scholarly journals Security of Prime Field Pairing Cryptoprocessor against Differential Power Attack

2011 ◽  
pp. 16-29 ◽  
Author(s):  
Santosh Ghosh ◽  
Dipanwita Roychowdhury
Keyword(s):  
Prime Field

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 the discrete logarithm problem for prime-field elliptic curves

2018 ◽  
Vol 51 ◽  
pp. 168-182 ◽  
Author(s):  
Alessandro Amadori ◽  
Federico Pintore ◽  
Massimiliano Sala
Keyword(s):  
Elliptic Curves ◽  
Discrete Logarithm ◽  
Discrete Logarithm Problem ◽  
Prime Field

scholarly journals A High-Speed Elliptic Curve Cryptography Processor for Teleoperated Systems Security

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Yong Xiao ◽  
Weibin Lin ◽  
Yun Zhao ◽  
Chao Cui ◽  
Ziwen Cai
Keyword(s):  
Elliptic Curve ◽  
Elliptic Curve Cryptography ◽  
High Speed ◽  
High Performance ◽  
Prime Field ◽  
Practical Applications ◽  
Cell Library ◽  
Cryptographic Algorithm ◽  
Human Operators ◽  
Key Length

Teleoperated robotic systems are those in which human operators control remote robots through a communication network. The deployment and integration of teleoperated robot’s systems in the medical operation have been hampered by many issues, such as safety concerns. Elliptic curve cryptography (ECC), an asymmetric cryptographic algorithm, is widely applied to practical applications because its far significantly reduced key length has the same level of security as RSA. The efficiency of ECC on GF (p) is dictated by two critical factors, namely, modular multiplication (MM) and point multiplication (PM) scheduling. In this paper, the high-performance ECC architecture of SM2 is presented. MM is composed of multiplication and modular reduction (MR) in the prime field. A two-stage modular reduction (TSMR) algorithm in the SCA-256 prime field is introduced to achieve low latency, which avoids more iterative subtraction operations than traditional algorithms. To cut down the run time, a schedule is put forward when exploiting the parallelism of multiplication and MR inside PM. Synthesized with a 0.13 um CMOS standard cell library, the proposed processor consumes 341.98k gate areas, and each PM takes 0.092 ms.

A high-speed RSD-based flexible ECC processor for arbitrary curves over general prime field

2018 ◽  
Vol 46 (10) ◽  
pp. 1858-1878 ◽  
Author(s):  
Yasir Ali Shah ◽  
Khalid Javeed ◽  
Shoaib Azmat ◽  
Xiaojun Wang
Keyword(s):  
High Speed ◽  
Prime Field

On the hit problem for the unstable $\mathscr A$-module $\mathcal P_5 = H^*((K(\mathbb F_2, 1))^{\times 5}, \mathbb F_2)$ and application

2021 ◽  
Author(s):  
Đặng Võ Phúc
Keyword(s):  
Algebraic Topology ◽  
Homotopy Theory ◽  
Polynomial Algebra ◽  
Steenrod Algebra ◽  
Minimal Set ◽  
Open Problems ◽  
Polynomial Algebras ◽  
Prime Field ◽  
Trivial Element ◽  
Hit Problem

Let us consider the prime field of two elements, $\mathbb F_2.$ One of the open problems in Algebraic topology is the hit problem for a module over the mod 2 Steenrod algebra $\mathscr A$. More specifically, this problem asks a minimal set of generators for the polynomial algebra $\mathcal P_m:=\mathbb F_2[x_1, x_2, \ldots, x_m]$ regarded as a connected unstable $\mathscr A$-module on $m$ variables $x_1, \ldots, x_m,$ each of degree one. The algebra $\mathcal P_m$ is the cohomology with $\mathbb F_2$-coefficients of the product of $m$ copies of the Eilenberg-MacLan space of type $(\mathbb F_2, 1).$ The hit problem has been thoroughly studied for 35 years in a variety of contexts by many authors and completely solved for $m\leq 4.$ Furthermore, it has been closely related to some classical problems in the homotopy theory and applied in studying the $m$-th Singer algebraic transfer $Tr^{\mathscr A}_m$ \cite{W.S1}. This transfer is one of the useful tools for studying the Adams $E^{2}$-term, ${\rm Ext}_{\mathscr A}^{*, *}(\mathbb F_2, \mathbb F_2) = H^{*, *}(\mathscr A, \mathbb F_2).$The aim of this work is to continue our study of the hit problem of five variables. At the same time, this result will be applied to the investigation of the fifth transfer of Singer and the modular representation of the general linear group of rank 5 over $\mathbb F_2.$ More precisely, we grew out of a previous result of us in \cite{D.P3} on the hit problem for $\mathscr A$-module $\mathcal P_5$ in the generic degree $5(2^t-1) + 18.2^t$ with $t$ an arbitrary non-negative integer. The result confirms Sum's conjecture \cite{N.S2} on the relation between the minimal set of $\mathscr A$-generators for the polynomial algebras $\mathcal P_{m-1}$ and $\mathcal P_{m}$ in the case $m=5$ and the above generic degree. Moreover, by using our result \cite{D.P3} and a presentation in the $\lambda$-algebra of $Tr_5^{\mathscr A}$, we show that the non-trivial element $h_1e_0 = h_0f_0\in {\rm Ext}_{\mathscr A}^{5, 5+(5(2^0-1) + 18.2^0)}(\mathbb F_2, \mathbb F_2)$ is in the image of the fifth transfer and that $Tr^{\mathscr A}_5$ is an isomorphism in the bidegree $(5, 5+(5(2^0-1) + 18.2^0)).$ In addition, the behavior of $Tr^{\mathscr A}_5$ in the bidegree $(5, 5+(5(2^t-1) + 18.2^t))$ when $t\geq 1$ was also discussed. This method is different from that of Singer in studying the image of the algebraic transfer.

Subgroups of Central Separable Algebras

1973 ◽  
Vol 25 (4) ◽  
pp. 881-887 ◽  
Author(s):  
E. D. Elgethun
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Division Ring ◽  
Multiplicative Group ◽  
Prime Field ◽  
Odd Order ◽  
Multiplicative Subgroup ◽  
A Finite Group ◽  
Separable Algebras

In [8] I. N. Herstein conjectured that all the finite odd order sub-groups of the multiplicative group in a division ring are cyclic. This conjecture was proved false in general by S. A. Amitsur in [1]. In his paper Amitsur classifies all finite groups which can appear as a multiplicative subgroup of a division ring. Let D be a division ring with prime field k and let G be a finite group isomorphic to a multiplicative subgroup of D.

Sign in / Sign up

Export Citation Format

Share Document