Trac #34614: add missing EllipticCurveHom methods to Îlu isogenies

The `EllipticCurveHom_velusqrt` class introduced in #34303 inherits from
`EllipticCurveHom` (#32388, #32502), but it doesn't yet implement all of
the required methods. In this patch, we add them.

Diff without the dependency: https://git.sagemath.org/sage.git/diff?id2=
7326051&id=8cb0f29aa2cd5be18506e1d65eac
954481d1ec8a

URL: https://trac.sagemath.org/34614
Reported by: lorenz
Ticket author(s): Lorenz Panny
Reviewer(s): Travis Scrimshaw, Kwankyu Lee

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -18,14 +18,18 @@
     sage: Q = E(6,5)
     sage: phi = E.isogeny(Q)
     sage: phi
-    Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x + 1 over Finite Field of size 11 to Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 11
+    Isogeny of degree 7 from Elliptic Curve defined by y^2 = x^3 + x + 1 over
+    Finite Field of size 11 to Elliptic Curve defined by y^2 = x^3 + 7*x + 8
+    over Finite Field of size 11
     sage: P = E(4,5)
     sage: phi(P)
     (10 : 0 : 1)
     sage: phi.codomain()
     Elliptic Curve defined by y^2 = x^3 + 7*x + 8 over Finite Field of size 11
     sage: phi.rational_maps()
-    ((x^7 + 4*x^6 - 3*x^5 - 2*x^4 - 3*x^3 + 3*x^2 + x - 2)/(x^6 + 4*x^5 - 4*x^4 - 5*x^3 + 5*x^2), (x^9*y - 5*x^8*y - x^7*y + x^5*y - x^4*y - 5*x^3*y - 5*x^2*y - 2*x*y - 5*y)/(x^9 - 5*x^8 + 4*x^6 - 3*x^4 + 2*x^3))
+    ((x^7 + 4*x^6 - 3*x^5 - 2*x^4 - 3*x^3 + 3*x^2 + x - 2)/(x^6 + 4*x^5 - 4*x^4
+    - 5*x^3 + 5*x^2), (x^9*y - 5*x^8*y - x^7*y + x^5*y - x^4*y - 5*x^3*y -
+    5*x^2*y - 2*x*y - 5*y)/(x^9 - 5*x^8 + 4*x^6 - 3*x^4 + 2*x^3))
 
 The methods directly accessible from an elliptic curve ``E`` over a
 field are
@@ -337,7 +341,6 @@ def compute_vw_kohel_even_deg3(b2, b4, s1, s2, s3):
     w = 3*(s1**3 - 3*s1*s2 + 3*s3) + (b2*temp1 + b4*s1)/2
     return v, w
 
-
 def compute_vw_kohel_odd(b2, b4, b6, s1, s2, s3, n):
     r"""
     Compute Vélu's `(v,w)` using Kohel's formulas for isogenies of odd
@@ -374,7 +377,6 @@ def compute_vw_kohel_odd(b2, b4, b6, s1, s2, s3, n):
     w = 10*(s1**3 - 3*s1*s2 + 3*s3) + 2*b2*(s1**2 - 2*s2) + 3*b4*s1 + n*b6
     return v, w
 
-
 def compute_codomain_kohel(E, kernel):
     r"""
     Compute the codomain from the kernel polynomial using Kohel's
@@ -481,7 +483,6 @@ def compute_codomain_kohel(E, kernel):
 
     return compute_codomain_formula(E, v, w)
 
-
 def two_torsion_part(E, psi):
     r"""
     Return the greatest common divisor of ``psi`` and the 2-torsion
@@ -516,6 +517,7 @@ def two_torsion_part(E, psi):
     psi_2 = E.two_division_polynomial(x)
     return psi.gcd(psi_2)
 
+
 class EllipticCurveIsogeny(EllipticCurveHom):
     r"""
     This class implements separable isogenies of elliptic curves.
@@ -1739,7 +1741,6 @@ def __setup_post_isomorphism(self, codomain, model):
         post_isom = oldE2.isomorphism_to(codomain)
         self.__set_post_isomorphism(codomain, post_isom)
 
