In this paper we present a right version of the algorithms developed for to
compute Gröbner bases over bijective skew PBW extensions in the left case
given in [3]. In particular, we adapt the theory of reduction and we build a
right division algorithm and generate a right version of Buchberger algorithm
over bijective skew PBW extensions, finally we illustrate some examples using
the SPBWE.lib library implemented in Maple (see [1], [4]). It is important to
note that the development of this theory is fundamental to complete the
SPBWE.lib library and to be able to develop many of the homological
applications that arise as result of obtaining the right Gröbner bases over
skew PBW extensions.
This paper presents method for obtaining high-degree compression functions
using natural symmetries in a given model of an elliptic curve. Such symmetries
may be found using symmetry of involution $[-1]$ and symmetry of translation
morphism $\tau_T=P+T$, where $T$ is the $n$-torsion point which naturally
belongs to the $E(\mathbb K)$ for a given elliptic curve model. We will study
alternative models of elliptic curves with points of order $2$ and $4$, and
specifically Huff's curves and the Hessian family of elliptic curves (like
Hessian, twisted Hessian and generalized Hessian curves) with a point of order
$3$. We bring up some known compression functions on those models and present
new ones as well. For (almost) every presented compression function,
differential addition and point doubling formulas are shown. As in the case of
high-degree compression functions manual investigation of differential addition
and doubling formulas is very difficult, we came up with a Magma program which
relies on the Gröbner basis. We prove that if for a model $E$ of an elliptic
curve exists an isomorphism $\phi:E \to E_M$, where $E_M$ is the Montgomery
curve and for any $P \in E(\mathbb K)$ holds that $\phi(P)=(\phi_x(P),
\phi_y(P))$, then for a model $E$ one may find compression function of degree
$2$. Moreover, one may find, defined for this compression function,
differential addition and doubling formulas of the same efficiency as
Montgomery's. However, it seems that for the family of […]
We develop a multiset query and update language executable in a term
rewriting system. Its most remarkable feature, besides non-standard approach to
quantification and introduction of fresh values, is non-determinism - a query
result is not uniquely determined by the database. We argue that this feature
is very useful, e.g., in modelling user choices during simulation or
reachability analysis of a data-centric business process - the intended
application of our work. Query evaluation is implemented by converting the
query into a terminating term rewriting system and normalizing the initial term
which encapsulates the current database. A normal form encapsulates a query
result. We prove that our language can express any relational algebra query.
Finally, we present a simple business process specification framework (and an
example specification). Both syntax and semantics of our query language is
implemented in Maude.