelem.field)
669 
670  def card(self):
671  """
672  Return the cardinality of the field
673  """
674  return self.char ** self.degree
675 
676  def createElement(self, seed):
677  """
678  Create an element of the field.
679  """
680  if isinstance(seed, (int, long)):
681  expansion = arith1.expand(seed, card(self.basefield))
682  return ExtendedFieldElement(
683  uniutil.FiniteFieldPolynomial(enumerate(expansion), self.basefield),
684  self)
685  elif seed in self.basefield:
686  return ExtendedFieldElement(
687  uniutil.FiniteFieldPolynomial([(0, seed)], self.basefield),
688  self)
689  elif seed in self:
690  # seed is in self, return only embedding
691  return self.zero + seed
692  elif (isinstance(seed, uniutil.FiniteFieldPolynomial) and
693  seed.getCoefficientRing() is self.basefield):
694  return ExtendedFieldElement(seed, self)
695  else:
696  try:
697  # lastly check sequence
698  return ExtendedFieldElement(
699  uniutil.FiniteFieldPolynomial(enumerate(seed), self.basefield),
700  self)
701  except TypeError:
702  raise TypeError("seed %s is not an appropriate object." % str(seed))
703 
704  def __repr__(self):
705  return "%s(%d, %d)" % (self.__class__.__name__, self.char, self.degree)
706 
707  def __str__(self):
708  return "F_%d @(%s)" % (card(self), str(self.modulus))
709 
710  def __hash__(self):
711  return (self.char ** self.degree) & 0xFFFFFFFF
712 
713  def issuperring(self, other):
714  """
715  Report whether the field is a superring of another ring.
716  """
717  if self is other:
718  return True
719  elif isinstance(other, ExtendedField):
720  if self.char == other.char and not (self.degree % other.degree):
721  return True
722  return False
723  elif isinstance(other, FinitePrimeField):
724  if self.char == other.getCharacteristic():
725  return True
726  return False
727  try:
728  return other.issubring(self)
729  except:
730  return False
731 
732  def issubring(self, other):
733  """
734  Report whether the field is a subring of another ring.
735  """
736  if self is other:
737  return True
738  elif isinstance(other, FinitePrimeField):
739  return False
740  elif isinstance(other, ExtendedField):
741  if self.char == other.char and not (other.degree % self.degree):
742  return True
743  return False
744  try:
745  return other.issuperring(self)
746  except:
747  return False
748 
749  def __contains__(self, elem):
750  """
751  Report whether elem is in field.
752  """
753  if (isinstance(elem, ExtendedFieldElement) and
754  elem.getRing().modulus == self.modulus):
755  return True
756  elif (isinstance(elem, FinitePrimeFieldElement) and
757  elem.getRing().getCharacteristic() == self.getCharacteristic()):
758  return True
759  return False
760 
761  def __eq__(self, other):
762  """
763  Equality test.
764  """
765  if isinstance(other, ExtendedField):
766  return self.char == other.char and self.degree == other.degree
767  return False
768 
769  def primitive_element(self):
770  """
771  Return a primitive element of the field, i.e., a generator of
772  the multiplicative group.
773  """
774  fullorder = card(self) - 1
775  if self._orderfactor is None:
776  self._orderfactor = factor_misc.FactoredInteger(fullorder)
777  for i in bigrange.range(self.char, card(self)):
778  g = self.createElement(i)
779  for p in self._orderfactor.prime_divisors():
780  if g ** (fullorder // p) == self.one:
781  break
782  else:
783  return g
784 
785  # properties
786  def _getOne(self):
787  "getter for one"
788  if self._one is None:
790  uniutil.FiniteFieldPolynomial([(0, 1)], self.basefield),
791  self)
792  return self._one
793 
794  one = property(_getOne, None, None, "multiplicative unit.")
795 
796  def _getZero(self):
797  "getter for zero"
798  if self._zero is None:
800  uniutil.FiniteFieldPolynomial([], self.basefield),
801  self)
802  return self._zero
803 
804  zero = property(_getZero, None, None, "additive unit.")
