scholarly journals Some Identities on the Character Sum Containing x(x - 1)(x - λ)

1971 ◽  
Vol 42 ◽  
pp. 109-113 ◽  
Author(s):  
Masatoshi Yamauchi
Keyword(s):  
Character Sum ◽  
Legendre Symbol ◽  
Prime Field

Let Fp be the prime field of characteristic p (p: an odd prime), and put = Fp - {0, 1}. Then for λ ∈ we define,where denotes the Legendre symbol, and consider the sum

scholarly journals A note on extensions of multilinear maps defined on multilinear varieties

2021 ◽  
pp. 1-26
Author(s):  
W. T. Gowers ◽  
L. Milićević
Keyword(s):  
Finite Fields ◽  
Vector Spaces ◽  
Restriction Map ◽  
Zero Set ◽  
Dimensional Vector ◽  
Prime Field ◽  
Finite Dimensional ◽  
Multilinear Maps ◽  
Multilinear Map

Abstract Let $G_1, \ldots , G_k$ be finite-dimensional vector spaces over a prime field $\mathbb {F}_p$ . A multilinear variety of codimension at most $d$ is a subset of $G_1 \times \cdots \times G_k$ defined as the zero set of $d$ forms, each of which is multilinear on some subset of the coordinates. A map $\phi$ defined on a multilinear variety $B$ is multilinear if for each coordinate $c$ and all choices of $x_i \in G_i$ , $i\not =c$ , the restriction map $y \mapsto \phi (x_1, \ldots , x_{c-1}, y, x_{c+1}, \ldots , x_k)$ is linear where defined. In this note, we show that a multilinear map defined on a multilinear variety of codimension at most $d$ coincides on a multilinear variety of codimension $O_{k}(d^{O_{k}(1)})$ with a multilinear map defined on the whole of $G_1\times \cdots \times G_k$ . Additionally, in the case of general finite fields, we deduce similar (but slightly weaker) results.

Construction of systems of polynomial equations with exact p-Divisibility via the covering method

2014 ◽  
Vol 13 (06) ◽  
pp. 1450013 ◽  
Author(s):  
Francis N. Castro ◽  
Ivelisse M. Rubio
Keyword(s):  
Exponential Sums ◽  
Polynomial Equations ◽  
Prime Field ◽  
Elementary Method ◽  
Covering Method ◽  
Systems Of Polynomial Equations

We present an elementary method to compute the exact p-divisibility of exponential sums of systems of polynomial equations over the prime field. Our results extend results by Carlitz and provide concrete and simple conditions to construct families of polynomial equations that are solvable over the prime field.

scholarly journals On algebraic closure in pseudofinite fields

2012 ◽  
Vol 77 (4) ◽  
pp. 1057-1066 ◽  
Author(s):  
Özlem Beyarslan ◽  
Ehud Hrushovski
Keyword(s):  
Automorphism Group ◽  
Finite Field ◽  
Algebraic Closure ◽  
Roots Of Unity ◽  
Prime Field ◽  
Almost All ◽  
Pseudofinite Fields

AbstractWe study the automorphism group of the algebraic closure of a substructureAof a pseudo-finite fieldF. We show that the behavior of this group, even whenAis large, depends essentially on the roots of unity inF. For almost all completions of the theory of pseudofinite fields, we show that overA, algebraic closure agrees with definable closure, as soon asAcontains the relative algebraic closure of the prime field.

MicroBlaze-Based Multiprocessor Embedded Cryptosystem on FPGA for Elliptic Curve Scalar Multiplication Over Fp

2019 ◽  
Vol 28 (03) ◽  
pp. 1950037 ◽  
Author(s):  
A. Bellemou ◽  
N. Benblidia ◽  
M. Anane ◽  
M. Issad
Keyword(s):  
Elliptic Curve ◽  
Scalar Multiplication ◽  
Security Level ◽  
Prime Field ◽  
Montgomery Modular Multiplication ◽  
Parallel Implementations ◽  
High Radix ◽  
System On Programmable Chip ◽  
Xilinx Fpga ◽  
Elliptic Curve Scalar Multiplication

In this paper, we present Microblaze-based parallel architectures of Elliptic Curve Scalar Multiplication (ECSM) computation for embedded Elliptic Curve Cryptosystem (ECC) on Xilinx FPGA. The proposed implementations support arbitrary Elliptic Curve (EC) forms defined over large prime field ([Formula: see text]) with different security-level sizes. ECSM is performed using Montgomery Power Ladder (MPL) algorithm in Chudnovsky projective coordinates system. At the low abstraction level, Montgomery Modular Multiplication (MMM) is considered as the critical operation. It is implemented within a hardware Accelerator MMM (AccMMM) core based on the modified high radix, [Formula: see text] MMM algorithm. The efficiency of our parallel implementations is achieved by the combination of the mixed SW/HW approach with Multi Processor System on Programmable Chip (MPSoPC) design. The integration of multi MicroBlaze processor in single architecture allows not only the flexibility of the overall system but also the exploitation of the parallelism in ECSM computation with several degrees. The Virtex-5 parallel implementations of 256-bit and 521-bis ECSM computations run at 100[Formula: see text]MHZ frequency and consume between 2,739 and 6,533 slices, 22 and 72 RAMs and between 16 and 48 DSP48E cores. For the considered security-level sizes, the delays to perform single ECSM are between 115[Formula: see text]ms and 14.72[Formula: see text]ms.

scholarly journals ON SOLUTIONS TO SOME POLYNOMIAL CONGRUENCES IN SMALL BOXES

2013 ◽  
Vol 89 (2) ◽  
pp. 300-307
Author(s):  
IGOR E. SHPARLINSKI
Keyword(s):  
Exponential Sums ◽  
Polynomial Systems ◽  
Side Length ◽  
Character Sum ◽  
Number Of Solutions ◽  
Polynomial Congruences

AbstractWe use bounds of mixed character sum to study the distribution of solutions to certain polynomial systems of congruences modulo a prime $p$. In particular, we obtain nontrivial results about the number of solutions in boxes with the side length below ${p}^{1/ 2} $, which seems to be the limit of more general methods based on the bounds of exponential sums along varieties.

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