Scalar multiplication

About: Scalar multiplication is a research topic. Over the lifetime, 1974 publications have been published within this topic receiving 31898 citations.

Efficient Algorithms for Pairing-Based Cryptosystems

Abstract: We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairing-based cryptography

Efficient algorithms for pairing-based cryptosystems

Abstract: We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics. We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over F p m, the latter technique being also useful in contexts other than that of pairing-based cryptography.

Linear Topological Spaces

Abstract: This chapter is largely preliminary in nature; it consists of a brief review of some of the terminology and the elementary theorems of general topology, an examination of the new concept “linear topological space” in terms of more familiar notions, and a comparison of this new concept with the mathematical objects of which it is an abstraction. After an introductory section on topology, we consider linear topological spaces, subspaces, quotient spaces, product spaces, and linear functions. With the exception of a few simple propositions relating to circled sets, these theorems are specializations of familiar results on topological groups (in other words, little use is made of scalar multiplication).

Geometric means in a novel vector space structure on symmetric positive-definite matrices

Abstract: In this work we present a new generalization of the geometric mean of positive numbers on symmetric positive‐definite matrices, called Log‐Euclidean. The approach is based on two novel algebraic structures on symmetric positive‐definite matrices: first, a lie group structure which is compatible with the usual algebraic properties of this matrix space; second, a new scalar multiplication that smoothly extends the Lie group structure into a vector space structure. From bi‐invariant metrics on the Lie group structure, we define the Log‐Euclidean mean from a Riemannian point of view. This notion coincides with the usual Euclidean mean associated with the novel vector space structure. Furthermore, this means corresponds to an arithmetic mean in the domain of matrix logarithms. We detail the invariance properties of this novel geometric mean and compare it to the recently introduced affine‐invariant mean. The two means have the same determinant and are equal in a number of cases, yet they are not identical in g...

Dataflow analysis of array and scalar references

Abstract: Given a program written in a simple imperative language (assignment statements,for loops, affine indices and loop limits), this paper presents an algorithm for analyzing the patterns along which values flow as the execution proceeds. For each array or scalar reference, the result is the name and iteration vector of the source statement as a function of the iteration vector of the referencing statement. The paper discusses several applications of the method: conversion of a program to a set of recurrence equations, array and scalar expansion, program verification and parallel program construction.