Hi,
I find this discussion interesting since this was a project I
proposed
for magma, implemented by Paulette Lieby. The idea was to take any
finite abelian group G, create a wrapper A = GenericAbelianGroup(G),
passing in various parameters:
magma> GenericAbelianGroup;
Intrinsic 'GenericAbelianGroup'
Signatures:
(<Any> O) -> GrpAbGen
[
IdIntrinsic: MonStgElt,
AddIntrinsic: MonStgElt,
InverseIntrinsic: MonStgElt,
UseRepresentation: BoolElt,
Order: RngIntElt,
UserGenerators: SetEnum,
ProperSubset: BoolElt,
RandomIntrinsic: MonStgElt,
ComputeStructure: BoolElt,
UseUserGenerators: BoolElt,
PollardRhoRParam: RngIntElt,
PollardRhoTParam: RngIntElt,
PollardRhoVParam: RngIntElt
]
Creates a generic abelian group whose universe is O.
Additional information can be passed using the parameters.
Such a wrapper should allow the translation between multiplicative and
additive notation,
with the challenge to do so efficiently (any overhead could kill
efficiency). The idea I had
was precisely the waste I saw in dozens of independent implementations
of baby-step,
giant-step and Pollard rho algorithms (not called generic black box
groups for nothing!),
and their excessive support. Write once and re-use.
Creation of subclasses in SAGE or specific constructors should make it
easier to manage
the large number of parameters which can be passed in to optimise a
given instance.
An important note is that this infrastructure is used for group
calculations for Jacobians
of hyperelliptic curves over finite fields but it fails to take into
account the Weil pairing
which reduces the problem to cyclic group calculations! Discrete
logarithms in non-cyclic
groups are very expensive [relative to the group exponent]. They are
potentially also
dangerous since implementations assuming cyclicity can fail to
terminate on non-cyclic
input.
Thus, as a warning, ignoring useful information like the Weil pairing
and MOV reductions,
smooth factors in the group order, and pathologies like non-cyclic
groups, should not be
omitted by focusing on large prime order groups.
--David
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