805 
806 
807 def fqiso(f_q, gfq):
808  """
809  Return isomorphism function of extended finite fields from f_q to gfq.
810  """
811  if f_q is gfq:
812  return lambda x: x
813  if card(f_q) != card(gfq):
814  raise TypeError("both fields must have the same cardinality.")
815 
816  # find a root of f_q's defining polynomial in gfq.
817  p = f_q.getCharacteristic()
818  q = card(f_q)
819  for i in bigrange.range(p, q):
820  root = gfq.createElement(i)
821  if not f_q.modulus(root):
822  break
823 
824  # finally, define a function
825  def f_q_to_gfq_iso(f_q_elem):
826  """
827  Return the image of the isomorphism of the given element.
828  """
829  if not f_q_elem:
830  return gfq.zero
831  if f_q_elem.rep.degree() == 0:
832  # F_p elements
833  return gfq.createElement(f_q_elem.rep)
834  return f_q_elem.rep(root)
835 
836  return f_q_to_gfq_iso
837 
838 
839 def embedding(f_q1, f_q2):
840  """
841  Return embedding homomorphism function from f_q1 to f_q2,
842  where q1 = p ** k1, q2 = p ** k2 and k1 divides k2.
843  """
844  if card(f_q1) == card(f_q2):
845  return fqiso(f_q1, f_q2)
846  # search multiplicative generators of both fields and relate them.
847  # 0. initialize basic variables
848  q1, q2 = card(f_q1), card(f_q2)
849 
850  # 1. find a multiplicative generator of f_q2
851  f_q2_gen = f_q2.primitive_element()
852  f_q2_subgen = f_q2_gen ** ((q2 - 1) // (q1 - 1))
853 
854  # 2. find a root of defining polynomial of f_q1 in f_q2
855  image_of_x_1 = _findroot(f_q1, f_q2, f_q2_subgen)
856 
857  # 3. finally, define a function
858  def f_q1_to_f_q2_homo(f_q1_elem):
859  """
860  Return the image of the isomorphism of the given element.
861  """
862  if not f_q1_elem:
863  return f_q2.zero
864  if f_q1_elem.rep.degree() == 0:
865  # F_p elements
866  return f_q2.createElement(f_q1_elem.rep)
867  return f_q1_elem.rep(image_of_x_1)
868 
869  return f_q1_to_f_q2_homo
870 
871 def double_embeddings(f_q1, f_q2):
872  """
873  Return embedding homomorphism functions from f_q1 and f_q2
874  to the composite field.
875  """
876  identity = lambda x: x
877  if f_q1 is f_q2:
878  return (identity, identity)
879  p = f_q2.getCharacteristic()
880  k1, k2 = f_q1.degree, f_q2.degree
881  if not k2 % k1:
882  return (embedding(f_q1, f_q2), identity)
883  if not k1 % k2:
884  return (identity, embedding(f_q2, f_q1))
885  composite = FiniteExtendedField(p, gcd.lcm(k1, k2))
886  return (embedding(f_q1, composite), embedding(f_q2, composite))
887 
888 
889 def _findroot(f_q1, f_q2, f_q2_subgen):
890  """
891  Find root of the defining polynomial of f_q1 in f_q2
892  """
893  if card(f_q1) > 50: # 50 is small, maybe
894  _log.debug("by affine multiple (%d)" % card(f_q1))
895  return affine_multiple_method(f_q1.modulus, f_q2)
896  root = f_q2_subgen
897  for i in range(1, card(f_q1)):
898  if not f_q1.modulus(root):
899  image_of_x_1 = root
900  break
901  root *= f_q2_subgen
902  return image_of_x_1
903 
904 def affine_multiple_method(lhs, field):
905  """
906  Find and return a root of the equation lhs = 0 by brute force
907  search in the given field. If there is no root in the field,
908  ValueError is raised.
909 
910  The first argument lhs is a univariate polynomial with
911  coefficients in a finite field. The second argument field is
912  an extension field of the field of coefficients of lhs.
