The bottleneck in the proving algorithm of most elliptic-curve-based SNARK proof systems is Multi-Scalar-Multiplication (MSM) algorithm. In this paper we give an overview a variant Pippenger MSM together with set optimizations tailored for curves that admit twisted Edwards form. We prove case SNARK-friendly chains and cycles elliptic curves, which are useful recursive constructions. Our contrib...

let $r$ be a commutative ring and let $m$ be an $r$-module. we define the small intersection graph $g(m)$ of $m$ with all non-small proper submodules of $m$ as vertices and two distinct vertices $n, k$ are adjacent if and only if $ncap k$ is a non-small submodule of $m$. in this article, we investigate the interplay between the graph-theoretic properties of $g(m)$ and algebraic properties of $m...

Along with the rapid growth in the number of electric vehicles, there is urgent need to construct electric vehicles charging stations (EVCSs) to satisfy the charging demand. However, during the process of carrying out quantitative and qualitative analysis on location decisions, it is necessary to make clear the relationships and role between various factors which make impacts on charging statio...

let r be a commutative ring with non-zero identity and m be a unital r-module. then the concept of quasi-secondary submodules of m is introduced and some results concerning this class of submodules is obtained

let $g$ be a group with identity $e$. let $r$ be a $g$-graded commutative ring with a non-zero identity and $m$ be a graded $r$-module. in this article, we introduce the concept of graded almost semiprime submodules. also, we investigate some basic properties of graded almost semiprime and graded weakly semiprime submodules and give some characterizations of them.

‎let $pa$ be a commutative banach algebra and $ex$ be a left banach $pa$-module‎. ‎we study the set‎ ‎$dec_{pa}(ex)$ of all elements in $pa$ which induce a decomposable multiplication operator on $ex$‎. ‎in the case $ex=pa$‎, ‎$dec_{pa}(pa)$ is the well-known apostol algebra of $pa$‎. ‎we show that $dec_{pa}(ex)$ is intimately related with the largest spectrally separable subalgebra of $pa$ and...

In this paper secured wireless communication using fuzzy logic based high speed public-key cryptography (FLHSPKC) has been proposed by satisfying the major issues likes computational safety, power management and restricted usage of memory in wireless communication. Wireless Sensor Network (WSN) has several major constraints likes’ inadequate source of energy, restricted computational potentiali...

$r$-module. in this paper, we explore more properties of $max$-injective modules and we study some conditions under which the maximal spectrum of $m$ is a $max$-spectral space for its zariski topology.