Trac #23402: faster hash of number field elements

Hashing of nf element go through `self.polynomial()` which is damn slow!

The attached branch provides a quick hash version (that is more robust
than the one for polynomial). A x10 speed up is seen on the following
example (425 ms vs 40ms)
{{{
sage: K.<b> = NumberField(x^3 - 2)
sage: l = [i * b + j * b^2 for i in range(100) for j in range(100)]
sage: l.sort()
sage: %time s = set(l[i+1] - l[i] for i in range(len(l)-1))
}}}

URL: https://trac.sagemath.org/23402
Reported by: vdelecroix
Ticket author(s): Vincent Delecroix
Reviewer(s): Travis Scrimshaw

src/sage/groups/matrix_gps/finitely_generated.py

@@ -1270,10 +1270,10 @@ def invariants_of_degree(self, deg, chi=None, R=None):
             sage: chi = G.character(G.character_table()[1])
             sage: R.<x,y,z> = K[]
             sage: G.invariants_of_degree(2, R=R, chi=chi)
-            [x^2 + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*y^2 + (2*izeta3^3 +
-            3*izeta3^2 + 8*izeta3 + 3)*z^2,
-             x*y + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*x*z + (-2*izeta3^3 -
-            3*izeta3^2 - 8*izeta3 - 4)*y*z]
+            [x*y + (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*x*z +
+             (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*y*z,
+             x^2 + (-2*izeta3^3 - 3*izeta3^2 - 8*izeta3 - 4)*y^2 +
+             (2*izeta3^3 + 3*izeta3^2 + 8*izeta3 + 3)*z^2]
 
         ::
 

src/sage/modular/modform_hecketriangle/hecke_triangle_group_element.py

@@ -2020,9 +2020,10 @@ def simple_fixed_point_set(self, extended=True):
 
             sage: el = G.V(3)*G.V(2)^(-1)*G.V(1)*G.V(6)
             sage: el.simple_fixed_point_set()
-            {(-lam + 3/2)*e - 1/2*lam + 1, 1/2*e + 1/2*lam, (-lam + 3/2)*e + 1/2*lam - 1, 1/2*e - 1/2*lam}
+            {(-lam + 3/2)*e + 1/2*lam - 1, (-lam + 3/2)*e - 1/2*lam + 1, 1/2*e - 1/2*lam, 1/2*e + 1/2*lam}
+
             sage: el.simple_fixed_point_set(extended=False)
-            {1/2*e + 1/2*lam, 1/2*e - 1/2*lam}
+            {1/2*e - 1/2*lam, 1/2*e + 1/2*lam}
         """
         if self.parent().n() == infinity:
             from warnings import warn
@@ -2507,7 +2508,7 @@ def is_hecke_symmetric(self):
             sage: el.is_hecke_symmetric()
             False
             sage: (el.simple_fixed_point_set(), el.inverse().simple_fixed_point_set())
-            ({1/2*e, (-1/2*lam + 1/2)*e}, {(1/2*lam - 1/2)*e, -1/2*e})
+            ({1/2*e, (-1/2*lam + 1/2)*e}, {-1/2*e, (1/2*lam - 1/2)*e})
 
             sage: el = G.V(3)*G.V(2)^(-1)*G.V(1)*G.V(6)
             sage: el.is_hecke_symmetric()
@@ -2519,7 +2520,7 @@ def is_hecke_symmetric(self):
             sage: el.is_hecke_symmetric()
             True
             sage: el.simple_fixed_point_set()
-            {(lam - 3/2)*e + 1/2*lam - 1, (-lam + 3/2)*e + 1/2*lam - 1, (lam - 3/2)*e - 1/2*lam + 1, (-lam + 3/2)*e - 1/2*lam + 1}
+            {(lam - 3/2)*e + 1/2*lam - 1, (-lam + 3/2)*e - 1/2*lam + 1, (lam - 3/2)*e - 1/2*lam + 1, (-lam + 3/2)*e + 1/2*lam - 1}
             sage: el.simple_fixed_point_set() == el.inverse().simple_fixed_point_set()
             True
         """

src/sage/modular/modform_hecketriangle/readme.py

@@ -376,7 +376,7 @@
       sage: el.is_hecke_symmetric()
       False
       sage: (el.simple_fixed_point_set(), el.inverse().simple_fixed_point_set())
-      ({1/2*e, (-1/2*lam + 1/2)*e}, {(1/2*lam - 1/2)*e, -1/2*e})
+      ({1/2*e, (-1/2*lam + 1/2)*e}, {-1/2*e, (1/2*lam - 1/2)*e})
       sage: el = G.V(2)*G.V(3)
       sage: el.is_hecke_symmetric()
       True