913 
914  Affine multiple A(X) is $\sum_{i=0}^{n} a_i X^{q^i} - a$ for some
915  a_i's and a in the coefficient field of lhs, which is a multiple
916  of the lhs.
917  """
918  polynomial_ring = lhs.getRing()
919  coeff_field = lhs.getCoefficientRing()
920  q = card(coeff_field)
921  n = lhs.degree()
922 
923  # residues = [x, x^q, x^{q^2}, ..., x^{q^{n-1}}]
924  residues = [lhs.mod(polynomial_ring.one.term_mul((1, 1)))] # x
925  for i in range(1, n):
926  residues.append(pow(residues[-1], q, lhs)) # x^{q^i}
927 
928  # find a linear relation among residues and a constant
929  coeff_matrix = matrix.createMatrix(n, n, [coeff_field.zero] * (n**2), coeff_field)
930  for j, residue in enumerate(residues):
931  for i in range(residue.degree() + 1):
932  coeff_matrix[i + 1, j + 1] = residue[i]
933  constant_components = [coeff_field.one] + [coeff_field.zero] * (n - 1)
934  constant_vector = vector.Vector(constant_components)
935  try:
936  relation_vector, kernel = coeff_matrix.solve(constant_vector)
937  for j in range(n, 0, -1):
938  if relation_vector[j]:
939  constant = relation_vector[j].inverse()
940  relation = [constant * c for c in relation_vector]
941  break
942  except matrix.NoInverseImage:
943  kernel_matrix = coeff_matrix.kernel()
944  relation_vector = kernel_matrix[1]
945  assert type(relation_vector) is vector.Vector
946  for j in range(n, 0, -1):
947  if relation_vector[j]:
948  normalizer = relation_vector[j].inverse()
949  relation = [normalizer * c for c in relation_vector]
950  constant = coeff_field.zero
951  break
952 
953  # define L(X) = A(X) + constant
954  coeffs = {}
955  for i, relation_i in enumerate(relation):
956  coeffs[q**i] = relation_i
957  linearized = uniutil.polynomial(coeffs, coeff_field)
958 
959  # Fq basis [1, X, ..., X^{s-1}]
960  qbasis = [1]
961  root = field.createElement(field.char)
962  s = arith1.log(card(field), q)
963  qbasis += [root**i for i in range(1, s)]
964  # represent L as Matrix
965  lmat = matrix.createMatrix(s, s, field.basefield)
966  for j, base in enumerate(qbasis):
967  imagei = linearized(base)
968  if imagei.getRing() == field.basefield:
969  lmat[1, j + 1] = imagei
970  else:
971  for i, coeff in imagei.rep.iterterms():
972  if coeff:
973  lmat[i + 1, j + 1] = coeff
974  # solve L(X) = the constant
975  constant_components = [constant] + [coeff_field.zero] * (s - 1)
976  constant_vector = vector.Vector(constant_components)
977  solution, kernel = lmat.solve(constant_vector)
978  assert lmat * solution == constant_vector
979  solutions = [solution]
980  for v in kernel:
981  for i in range(card(field.basefield)):
982  solutions.append(solution + i * v)
983 
984  # roots of A(X) contains the solutions of lhs = 0
985  for t in bigrange.multirange([(card(field.basefield),)] * len(kernel)):
986  affine_root_vector = solution
987  for i, ti in enumerate(t):
988  affine_root_vector += ti * kernel[i]
989  affine_root = field.zero
990  for i, ai in enumerate(affine_root_vector):
991  affine_root += ai * qbasis[i]
992  if not lhs(affine_root):
993  return affine_root
994 
995  raise ValueError("no root found")
996 
997 
998 def FiniteExtendedField(characteristic, n_or_modulus):
999  """
1000  Return ExtendedField F_{p^n} or F_p[]/(modulus).
1001 
1002  This is a convenience wrapper for backward compatibility.
1003  """
1004  return ExtendedField.prime_extension(characteristic, n_or_modulus)
1005 
1006 FiniteExtendedFieldElement = ExtendedFieldElement
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