We show by an example that in a Verdier triangulated category, there may exist two mutually nonisomorphic Verdier octahedra containing the same commutative triangle.

The different methods used in the multiplication of numbers are long method (column method) multiplication, grid and addition multiplication. utilization mentioned to solve power (index) base number (number that multiply itself a times) horrendous, outrageous, tedious, time consuming too be carried out. This study develops computation using Kifilideen (matrix, combination distributive (MCD)) ap...

0 Introduction 7 0.1 Is being a 3-triangle characterised by 2-triangles? . . . . . . . . . . . . . . . . . . . 7 0.2 Is being an n-triangle characterised by (n− 1)-triangles? . . . . . . . . . . . . . . . 8 0.3 An appendix on transport of structure . . . . . . . . . . . . . . . . . . . . . . . . . 9 0.4 Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 ...

In the Grothendieck cohomology of a flag variety G/H there are two canonical additive bases, namely, the Demazure basis [D] and the Grothendieck basis [KK]. We present explicit formulae that reduce the multiplication of these basis elements to the Cartan numbers of G.

This paper investigates performance and energy characteristics of software algorithms for long integer arithmetic. We analyze and compare the number of RISC-like processor instructions (e.g. singleprecision multiplication, addition, load, and store instructions) required for the execution of different algorithms such as Schoolbook multiplication, Karatsuba and Comba multiplication, as well as M...

In this paper we present a new multiplication algorithm for residues modulo the Mersenne prime 2 − 1. Using this approach, on an Intel Haswell Core i7-4770, constant-time variable-base scalar multiplication on NIST’s (and SECG’s) curve P-521 requires 1,108,000 cycles, while on the recently proposed Edwards curve E-521 it requires just 943,000 cycles. As a comparison, on the same architecture op...

In this work we introduce a novel paradigm for the construction of ramp schemes with strong multiplication that allows the secret to be chosen in an extension field, whereas the shares lie in a base field. When applied to the setting of Shamir’s scheme, for example, this leads to a ramp scheme with strong multiplication from which protocols can be constructed for atomic secure multiplication wi...

In this work, we retake an old idea that Koblitz presented in his landmark paper [21], where he suggested the possibility of defining anomalous elliptic curves over the base field F4. We present a careful implementation of the base and quadratic field arithmetic required for computing the scalar multiplication operation in such curves. In order to achieve a fast reduction procedure, we adopted ...

This paper proposes a fast parallel Montgomery multiplication algorithm based on Residue Number Systems (RNS). It is easy to construct a fast modular exponentiation by applying the algorithm repeatedly. To realize an efficient RNS Montgomery multiplication, the main contribution of this paper is to provide a new RNS base extension algorithm. Cox-Rower Architecture described in this paper is a h...

Among the various arithmetic operations required in implementing public key cryptographic algorithms, the elliptic curve point multiplication has probably received the maximum attention from the research community in the last decade. Many methods for efficient and secure implementation of point multiplication have been proposed. The efficiency of these methods mainly depends on the representati...