-
     ###########################
     # Velu's Formula Functions
     ###########################
@@ -1873,7 +1874,6 @@ def __compute_codomain_via_velu(self):
         """
         return compute_codomain_formula(self._domain, self.__v, self.__w)
 
-
     @staticmethod
     def __velu_sum_helper(xQ, Qvalues, a1, a3, x, y):
         r"""
@@ -1922,7 +1922,6 @@ def __velu_sum_helper(xQ, Qvalues, a1, a3, x, y):
 
         return tX, tY
 
-
     def __compute_via_velu_numeric(self, xP, yP):
         r"""
         Private function that sorts the list of points in the kernel
@@ -1952,7 +1951,6 @@ def __compute_via_velu_numeric(self, xP, yP):
 
         return self.__compute_via_velu(xP,yP)
 
-
     def __compute_via_velu(self, xP, yP):
         r"""
         Private function for Vélu's formulas, to perform the summation.
@@ -2018,7 +2016,6 @@ def __compute_via_velu(self, xP, yP):
 
         return X, Y
 
-
     def __initialize_rational_maps_via_velu(self):
         r"""
         Private function for Vélu's formulas, helper function to
@@ -2040,7 +2037,6 @@ def __initialize_rational_maps_via_velu(self):
         y = self.__mpoly_ring.gen(1)
         return self.__compute_via_velu(x,y)
 
-
     def __init_kernel_polynomial_velu(self):
         r"""
         Private function for Vélu's formulas, helper function to
@@ -2070,7 +2066,6 @@ def __init_kernel_polynomial_velu(self):
 
         self.__kernel_polynomial = psi
 
-
     ###################################
     # Kohel's Variant of Velu's Formula
     ###################################
@@ -2262,7 +2257,6 @@ def __init_even_kernel_polynomial(self, E, psi_G):
 
         return phi, omega, v, w, n, d
 
-
     def __init_odd_kernel_polynomial(self, E, psi):
         r"""
         Return the isogeny parameters for a cyclic isogeny of odd degree.
@@ -2363,7 +2357,6 @@ def __init_odd_kernel_polynomial(self, E, psi):
 
         return phi, omega, v, w, n, d
 
-
     #
     # This is the fast omega computation that works when characteristic is not 2
     #
@@ -2692,36 +2685,6 @@ def scaling_factor(self):
             sc *= self.__post_isomorphism.scaling_factor()
         return sc
 
-    def as_morphism(self):
-        r"""
-        Return this isogeny as a morphism of projective schemes.
-
-        EXAMPLES::
-
-            sage: k = GF(11)
-            sage: E = EllipticCurve(k, [1,1])
-            sage: Q = E(6,5)
-            sage: phi = E.isogeny(Q)
-            sage: mor = phi.as_morphism()
-            sage: mor.domain() == E
-            True
-            sage: mor.codomain() == phi.codomain()
-            True
-            sage: mor(Q) == phi(Q)
-            True
-
-        TESTS::
-
-            sage: mor(0*Q)
-            (0 : 1 : 0)
-            sage: mor(1*Q)
-            (0 : 1 : 0)
-        """
-        from sage.schemes.curves.constructor import Curve
-        X_affine = Curve(self.domain()).affine_patch(2)
-        Y_affine = Curve(self.codomain()).affine_patch(2)
-        return X_affine.hom(self.rational_maps(), Y_affine).homogenize(2)
-
     def kernel_polynomial(self):
         r"""
         Return the kernel polynomial of this isogeny.
@@ -3246,7 +3209,6 @@ def split_kernel_polynomial(poly):
     from sage.misc.misc_c import prod
     return prod([p for p,e in poly.squarefree_decomposition()])
 
