Trac #32826: scalar-multiplication endomorphisms of elliptic curves

This ticket adds `EllipticCurveHom_scalar`, a new class encapsulating
scalar multiplications on elliptic curves. This serves two main
purposes:

1. It solves one of the motivations behind #8014 (faster
`multiplication_by_m_isogeny`).
2. Wrapping scalar multiplications as an `EllipticCurveHom` is an
important step towards implementing endomorphism rings (see #7368).

We also deprecate `.multiplication_by_m_isogeny()`: It should be
replaceable by `.scalar_multiplication()` in all cases.

URL: https://trac.sagemath.org/32826
Reported by: lorenz
Ticket author(s): Lorenz Panny
Reviewer(s): John Cremona

src/doc/en/reference/arithmetic_curves/index.rst

@@ -22,6 +22,7 @@ Maps between them
    sage/schemes/elliptic_curves/ell_curve_isogeny
    sage/schemes/elliptic_curves/hom_velusqrt
    sage/schemes/elliptic_curves/hom_composite
+   sage/schemes/elliptic_curves/hom_scalar
    sage/schemes/elliptic_curves/isogeny_small_degree
 
 

src/sage/schemes/elliptic_curves/ell_curve_isogeny.py

@@ -92,7 +92,7 @@
 from sage.schemes.elliptic_curves.ell_generic import is_EllipticCurve
 
 from sage.schemes.elliptic_curves.weierstrass_morphism \
-        import WeierstrassIsomorphism, _isomorphisms, baseWI
+        import WeierstrassIsomorphism, _isomorphisms, baseWI, negation_morphism
 
 #
 # Private function for parsing input to determine the type of
@@ -1336,9 +1336,7 @@ def __neg__(self):
             ((x^2 + (-a)*x - 2)/(x + (-a)), (-x^2*y + (2*a)*x*y - y)/(x^2 + (-2*a)*x - 1))
         """
         output = copy(self)
-        E2 = output._codomain
-        iso = WeierstrassIsomorphism(E2, (-1, 0, -E2.a1(), -E2.a3()))
-        output._set_post_isomorphism(iso)
+        output._set_post_isomorphism(negation_morphism(output._codomain))
         return output
 
     #
@@ -1402,8 +1400,8 @@ def __clear_cached_values(self):
             sage: E = EllipticCurve(QQ, [0,0,0,1,0])
             sage: phi = EllipticCurveIsogeny(E, x)
             sage: old_ratl_maps = phi.rational_maps()
-            sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
-            sage: phi._set_post_isomorphism(WeierstrassIsomorphism(phi.codomain(), (-1,0,0,0)))
+            sage: from sage.schemes.elliptic_curves.weierstrass_morphism import negation_morphism
+            sage: phi._set_post_isomorphism(negation_morphism(phi.codomain()))
             sage: old_ratl_maps == phi.rational_maps()
             False
             sage: old_ratl_maps[1] == -phi.rational_maps()[1]
@@ -2729,6 +2727,30 @@ def kernel_polynomial(self):
         return self.__kernel_polynomial
 
 
+    def is_separable(self):
+        r"""
+        Determine whether or not this isogeny is separable.
+
+        Since :class:`EllipticCurveIsogeny` only implements
+        separable isogenies, this method always returns ``True``.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
+            sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
+            sage: phi.is_separable()
+            True
+
+        ::
+
+            sage: E = EllipticCurve('11a1')
+            sage: phi = EllipticCurveIsogeny(E, E.torsion_points())
+            sage: phi.is_separable()
+            True
+        """
+        return True
+
+
     def _set_pre_isomorphism(self, preWI):
         """
         Modify this isogeny by pre-composing with a

src/sage/schemes/elliptic_curves/ell_generic.py

@@ -2324,6 +2324,7 @@ def multiplication_by_m_isogeny(self, m):
 
             sage: E = EllipticCurve('11a1')
             sage: E.multiplication_by_m_isogeny(7)
+            doctest:warning ... DeprecationWarning: ...
             Isogeny of degree 49 from Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field to Elliptic Curve defined by y^2 + y = x^3 - x^2 - 10*x - 20 over Rational Field
 
         TESTS:
@@ -2363,6 +2364,9 @@ def multiplication_by_m_isogeny(self, m):
             sage: all(mu(-m) == -mu(m) for m in (1,2,3,5,7))
             True
         """
+        from sage.misc.superseded import deprecation
+        deprecation(32826, 'The .multiplication_by_m_isogeny() method is superseded by .scalar_multiplication().')
+
         mx, my = self.multiplication_by_m(m)
 
         torsion_poly = self.torsion_polynomial(abs(m)).monic()
@@ -2377,6 +2381,29 @@ def multiplication_by_m_isogeny(self, m):
 
         assert False, 'bug in multiplication_by_m_isogeny()'
 