src/sage/rings/number_field/number_field_element.pyx

@@ -44,6 +44,8 @@ from sage.libs.gmp.mpq cimport *
 from sage.libs.mpfi cimport mpfi_t, mpfi_init, mpfi_set, mpfi_clear, mpfi_div_z, mpfi_init2, mpfi_get_prec, mpfi_set_prec
 from sage.libs.mpfr cimport mpfr_equal_p, mpfr_less_p, mpfr_greater_p, mpfr_greaterequal_p, mpfr_floor, mpfr_get_z, MPFR_RNDN
 from sage.libs.ntl.error import NTLError
+from sage.libs.gmp.pylong cimport mpz_pythonhash
+
 from cpython.object cimport Py_EQ, Py_NE, Py_LT, Py_GT, Py_LE, Py_GE
 from sage.structure.richcmp cimport rich_to_bool
 
@@ -2854,16 +2856,56 @@ cdef class NumberFieldElement(FieldElement):
 
     def __hash__(self):
         """
-        Return hash of this number field element, which is just the
-        hash of the underlying polynomial.
+        Return hash of this number field element.
+
+        It respects the hash values of rational numbers.
 
         EXAMPLES::
 
             sage: K.<b> = NumberField(x^3 - 2)
-            sage: hash(b^2 + 1) == hash((b^2 + 1).polynomial()) # indirect doctest
+            sage: hash(b^2 + 1)   # random
+            175247765440
+            sage: hash(K(13)) == hash(13)
+            True
+            sage: hash(K(-2/3)) == hash(-2/3)
+            True
+
+        No collisions (even on low bits)::
+
+            sage: from itertools import product
+            sage: elts = []
+            sage: for (i,j,k) in product((-1,0,1,2,3), repeat=3):
+            ....:     x = i + j*b + k*b^2
+            ....:     elts.append(x)
+            ....:     if gcd([2,i,j,k]) == 1:
+            ....:         elts.append(x / 2)
+            ....:     if gcd([3,i,j,k]) == 1:
+            ....:         elts.append(x / 3)
+            sage: len(set(map(hash, elts))) == len(elts)
+            True
+            sage: len(set(hash(x)%(2^18) for x in elts)) == len(elts)
             True
         """
-        return hash(self.polynomial())
+        cdef Py_hash_t h
+        cdef int i
+        cdef mpz_t z
+
+        mpz_init(z)
+        ZZX_getitem_as_mpz(z, &self.__numerator, 0)
+        h = mpz_pythonhash(z)
+
+        for i from 1 <= i <= ZZX_deg(self.__numerator):
+            ZZX_getitem_as_mpz(z, &self.__numerator, i)
+            # magic number below is floor(2^63 / (2+sqrt(2)))
+            h ^= mpz_pythonhash(z) + (<Py_hash_t> 2701463124188384701) + (h << 16) + (h >> 2)
+
+        ZZ_to_mpz(z, &self.__denominator)
+        # magic number below is floor((1+sqrt(5)) * 2^63)
+        h += (mpz_pythonhash(z) - 1) * (<Py_hash_t> 7461864723258187525)
+
+        mpz_clear(z)
+
+        return h
 
     cpdef list _coefficients(self):
         """

src/sage/schemes/curves/curve.py

@@ -265,9 +265,12 @@ def singular_points(self, F=None):
             + 67/3*x^2*y*z^2 + 117/4*y^5 + 9*x^5 + 6*x^3*z^2 + 393/4*x*y^4\
             + 145*x^2*y^3 + 115*x^3*y^2 + 49*x^4*y], P)
             sage: C.singular_points(K)
-            [(1/2*b^5 + 1/2*b^3 - 1/2*b - 1 : 1 : 0), (-2/3*b^4 + 1/3 : 0 : 1),
-            (2/3*b^4 - 1/3 : 0 : 1), (b^6 : -b^6 : 1), (-b^6 : b^6 : 1),
-            (-1/2*b^5 - 1/2*b^3 + 1/2*b - 1 : 1 : 0)]
+            [(b^6 : -b^6 : 1),
+             (-b^6 : b^6 : 1),
+             (1/2*b^5 + 1/2*b^3 - 1/2*b - 1 : 1 : 0),
+             (-1/2*b^5 - 1/2*b^3 + 1/2*b - 1 : 1 : 0),
+             (2/3*b^4 - 1/3 : 0 : 1),
+             (-2/3*b^4 + 1/3 : 0 : 1)]
         """
         if F is None:
             if not self.base_ring() in Fields():

