Fast Dot Product over Finite Field
نویسنده
چکیده
Finite fields have great applications in various areas as cryptography, that is why it is important to have fast ways of computation to manipulate them. A first approach developed in this report lies in representing integers of the field using floating-point numbers, which lead to efficient computations. Operations in our case are done by restricting the characteristic p of the field to a floating-point mantissa: p − 1 < 2M−1. Taking advantage of error-free transformations on modern architectures, one can manage quite large finite fields exactly with floating-point arithmetic. After returning back to the basic of floating-point numbers, we introduce slightly different approaches to compute the dot product in an efficient way. In a second part, we have the same calculations done in a Residue Number System (RNS) over both integer and floating-point numbers. We show how this system can be efficient for well-chosen basis and present experimental results. Finally, we discuss how we parallelized our algorithms on a GPU card. Skalärprodukt i ändliga kroppar
منابع مشابه
Dot Product Representations of Graphs
We introduce the concept of dot product representations of graphs, giving some motivations as well as surveying the previously known results. We extend these representations to more general fields, looking at the complex numbers, rational numbers, and finite fields. Finally, we study the behavior of dot product representations in field extensions.
متن کاملEfficient dot product over word-size finite fields
We want to achieve efficiency for the exact computation of the dot product of two vectors over word size finite fields. We therefore compare the practical behaviors of a wide range of implementation techniques using different representations. The techniques used include floating point representations, discrete logarithms, tabulations, Montgomery reduction, delayed modulus.
متن کاملPinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates
An analog of the Falconer distance problem in vector spaces over finite fields asks for the threshold α > 0 such that |∆(E)| & q whenever |E| & q, where E ⊂ Fq , the d-dimensional vector space over a finite field with q elements (not necessarily prime). Here ∆(E) = {(x1 − y1) 2 + · · · + (xd − yd) 2 : x, y ∈ E}. The fourth listed author and Misha Rudnev ([20]) established the threshold d+1 2 , ...
متن کاملClassical Wavelet Transforms over Finite Fields
This article introduces a systematic study for computational aspects of classical wavelet transforms over finite fields using tools from computational harmonic analysis and also theoretical linear algebra. We present a concrete formulation for the Frobenius norm of the classical wavelet transforms over finite fields. It is shown that each vector defined over a finite field can be represented as...
متن کاملClassical wavelet systems over finite fields
This article presents an analytic approach to study admissibility conditions related to classical full wavelet systems over finite fields using tools from computational harmonic analysis and theoretical linear algebra. It is shown that for a large class of non-zero window signals (wavelets), the generated classical full wavelet systems constitute a frame whose canonical dual are classical full ...
متن کامل