# Wrapper class for abelian groups#

This class is intended as a template for anything in Sage that needs the functionality of abelian groups. One can create an AdditiveAbelianGroupWrapper object from any given set of elements in some given parent, as long as an _add_ method has been defined.

EXAMPLES:

We create a toy example based on the Mordell-Weil group of an elliptic curve over $$\QQ$$:

sage: E = EllipticCurve('30a2')
sage: pts = [E(4,-7,1), E(7/4, -11/8, 1), E(3, -2, 1)]
sage: M = AdditiveAbelianGroupWrapper(pts.parent(), pts, [3, 2, 2])
sage: M
Additive abelian group isomorphic to Z/3 + Z/2 + Z/2 embedded in Abelian
group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - 19*x + 26
over Rational Field
sage: M.gens()
((4 : -7 : 1), (7/4 : -11/8 : 1), (3 : -2 : 1))
sage: 3*M.0
(0 : 1 : 0)
sage: 3000000000000001 * M.0
(4 : -7 : 1)
sage: M == loads(dumps(M))  # known bug, see https://github.com/sagemath/sage/issues/11599#comment:7
True


Todo

• Think about subgroups and quotients, which probably won’t work in the current implementation – some fiddly adjustments will be needed in order to be able to pass extra arguments to the subquotient’s init method.

AUTHORS:

This class is used to wrap a subgroup of an existing additive abelian group as a new additive abelian group.

EXAMPLES:

sage: G2 = AdditiveAbelianGroupWrapper(Zmod(42), , ); G2
Additive abelian group isomorphic to Z/21 embedded in Ring of integers modulo 42
sage: G6 = AdditiveAbelianGroupWrapper(Zmod(42), , ); G6
Additive abelian group isomorphic to Z/7 embedded in Ring of integers modulo 42
sage: G = AdditiveAbelianGroupWrapper(Zmod(42), [21,14,6], [2,3,7]); G
Additive abelian group isomorphic to Z/2 + Z/3 + Z/7 embedded in Ring of integers modulo 42
sage: G.invariants()
(42,)

sage: AdditiveAbelianGroupWrapper(QQbar, [sqrt(2), sqrt(3)], [0, 0])
Additive abelian group isomorphic to Z + Z embedded in Algebraic Field

sage: EllipticCurve(GF(419**2), [1,0]).abelian_group()  # indirect doctest
Additive abelian group isomorphic to Z/420 + Z/420 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + x over Finite Field in z2 of size 419^2

Element#
discrete_exp(v)#

Given a list (or other iterable) of length equal to the number of generators of this group, compute the element of the ambient group with those exponents in terms of the generators of self.

EXAMPLES:

sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(QQbar(2)), -1], [0, 0])
sage: v = G.discrete_exp([3, 5]); v
-0.7573593128807148?
sage: v.parent() is QQbar
True


This method is an inverse of discrete_log():

sage: orders = [2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*11]
sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), orders)
sage: el = A.random_element()
sage: A.discrete_exp(A.discrete_log(el)) == el
True

discrete_log(x, gens=None)#

Given an element of the ambient group, attempt to express it in terms of the generators of this group or the given generators of a subgroup.

ALGORITHM:

This reduces to p-groups, then calls _discrete_log_pgroup() which implements a basic version of the recursive algorithm from [Suth2008].

AUTHORS:

• Lorenz Panny (2017)

EXAMPLES:

sage: G = AdditiveAbelianGroup([2, 2*3, 2*3*5, 2*3*5*7, 2*3*5*7*11])
sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), [g.order() for g in G.gens()])
sage: A.discrete_log(A.discrete_exp([1,5,23,127,539]))
(1, 5, 23, 127, 539)

sage: x = polygen(ZZ, 'x')
sage: F.<t> = GF(1009**2, modulus=x**2+11); E = EllipticCurve(j=F(940))     # optional - sage.rings.finite_rings
sage: P, Q = E(900*t + 228, 974*t + 185), E(1007*t + 214, 865*t + 802)      # optional - sage.rings.finite_rings
sage: E.abelian_group().discrete_log(123 * P + 777 * Q, [P, Q])             # optional - sage.rings.finite_rings
(123, 777)

sage: V = Zmod(8)**2
sage: G = AdditiveAbelianGroupWrapper(V, [[2,2],[4,0]], [4, 2])
sage: G.discrete_log(V([6, 2]))
(1, 1)
sage: G.discrete_log(V([6, 4]))
Traceback (most recent call last):
...
TypeError: Not in group

sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(2)], )
sage: G.discrete_log(QQbar(2*sqrt(2)))
Traceback (most recent call last):
...
NotImplementedError: No black-box discrete log for infinite abelian groups

static from_generators(gens, universe=None)#

This method constructs the subgroup generated by a sequence of finite-order elements in an additive abelian group.

The elements need not be independent, hence this can be used to perform tasks such as finding relations between some given elements of an abelian group, computing the structure of the generated subgroup, enumerating all elements of the subgroup, and solving discrete-logarithm problems.