src/sage/schemes/projective/projective_morphism.py

@@ -3384,12 +3384,13 @@ def periodic_points(self, n, minimal=True, R=None, algorithm='variety', return_s
             sage: H = Hom(P,P)
             sage: f = H([x^2+z^2, y^2+x^2, z^2+y^2])
             sage: f.periodic_points(1)
-            [(-2*s^5 + 4*s^4 - 5*s^3 + 3*s^2 - 4*s : -2*s^5 + 5*s^4 - 7*s^3 + 6*s^2 - 7*s + 3 : 1),
-            (-s^5 + 3*s^4 - 4*s^3 + 4*s^2 - 4*s + 2 : -s^5 + 2*s^4 - 2*s^3 + s^2 - s : 1),
-            (2*s^5 - 6*s^4 + 9*s^3 - 8*s^2 + 7*s - 4 : 2*s^5 - 5*s^4 + 7*s^3 - 5*s^2 + 6*s - 2 : 1),
-            (s^5 - 2*s^4 + 2*s^3 + s : s^5 - 3*s^4 + 4*s^3 - 3*s^2 + 2*s - 1 : 1),
-            (s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 3*s - 1 : -s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 4*s + 2 : 1), (1 : 1 : 1),
-            (-s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 3*s + 1 : s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 4*s - 1 : 1)]
+            [(-s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 3*s + 1 : s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 4*s - 1 : 1),
+             (-2*s^5 + 4*s^4 - 5*s^3 + 3*s^2 - 4*s : -2*s^5 + 5*s^4 - 7*s^3 + 6*s^2 - 7*s + 3 : 1),
+             (-s^5 + 3*s^4 - 4*s^3 + 4*s^2 - 4*s + 2 : -s^5 + 2*s^4 - 2*s^3 + s^2 - s : 1),
+             (s^5 - 2*s^4 + 3*s^3 - 3*s^2 + 3*s - 1 : -s^5 + 3*s^4 - 5*s^3 + 4*s^2 - 4*s + 2 : 1),
+             (2*s^5 - 6*s^4 + 9*s^3 - 8*s^2 + 7*s - 4 : 2*s^5 - 5*s^4 + 7*s^3 - 5*s^2 + 6*s - 2 : 1),
+             (1 : 1 : 1),
+             (s^5 - 2*s^4 + 2*s^3 + s : s^5 - 3*s^4 + 4*s^3 - 3*s^2 + 2*s - 1 : 1)]
 
         ::
 
@@ -4788,9 +4789,18 @@ def all_rational_preimages(self, points):
             sage: H = End(P)
             sage: f = H([16*x^2 - 29*y^2, 16*y^2])
             sage: f.all_rational_preimages([P(16*w^2 - 29,16)])
-            [(w^2 + w - 25/16 : 1), (-w - 1/2 : 1), (w^2 - 29/16 : 1), (-w : 1), (w + 1/2 : 1), (w^2 - 21/16 : 1),
-            (w : 1), (-w^2 + 21/16 : 1), (-w^2 - w + 25/16 : 1), (w^2 + w - 33/16 : 1), (-w^2 + 29/16 : 1),
-            (-w^2 - w + 33/16 : 1)]
+            [(-w^2 + 21/16 : 1),
+             (w : 1),
+             (w + 1/2 : 1),
+             (w^2 + w - 33/16 : 1),
+             (-w^2 - w + 25/16 : 1),
+             (w^2 - 21/16 : 1),
+             (-w^2 - w + 33/16 : 1),
+             (-w : 1),
+             (-w - 1/2 : 1),
+             (-w^2 + 29/16 : 1),
+             (w^2 - 29/16 : 1),
+             (w^2 + w - 25/16 : 1)]
 
         ::
 
@@ -5085,12 +5095,12 @@ def connected_rational_component(self, P, n=0):
             sage: f.connected_rational_component(P)
             [(w : 1),
              (w^2 - 29/16 : 1),
-             (w^2 + w - 25/16 : 1),
              (-w^2 - w + 25/16 : 1),
+             (w^2 + w - 25/16 : 1),
              (-w : 1),
+             (-w^2 + 29/16 : 1),
              (w + 1/2 : 1),
              (-w - 1/2 : 1),
-             (-w^2 + 29/16 : 1),
              (-w^2 + 21/16 : 1),
              (w^2 - 21/16 : 1),
              (w^2 + w - 33/16 : 1),