-
 def compute_isogeny_kernel_polynomial(E1, E2, ell, algorithm="starks"):
     r"""
     Return the kernel polynomial of an isogeny of degree ``ell``
@@ -3487,7 +3449,6 @@ def compute_sequence_of_maps(E1, E2, ell):
 
     return pre_isom, post_isom, E1pr, E2pr, ker_poly
 
-
 # Utility functions for manipulating isogeny degree matrices
 
 def fill_isogeny_matrix(M):

src/sage/schemes/elliptic_curves/hom.py

@@ -20,7 +20,6 @@
 - Lorenz Panny (2021): Refactor isogenies and isomorphisms into
   the common :class:`EllipticCurveHom` interface.
 """
-
 from sage.misc.cachefunc import cached_method
 from sage.structure.richcmp import richcmp_not_equal, richcmp, op_EQ, op_NE
 
@@ -93,7 +92,6 @@ def _composition_(self, other, homset):
         from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
         return EllipticCurveHom_composite.from_factors([other, self])
 
-
     @staticmethod
     def _comparison_impl(left, right, op):
         """
@@ -194,7 +192,6 @@ def _richcmp_(self, other, op):
 
         return richcmp(self.rational_maps(), other.rational_maps(), op)
 
-
     def degree(self):
         r"""
         Return the degree of this elliptic-curve morphism.
@@ -327,11 +324,10 @@ def x_rational_map(self):
             ...
             NotImplementedError: ...
         """
-        #TODO: could have a default implementation that simply
-        #      returns the first component of rational_maps()
+        # TODO: could have a default implementation that simply
+        # returns the first component of rational_maps()
         raise NotImplementedError('children must implement')
 
-
     def scaling_factor(self):
         r"""
         Return the Weierstrass scaling factor associated to this
@@ -409,7 +405,6 @@ def formal(self, prec=20):
         assert th.valuation() == +1, f"th has valuation {th.valuation()} (should be +1)"
         return th
 
-
     def is_normalized(self):
         r"""
         Determine whether this morphism is a normalized isogeny.
@@ -489,7 +484,6 @@ def is_normalized(self):
         """
         return self.scaling_factor() == 1
 
-
     def is_separable(self):
         r"""
         Determine whether or not this morphism is separable.
@@ -657,6 +651,36 @@ def __hash__(self):
         """
         return hash((self.domain(), self.codomain(), self.kernel_polynomial()))
 
+    def as_morphism(self):
+        r"""
+        Return ``self`` as a morphism of projective schemes.
+
+        EXAMPLES::
+
+            sage: k = GF(11)
+            sage: E = EllipticCurve(k, [1,1])
+            sage: Q = E(6,5)
+            sage: phi = E.isogeny(Q)
+            sage: mor = phi.as_morphism()
+            sage: mor.domain() == E
+            True
+            sage: mor.codomain() == phi.codomain()
+            True
+            sage: mor(Q) == phi(Q)
+            True
+
+        TESTS::
+
+            sage: mor(0*Q)
+            (0 : 1 : 0)
+            sage: mor(1*Q)
+            (0 : 1 : 0)
+        """
+        from sage.schemes.curves.constructor import Curve
+        X_affine = Curve(self.domain()).affine_patch(2)
+        Y_affine = Curve(self.codomain()).affine_patch(2)
+        return X_affine.hom(self.rational_maps(), Y_affine).homogenize(2)
+
 
 def compare_via_evaluation(left, right):
     r"""
@@ -713,14 +737,13 @@ def compare_via_evaluation(left, right):
         EE = E.base_extend(F.extension(e))
         Ps = EE.gens()
         return all(left._eval(P) == right._eval(P) for P in Ps)
-
     elif isinstance(F, number_field_base.NumberField):
         for _ in range(100):
             P = E.lift_x(F.random_element(), extend=True)
             if not P.has_finite_order():
                 return left._eval(P) == right._eval(P)
         else:
             assert False, "couldn't find a point of infinite order"
-
     else:
         raise NotImplementedError('not implemented for this base field')
+