+    def scalar_multiplication(self, m):
+        r"""
+        Return the scalar-multiplication map `[m]` on this elliptic
+        curve as a
+        :class:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar`
+        object.
+
+        EXAMPLES::
+
+            sage: E = EllipticCurve('77a1')
+            sage: m = E.scalar_multiplication(-7); m
+            Scalar-multiplication endomorphism [-7] of Elliptic Curve defined by y^2 + y = x^3 + 2*x over Rational Field
+            sage: m.degree()
+            49
+            sage: P = E(2,3)
+            sage: m(P)
+            (-26/225 : -2132/3375 : 1)
+            sage: m.rational_maps() == E.multiplication_by_m(-7)
+            True
+        """
+        from sage.schemes.elliptic_curves.hom_scalar import EllipticCurveHom_scalar
+        return EllipticCurveHom_scalar(self, m)
+
     def isomorphism_to(self, other):
         """
         Given another weierstrass model ``other`` of self, return an
@@ -2489,9 +2516,9 @@ def automorphisms(self, field=None):
             ....:         continue
             ....:     break
             sage: Aut = E.automorphisms()
-            sage: Aut[0] == E.multiplication_by_m_isogeny(1)
+            sage: Aut[0] == E.scalar_multiplication(1)
             True
-            sage: Aut[1] == E.multiplication_by_m_isogeny(-1)
+            sage: Aut[1] == E.scalar_multiplication(-1)
             True
             sage: sorted(Aut) == Aut
             True

src/sage/schemes/elliptic_curves/hom.py

@@ -10,6 +10,7 @@
 - :class:`~sage.schemes.elliptic_curves.ell_curve_isogeny.EllipticCurveIsogeny`
 - :class:`~sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism`
 - :class:`~sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite`
+- :class:`~sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar`
 - :class:`~sage.schemes.elliptic_curves.hom_velusqrt.EllipticCurveHom_velusqrt`
 
 AUTHORS:
@@ -30,6 +31,8 @@
 from sage.rings.finite_rings import finite_field_base
 from sage.rings.number_field import number_field_base
 
+import sage.schemes.elliptic_curves.weierstrass_morphism as wm
+
 
 class EllipticCurveHom(Morphism):
     """
@@ -174,13 +177,14 @@ def _richcmp_(self, other, op):
 
         ::
 
-            sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
+            sage: from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism, identity_morphism
             sage: E = EllipticCurve([9,9])
             sage: F = E.change_ring(GF(71))
-            sage: wE = WeierstrassIsomorphism(E, (1,0,0,0))
-            sage: wF = WeierstrassIsomorphism(F, (1,0,0,0))
-            sage: mE = E.multiplication_by_m_isogeny(1)
+            sage: wE = identity_morphism(E)
+            sage: wF = identity_morphism(F)
+            sage: mE = E.scalar_multiplication(1)
             sage: mF = F.multiplication_by_m_isogeny(1)
+            doctest:warning ... DeprecationWarning: ...
             sage: [mE == wE, mF == wF]
             [True, True]
             sage: [a == b for a in (wE,mE) for b in (wF,mF)]
@@ -285,6 +289,7 @@ def kernel_polynomial(self):
         - :meth:`EllipticCurveIsogeny.kernel_polynomial`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.kernel_polynomial`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.kernel_polynomial`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.kernel_polynomial`
 
         TESTS::
 
@@ -305,6 +310,7 @@ def dual(self):
         - :meth:`EllipticCurveIsogeny.dual`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.dual`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.dual`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.dual`
 
         TESTS::
 
@@ -327,6 +333,7 @@ def rational_maps(self):
         - :meth:`EllipticCurveIsogeny.rational_maps`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.rational_maps`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.rational_maps`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.rational_maps`
 
         TESTS::
 
@@ -348,6 +355,7 @@ def x_rational_map(self):
         - :meth:`EllipticCurveIsogeny.x_rational_map`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.x_rational_map`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.x_rational_map`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.x_rational_map`
 
         TESTS::
 
@@ -377,6 +385,7 @@ def scaling_factor(self):
         - :meth:`EllipticCurveIsogeny.scaling_factor`
         - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.scaling_factor`
         - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.scaling_factor`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.scaling_factor`
 
         TESTS::
 
@@ -521,27 +530,22 @@ def is_separable(self):
         r"""
         Determine whether or not this morphism is separable.
 
