Make search for a random irreducible/primitive polynomial much more e…

…fficient

src/galois/_polys/_irreducible.py

@@ -12,7 +12,13 @@
 from .._domains import _factory
 from .._helper import export, verify_isinstance
 from .._prime import is_prime_power
-from ._poly import Poly, _deterministic_search_fixed_terms, _minimum_terms
+from ._poly import (
+ Poly,
+ _deterministic_search_fixed_terms,
+ _minimum_terms,
+ _random_search,
+ _random_search_fixed_terms,
+)
 
 if TYPE_CHECKING:
  from .._fields import FieldArray
@@ -113,21 +119,23 @@ def irreducible_poly(
 
  try:
  if method == "min":
- poly = next(irreducible_polys(order, degree, terms))
- elif method == "max":
- poly = next(irreducible_polys(order, degree, terms, reverse=True))
- else:
- if terms == "min":
- terms = _minimum_terms(order, degree, "is_irreducible")
- poly = _random_search(order, degree, terms)
+ return next(irreducible_polys(order, degree, terms))
+ if method == "max":
+ return next(irreducible_polys(order, degree, terms, reverse=True))
+
+ # Random search
+ if terms is None:
+ return next(_random_search(order, degree, "is_irreducible"))
+ if terms == "min":
+ terms = _minimum_terms(order, degree, "is_irreducible")
+ return next(_random_search_fixed_terms(order, degree, terms, "is_irreducible"))
+
  except StopIteration as e:
  terms_str = "any" if terms is None else str(terms)
  raise RuntimeError(
  f"No monic irreducible polynomial of degree {degree} over GF({order}) with {terms_str} terms exists."
  ) from e
 
- return poly
-
 
 @export
 def irreducible_polys(
@@ -262,22 +270,3 @@ def _deterministic_search(
  return poly
 
  return None
-
-
-def _random_search(order: int, degree: int, terms: int | None) -> Poly:
- """
- Searches for a random irreducible polynomial.
- """
- field = _factory.FIELD_FACTORY(order)
-
- # Only search monic polynomials of degree m over GF(p)
- start = order**degree
- stop = 2 * order**degree
-
- while True:
- integer = random.randint(start, stop - 1)
- poly = Poly.Int(integer, field=field)
- if terms is not None and poly.nonzero_coeffs.size != terms:
- continue
- if poly.is_irreducible():
- return poly

src/galois/_polys/_poly.py

@@ -4,6 +4,7 @@
 from __future__ import annotations
 
 import functools
+import random
 from typing import Callable, Iterable, Iterator, Sequence, Type, overload
 
