Digital expansions are one method for efficient implementations of scalar multiplication (or linear combinations) in Abelian groups, such as the point group of elliptic curves. One application is in public key cryptography, where scalar multiplication is a key ingredient. Let G be an Abelian group. A positive integer n ∈ Z can also be seen as an endomorphism of G by setting nQ = Q+ · · ·+Q (wit...

LetN be a zero-symmetric left near-ring, not necessarily with amultiplicative identity element; and letZ be its multiplicative center. DefineN to be 3-prime if for all a, b ∈ N\{0}, aNb / {0}; and callN 2-torsion-free if N, has no elements of order 2. A derivation onN is an additive endomorphism D of N such that D xy xD y D x y for all x, y ∈ N. A generalized derivation f with associated deriva...

We describe a method to perform scalar multiplication on two classes of ordinary elliptic curves, namely [Formula: see text] in prime characteristic [Formula: see text], and [Formula: see text] in prime characteristic [Formula: see text]. On these curves, the 4-th and 6-th roots of unity act as (computationally efficient) endomorphisms. In order to optimise the scalar multiplication, we conside...

In a recent paper,1 K. Asano gave a new proof of the theorem that a domain of integrity has a right quotient ring if and only if every pair of nonzero elements has a common nonzero right multiple. His method of proof is used in the present work to extend the centralizer of a ring over a module to a system of semi-endomorphisms of the module. From this extension, necessary and sufficient conditi...

By a careful investigation of the model theory of modules over a special class of uniserial domains we give some (counter) examples to a decomposition of a serial module. For instance there is a uniserial module M over a uniserial domain that is not quasi-small. Also there is a projective non–free countably generated module over the endomorphism ring of M . MSC: 16D70; 16D99; 03C60

A finite module M over a noetherian local ring R is said to be Gorenstein if Ext(k, M) = 0 for all i 6= dimR. A endomorphism φ : R → R of rings is called contracting if φ(m) ⊆ m for some i ≥ 1. Letting R denote the R-module R with action induced by φ, we prove: A finite R-module M is Gorenstein if and only if HomR( R,M) ∼= M and ExtiR( R,M) = 0 for 1 ≤ i ≤ depthR.

A depth two extension A | B is shown to be weak depth two over its double centralizer V A (V A (B)) if this is separable over B. We introduce a notion of codepth two coalgebra homomorphism g : C → D, dual to a depth two algebra homomorphism. It is shown that the endomorphism ring of bicomodule endomorphisms End D C D forms a right bialgebroid over the centralizer subalgebra g * : D * → C * of t...