-        .. NOTE::
-
-            This method currently always returns ``True`` as Sage does
-            not yet implement inseparable isogenies. This will probably
-            change in the future.
-
-        EXAMPLES::
+        Implemented by child classes. For examples, see:
 
-            sage: E = EllipticCurve(GF(17), [0,0,0,3,0])
-            sage: phi = EllipticCurveIsogeny(E,  E((0,0)))
-            sage: phi.is_separable()
-            True
+        - :meth:`EllipticCurveIsogeny.is_separable`
+        - :meth:`sage.schemes.elliptic_curves.weierstrass_morphism.WeierstrassIsomorphism.is_separable`
+        - :meth:`sage.schemes.elliptic_curves.hom_composite.EllipticCurveHom_composite.is_separable`
+        - :meth:`sage.schemes.elliptic_curves.hom_scalar.EllipticCurveHom_scalar.is_separable`
 
-        ::
+        TESTS::
 
-            sage: E = EllipticCurve('11a1')
-            sage: phi = EllipticCurveIsogeny(E, E.torsion_points())
-            sage: phi.is_separable()
-            True
+            sage: from sage.schemes.elliptic_curves.hom import EllipticCurveHom
+            sage: EllipticCurveHom.is_separable(None)
+            Traceback (most recent call last):
+            ...
+            NotImplementedError: ...
         """
-        return True
+        raise NotImplementedError('children must implement')
 
     def is_surjective(self):
         r"""
@@ -648,9 +652,7 @@ def __neg__(self):
             sage: psi.rational_maps() == (f, -g)
             True
         """
-        import sage.schemes.elliptic_curves.weierstrass_morphism as wm
-        a1,_,a3,_,_ = self.codomain().a_invariants()
-        return wm.WeierstrassIsomorphism(self.codomain(), (-1,0,-a1,-a3)) * self
+        return wm.negation_morphism(self.codomain()) * self
 
     @cached_method
     def __hash__(self):
@@ -745,7 +747,7 @@ def compare_via_evaluation(left, right):
         sage: from sage.schemes.elliptic_curves.hom_composite import EllipticCurveHom_composite
         sage: mu = EllipticCurveHom_composite.from_factors([phi, psi])
         sage: from sage.schemes.elliptic_curves.hom import compare_via_evaluation
-        sage: compare_via_evaluation(mu, E.multiplication_by_m_isogeny(7))
+        sage: compare_via_evaluation(mu, E.scalar_multiplication(7))
         True
 
     .. SEEALSO::

src/sage/schemes/elliptic_curves/hom_composite.py

@@ -87,7 +87,7 @@
 from sage.schemes.elliptic_curves.ell_generic import EllipticCurve_generic
 from sage.schemes.elliptic_curves.hom import EllipticCurveHom, compare_via_evaluation
 from sage.schemes.elliptic_curves.ell_curve_isogeny import EllipticCurveIsogeny
-from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism
+from sage.schemes.elliptic_curves.weierstrass_morphism import WeierstrassIsomorphism, identity_morphism
 
 # TODO: Implement sparse strategies? (cf. the SIKE cryptosystem)
 
@@ -272,7 +272,7 @@ def __init__(self, E, kernel, codomain=None, model=None):
         self._phis = _compute_factored_isogeny(kernel)
 
         if not self._phis:
-            self._phis = [WeierstrassIsomorphism(E, (1, 0, 0, 0))]
+            self._phis = [identity_morphism(E)]
 
         if model is not None:
             if codomain is not None:
@@ -350,7 +350,7 @@ def from_factors(cls, maps, E=None, strict=True):
         TESTS::
 
             sage: E = EllipticCurve('4730k1')
-            sage: EllipticCurveHom_composite.from_factors([], E) == E.multiplication_by_m_isogeny(1)
+            sage: EllipticCurveHom_composite.from_factors([], E) == E.scalar_multiplication(1)
             True
 
         ::
@@ -375,7 +375,7 @@ def from_factors(cls, maps, E=None, strict=True):
             E = phi.codomain()
 
         if not maps:
-            maps = (WeierstrassIsomorphism(E, (1,0,0,0)),)
+            maps = (identity_morphism(E),)
 
         if len(maps) == 1 and not strict:
             return maps[0]
@@ -563,7 +563,7 @@ def _comparison_impl(left, right, op):
             sage: psi = phi.codomain().isogeny(phi(Q))
             sage: psi = psi.codomain().isomorphism_to(E) * psi
             sage: comp = psi * phi
-            sage: mu = E.multiplication_by_m_isogeny(phi.degree())
+            sage: mu = E.scalar_multiplication(phi.degree())
             sage: sum(a*comp == mu for a in E.automorphisms())
             1
 
@@ -680,9 +680,9 @@ def dual(self):
             Composite morphism of degree 9 = 3^2:
               From: Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 28339*x + 59518 over Finite Field of size 65537
               To:   Elliptic Curve defined by y^2 + x*y + 3*y = x^3 + 2*x^2 + 4*x + 5 over Finite Field of size 65537
-            sage: psi * phi == phi.domain().multiplication_by_m_isogeny(phi.degree())
+            sage: psi * phi == phi.domain().scalar_multiplication(phi.degree())
             True
-            sage: phi * psi == psi.domain().multiplication_by_m_isogeny(psi.degree())
+            sage: phi * psi == psi.domain().scalar_multiplication(psi.degree())
             True
         """
         phis = (phi.dual() for phi in self._phis[::-1])