let r be a commutative ring with identity. let n and k be two submodules of a multiplication r-module m. thenn=im and k=jm for some ideals i and j of r. the product of n and k denoted by nk is defined by nk=ijm. inthis paper we characterize some particular cases of multiplication modules by using the product of submodules.

This paper analyzes the best speeds that can be obtained for single-scalar multiplication with variable base point by combining a huge range of options: – many choices of coordinate systems and formulas for individual group operations, including new formulas for tripling on Edwards curves; – double-base chains with many different doubling/tripling ratios, including standard base-2 chains as an ...

A positional numeration system is given by a base and by a set of digits. The base is a real or complex number β such that |β| > 1, and the digit set A is a finite set of real or complex digits (including 0). Thus a number can be seen as a finite or infinite string of digits. An on-line algorithm processes the input piece-by-piece in a serial fashion. On-line arithmetic, introduced by Trivedi a...

Two-tone intermodulation in ultrahigh vacuum/chemical vapor deposition SiGe heterojunction bipolar transistors (HBTs) were analyzed using a Volterra-series-based approach that completely distinguishes individual nonlinearities. Avalanche multiplication and collector–base (CB) capacitance were shown to be the dominant nonlinearities in a single-stage common-emitter amplifier. At a given , an opt...

The aim of this research was to determine whether the well-documented link between heterosexual parents’ secure base support (i.e., sensitivity) and child behavior security) present among Mexican same-sex families with 1-to 6-year-old-children. The sample included 22 child-caregiver dyads from four ...

We present a new RNS modular multiplication for very large operands. The algorithm is based on Montgomery’s method adapted to residue arithmetic. By choosing the moduli of the RNS system reasonably large, an effect corresponding to a redundant high-radix implementation is achieved, due to the carry-free nature of residue arithmetic. The actual computation in the multiplication takes place in co...

This paper considers efficient scalar multiplication of elliptic curves over binary fields with a twofold purpose. Firstly, we derive the most efficient 3P formula in λ-projective coordinates and 5P formula in both affine and λ-projective coordinates. Secondly, extensive experiments have been conducted to test various multi-base scalar multiplication methods (e.g., greedy, ternary/binary, multi...

Multiplication of two numbers represented in base is shown to be computable by an on-line algorithm when is a negative integer, a positive non-integer real number, or a complex number of the form ipr, where r is a positive integer.

We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x −mx − x−1, m ∈ N, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas und...

We describe an algorithm for point multiplication on generic elliptic curves, based on a representation of the scalar as a sum of mixed powers of 2 and 3. The sparseness of this so-called double-base number system, combined with some efficient point tripling formulae, lead to efficient point multiplication algorithms for curves defined over both prime and binary fields. Side-channel resistance ...