 import numpy as np
@@ -2318,7 +2319,7 @@ def _deterministic_search_fixed_terms(
  reverse: bool = False,
 ) -> Iterator[Poly]:
  """
- Iterates over all polynomials of the given degree and number of non-zero terms in lexicographical
+ Iterates over all monic polynomials of the given degree and number of non-zero terms in lexicographical
  order, only yielding those that pass the specified test (either 'is_irreducible()' or 'is_primitive()').
  """
  assert test in ["is_irreducible", "is_primitive"]
@@ -2344,7 +2345,7 @@ def _deterministic_search_fixed_terms_recursive(
  direction: Callable[[Iterable[int]], Iterable[int]],
 ) -> Iterator[Poly]:
  """
- Recursively finds all polynomials having non-zero coefficients `coeffs` with degree `degrees` with `terms`
+ Recursively finds all monic polynomials having non-zero coefficients `coeffs` with degree `degrees` with `terms`
  additional non-zero terms. The polynomials are found in lexicographical order, only yielding those that pass
  the specified test (either 'is_irreducible()' or 'is_primitive()').
  """
@@ -2371,3 +2372,53 @@ def _deterministic_search_fixed_terms_recursive(
  yield from _deterministic_search_fixed_terms_recursive(
  next_degrees, next_coeffs, terms - 1, test, field, direction
  )
+
+
+def _random_search(order: int, degree: int, test: str) -> Iterator[Poly]:
+ """
+ Searches for a random monic polynomial of specified degree, only yielding those that pass the specified test
+ (either 'is_irreducible()' or 'is_primitive()').
+ """
+ assert test in ["is_irreducible", "is_primitive"]
+ field = _factory.FIELD_FACTORY(order)
+
+ # Only search monic polynomials of degree m over GF(q)
+ start = order**degree
+ stop = 2 * order**degree
+
+ while True:
+ integer = random.randint(start, stop - 1)
+ poly = Poly.Int(integer, field=field)
+ if getattr(poly, test)():
+ yield poly
+
+
+def _random_search_fixed_terms(
+ order: int,
+ degree: int,
+ terms: int,
+ test: str,
+) -> Iterator[Poly]:
+ """
+ Searches for a random monic polynomial of specified degree and number of non-zero terms, only yielding those that
+ pass the specified test (either 'is_irreducible()' or 'is_primitive()').
+ """
+ assert test in ["is_irreducible", "is_primitive"]
+ field = _factory.FIELD_FACTORY(order)
+
+ if terms == 1:
+ # The x^m term is always 1. If there's only one term, then the x^m is the polynomial.
+ poly = Poly.Degrees([degree], [1], field=field)
+ if getattr(poly, test)():
+ yield poly
+ else:
+ while True:
+ # The x^m term is always 1 and the x^0 term is always non-zero.
+ mid_degrees = random.sample(range(1, degree), terms - 2)
+ mid_coeffs = np.random.randint(1, field.order, terms - 2)
+ x0_coeff = np.random.randint(1, field.order)
+ degrees = (degree, *mid_degrees, 0)
+ coeffs = (1, *mid_coeffs, x0_coeff)
+ poly = Poly.Degrees(degrees, coeffs, field=field)
+ if getattr(poly, test)():
+ yield poly

src/galois/_polys/_primitive.py

@@ -13,7 +13,13 @@
 from .._domains import _factory
 from .._helper import export, verify_isinstance
 from .._prime import is_prime, is_prime_power
-from ._poly import Poly, _deterministic_search_fixed_terms, _minimum_terms
+from ._poly import (
+ Poly,
+ _deterministic_search_fixed_terms,
+ _minimum_terms,
+ _random_search,
+ _random_search_fixed_terms,
+)
 
 if TYPE_CHECKING:
  from .._fields import FieldArray
@@ -132,21 +138,23 @@ def primitive_poly(
 
  try:
  if method == "min":
- poly = next(primitive_polys(order, degree, terms))
- elif method == "max":
- poly = next(primitive_polys(order, degree, terms, reverse=True))
- else:
- if terms == "min":
- terms = _minimum_terms(order, degree, "is_primitive")
- poly = _random_search(order, degree, terms)
+ return next(primitive_polys(order, degree, terms))
+ if method == "max":
+ return next(primitive_polys(order, degree, terms, reverse=True))
+
+ # Random search
+ if terms is None:
+ return next(_random_search(order, degree, "is_primitive"))
+ if terms == "min":
+ terms = _minimum_terms(order, degree, "is_primitive")
+ return next(_random_search_fixed_terms(order, degree, terms, "is_primitive"))
+
  except StopIteration as e:
  terms_str = "any" if terms is None else str(terms)
  raise RuntimeError(
  f"No monic primitive polynomial of degree {degree} over GF({order}) with {terms_str} terms exists."
  ) from e
 
- return poly
-
 
 @export
 def primitive_polys(
@@ -285,25 +293,6 @@ def _deterministic_search(
  return None
 
 
-def _random_search(order: int, degree: int, terms: int | None) -> Poly:
- """
- Searches for a random primitive polynomial.
- """
- field = _factory.FIELD_FACTORY(order)
-
- # Only search monic polynomials of degree m over GF(p)
- start = order**degree
- stop = 2 * order**degree
-
- while True:
- integer = random.randint(start, stop - 1)
- poly = Poly.Int(integer, field=field)
- if terms is not None and poly.nonzero_coeffs.size != terms:
- continue
- if poly.is_primitive():
- return poly
-
-
 @export
 def conway_poly(characteristic: int, degree: int) -> Poly:
  r"""