src/sage/schemes/elliptic_curves/hom_composite.py

@@ -24,7 +24,9 @@
     sage: EllipticCurveHom_composite(E, P)
     Composite morphism of degree 11150372599265311570767859136324180752990208 = 2^143:
       From: Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 33451117797795934712303577408972542258970623^2
-      To:   Elliptic Curve defined by y^2 = x^3 + (18676616716352953484576727486205473216172067*z2+32690199585974925193292786311814241821808308)*x + (3369702436351367403910078877591946300201903*z2+15227558615699041241851978605002704626689722) over Finite Field in z2 of size 33451117797795934712303577408972542258970623^2
+      To:   Elliptic Curve defined by y^2 = x^3 + (18676616716352953484576727486205473216172067*z2+32690199585974925193292786311814241821808308)*x
+    + (3369702436351367403910078877591946300201903*z2+15227558615699041241851978605002704626689722)
+    over Finite Field in z2 of size 33451117797795934712303577408972542258970623^2
 
 Yet, the interface provided by :class:`EllipticCurveHom_composite`
 is identical to :class:`EllipticCurveIsogeny` and other instantiations
@@ -40,10 +42,15 @@
     sage: psi(E.lift_x(11))
     (352 : 73 : 1)
     sage: psi.rational_maps()
-    ((x^35 + 162*x^34 + 186*x^33 + 92*x^32 - ... + 44*x^3 + 190*x^2 + 80*x - 72)/(x^34 + 162*x^33 - 129*x^32 + 41*x^31 + ... + 66*x^3 - 191*x^2 + 119*x + 21),
-     (x^51*y - 176*x^50*y + 115*x^49*y - 120*x^48*y + ... + 72*x^3*y + 129*x^2*y + 163*x*y + 178*y)/(x^51 - 176*x^50 + 11*x^49 + 26*x^48 - ... - 77*x^3 + 185*x^2 + 169*x - 128))
+    ((x^35 + 162*x^34 + 186*x^33 + 92*x^32 - ... + 44*x^3 + 190*x^2 + 80*x -
+    72)/(x^34 + 162*x^33 - 129*x^32 + 41*x^31 + ... + 66*x^3 - 191*x^2 + 119*x
+    + 21), (x^51*y - 176*x^50*y + 115*x^49*y - 120*x^48*y + ... + 72*x^3*y +
+    129*x^2*y + 163*x*y + 178*y)/(x^51 - 176*x^50 + 11*x^49 + 26*x^48 - ... -
+    77*x^3 + 185*x^2 + 169*x - 128))
     sage: psi.kernel_polynomial()
-    x^17 + 81*x^16 + 7*x^15 + 82*x^14 + 49*x^13 + 68*x^12 + 109*x^11 + 326*x^10 + 117*x^9 + 136*x^8 + 111*x^7 + 292*x^6 + 55*x^5 + 389*x^4 + 175*x^3 + 43*x^2 + 149*x + 373
+    x^17 + 81*x^16 + 7*x^15 + 82*x^14 + 49*x^13 + 68*x^12 + 109*x^11 + 326*x^10
+    + 117*x^9 + 136*x^8 + 111*x^7 + 292*x^6 + 55*x^5 + 389*x^4 + 175*x^3 +
+    43*x^2 + 149*x + 373
     sage: psi.dual()
     Composite morphism of degree 35 = 7*5:
       From: Elliptic Curve defined by y^2 = x^3 + 101*x + 285 over Finite Field of size 419
@@ -82,7 +89,7 @@
 from sage.schemes.elliptic_curves.ell_curve_isogeny import EllipticCurveIsogeny
 from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
 
-#TODO: implement sparse strategies? (cf. the SIKE cryptosystem)
+# TODO: Implement sparse strategies? (cf. the SIKE cryptosystem)
 
 def _eval_factored_isogeny(phis, P):
     """
@@ -468,7 +475,6 @@ def factors(self):
         """
         return self._phis
 
-
     # EllipticCurveHom methods
 
     @staticmethod
@@ -765,7 +771,6 @@ def is_injective(self):
         """
         return all(phi.is_injective() for phi in self._phis)
 
-
     @staticmethod
     def make_default():
         r"""