EXAMPLES:

sage: G = AdditiveAbelianGroup([15, 30, 45])
sage: gs = [G((1,2,3)), G((4,5,6)), G((7,7,7)), G((3,2,1))]
Additive abelian group isomorphic to Z/90 + Z/15 embedded in Additive abelian group isomorphic to Z/15 + Z/30 + Z/45
sage: H.gens()
((12, 13, 14), (1, 26, 21))

generator_orders()#

The orders of the generators with which this group was initialised. (Note that these are not necessarily a minimal set of generators.) Generators of infinite order are returned as 0. Compare self.invariants(), which returns the orders of a minimal set of generators.

EXAMPLES:

sage: V = Zmod(6)**2
sage: G = AdditiveAbelianGroupWrapper(V, [2*V.0, 3*V.1], [3, 2])
sage: G.generator_orders()
(3, 2)
sage: G.invariants()
(6,)

torsion_subgroup(n=None)#

Return the $$n$$-torsion subgroup of this additive abelian group when $$n$$ is given, and the torsion subgroup otherwise.

The [$$n$$-]torsion subgroup consists of all elements whose order is finite [and divides $$n$$].

EXAMPLES:

sage: ords = [2, 2*3, 2*3*5, 0, 2*3*5*7, 2*3*5*7*11]
sage: A = AdditiveAbelianGroupWrapper(G.0.parent(), G.gens(), ords)
sage: T = A.torsion_subgroup(5)
sage: T
Additive abelian group isomorphic to Z/5 + Z/5 + Z/5 embedded in Additive abelian group isomorphic to Z/2 + Z/6 + Z/30 + Z + Z/210 + Z/2310
sage: T.gens()
((0, 0, 6, 0, 0, 0), (0, 0, 0, 0, 42, 0), (0, 0, 0, 0, 0, 462))

sage: E = EllipticCurve(GF(487^2), [311,205])
sage: T = E.abelian_group().torsion_subgroup(42)
sage: T
Additive abelian group isomorphic to Z/42 + Z/6 embedded in Abelian group of points on Elliptic Curve defined by y^2 = x^3 + 311*x + 205 over Finite Field in z2 of size 487^2
sage: [P.order() for P in T.gens()]
[42, 6]

sage: E = EllipticCurve('574i1')
sage: pts = [E(103,172), E(61,18)]
sage: assert pts.order() == 7 and pts.order() == infinity
sage: M = AdditiveAbelianGroupWrapper(pts.parent(), pts, [7,0])
sage: M
Additive abelian group isomorphic to Z/7 + Z embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field
sage: M.torsion_subgroup()
Additive abelian group isomorphic to Z/7 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field
sage: M.torsion_subgroup(7)
Additive abelian group isomorphic to Z/7 embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field
sage: M.torsion_subgroup(5)
Trivial group embedded in Abelian group of points on Elliptic Curve defined by y^2 + x*y + y = x^3 - x^2 - 19353*x + 958713 over Rational Field


AUTHORS:

• Lorenz Panny (2022)

universe()#

The ambient group in which this abelian group lives.

EXAMPLES:

sage: G = AdditiveAbelianGroupWrapper(QQbar, [sqrt(QQbar(2)), sqrt(QQbar(3))], [0, 0])
sage: G.universe()
Algebraic Field


An element of an AdditiveAbelianGroupWrapper.

element()#

Return the underlying object that this element wraps.

EXAMPLES:

sage: T = EllipticCurve('65a').torsion_subgroup().gen(0)
sage: T; type(T)
(0 : 0 : 1)
<class 'sage.schemes.elliptic_curves.ell_torsion.EllipticCurveTorsionSubgroup_with_category.element_class'>
sage: T.element(); type(T.element())
(0 : 0 : 1)
<class 'sage.schemes.elliptic_curves.ell_point.EllipticCurvePoint_number_field'>


Bases: Morphism

The embedding into the ambient group. Used by the coercion framework.

Given a generating set of some finite abelian group (additively written), compute and return a basis of the group.

Note

A basis of a finite abelian group is a generating set $$\{g_1, \ldots, g_n\}$$ such that each element of the group can be written as a unique linear combination $$\alpha_1 g_1 + \cdots + \alpha_n g_n$$ with each $$\alpha_i \in \{0, \ldots, \mathrm{ord}(g_i)-1\}$$.

ALGORITHM: [Suth2007], Algorithm 9.1 & Remark 9.1

EXAMPLES:

sage: from sage.groups.additive_abelian.additive_abelian_wrapper import basis_from_generators
sage: E = EllipticCurve(GF(31337^6,'a'), j=37)
sage: E.order()
946988065073788930380545280
sage: (R,S), (ordR,ordS) = basis_from_generators(E.gens())
sage: ordR, ordS
(313157428926517503432720, 3024)
sage: R.order() == ordR
True
sage: S.order() == ordS
True
sage: ordR * ordS == E.order()
True
sage: R.weil_pairing(S, ordR).multiplicative_order() == ordS
True
sage: E.abelian_group().invariants()
(3024, 313157428926517503432720)