3 Arithmetic in Finite Fields
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Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography
Frequency Domain Finite Field Arithmetic for Elliptic Curve Cryptography by Sel»cukBakt³r A Dissertation Submitted to the Faculty of the Worcester Polytechnic Institute in partial ful¯llment of the requirements for the Degree of Doctor of Philosophy in Electrical and Computer Engineering by April, 2008 Approved: Dr. Berk Sunar Dr. Stanley Selkow Dissertation Advisor Dissertation Committee ECE Department Computer Science Department Dr. Kaveh Pahlavan Dr. Fred J. Looft Dissertation Committee Department Head ECE Department ECE Department °c Copyright by Sel»cukBakt³r All rights reserved. April, 2008 i To my family; to my parents Ay»seand Mehmet Bakt³r, to my sisters Elif and Zeynep, and to my brothers Selim and O¸guz ii Abstract E±cient implementation of the number theoretic transform (NTT), also known as the discrete Fourier transform (DFT) over a ¯nite ¯eld, has been studied actively for decades and found many applications in digital signal processing. In 1971 SchÄonhageand Strassen proposed an NTT based asymptotically fast multiplication method with the asymptotic complexity O(m log m log log m) for multiplication of m-bit integers or (m ¡ 1)st degree polynomials. SchÄonhageand Strassen's algorithm was known to be the asymptotically fastest multipli- cation algorithm until FÄurerimproved upon it in 2007. However, unfortunately, both al- gorithms bear signi¯cant overhead due to the conversions between the time and frequency domains which makes them impractical for small operands, e.g. less than 1000 bits in length as used in many applications. With this work we investigate for the ¯rst time the practical application of the NTT, which found applications in digital signal processing, to ¯nite ¯eld multiplication with an emphasis on elliptic curve cryptography (ECC). -
Convolution.Pdf
Convolution March 22, 2020 0.0.1 Setting up [23]: from bokeh.plotting import figure from bokeh.io import output_notebook, show,curdoc from bokeh.themes import Theme import numpy as np output_notebook() theme=Theme(json={}) curdoc().theme=theme 0.1 Convolution Convolution is a fundamental operation that appears in many different contexts in both pure and applied settings in mathematics. Polynomials: a simple case Let’s look at a simple case first. Consider the space of real valued polynomial functions P. If f 2 P, we have X1 i f(z) = aiz i=0 where ai = 0 for i sufficiently large. There is an obvious linear map a : P! c00(R) where c00(R) is the space of sequences that are eventually zero. i For each integer i, we have a projection map ai : c00(R) ! R so that ai(f) is the coefficient of z for f 2 P. Since P is a ring under addition and multiplication of functions, these operations must be reflected on the other side of the map a. 1 Clearly we can add sequences in c00(R) componentwise, and since addition of polynomials is also componentwise we have: a(f + g) = a(f) + a(g) What about multiplication? We know that X Xn X1 an(fg) = ar(f)as(g) = ar(f)an−r(g) = ar(f)an−r(g) r+s=n r=0 r=0 where we adopt the convention that ai(f) = 0 if i < 0. Given (a0; a1;:::) and (b0; b1;:::) in c00(R), define a ∗ b = (c0; c1;:::) where X1 cn = ajbn−j j=0 with the same convention that ai = bi = 0 if i < 0. -
Tensor Products of F-Algebras
Mediterr. J. Math. (2017) 14:63 DOI 10.1007/s00009-017-0841-x c The Author(s) 2017 This article is published with open access at Springerlink.com Tensor Products of f-algebras G. J. H. M. Buskes and A. W. Wickstead Abstract. We construct the tensor product for f-algebras, including proving a universal property for it, and investigate how it preserves algebraic properties of the factors. Mathematics Subject Classification. 06A70. Keywords. f-algebras, Tensor products. 1. Introduction An f-algebra is a vector lattice A with an associative multiplication , such that 1. a, b ∈ A+ ⇒ ab∈ A+ 2. If a, b, c ∈ A+ with a ∧ b = 0 then (ac) ∧ b =(ca) ∧ b =0. We are only concerned in this paper with Archimedean f-algebras so hence- forth, all f-algebras and, indeed, all vector lattices are assumed to be Archimedean. In this paper, we construct the tensor product of two f-algebras A and B. To be precise, we show that there is a natural way in which the Frem- lin Archimedean vector lattice tensor product A⊗B can be endowed with an f-algebra multiplication. This tensor product has the expected univer- sal property, at least as far as order bounded algebra homomorphisms are concerned. We also show that, unlike the case for order theoretic properties, algebraic properties behave well under this product. Again, let us emphasise here that algebra homomorphisms are not assumed to map an identity to an identity. Our approach is very much representational, however, much that might annoy some purists. In §2, we derive the representations needed, which are very mild improvements on results already in the literature. -
Convolution Products of Monodiffric Functions
View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Elsevier - Publisher Connector JOC‘RNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS 37, 271-287 (1972) Convolution Products of Monodiffric Functions GEORGE BERZSENYI Lamar University, Beaumont, Texas 77710 Submitted by R. J. Dufin As it has been repeatedly pointed out by mathematicians working in discrete function theory, it is both essential and difficult to find discrete analogs for the ordinary pointwise product operation of analytic functions of a complex variable. R. P. Isaacs [Ill, the initiator of monodiffric function theory, is to be credited with the first results in this direction. Later G. J. Kurowski [13] introduced some new monodiffric analogues of pointwise multiplication while the present author [2] succeeded in sharpening and generalizing Isaacs’ and Kurowski’s results. However, all of these attempts were mainly concerned with trying to preserve as many of the pointwise aspects as possible. Consequently, each of these operations proved to be less than satisfactory either from the algebraic or from the function theoretic viewpoint. In the present paper, a different approach is taken. Influenced by the suc- cess of R. J. Duffin and C. S. Duris [5] in a related area, we utilize the line integrals introduced in Ref. [l] and the conjoint line integral of Isaacs [l I] to define several convolution-type product operations. It is shown that if D is a suitably restricted discrete domain then monodiffricity is preserved by each of these operations. Further algebraic and function theoretic properties of these operations are investigated and some examples of their applicability are given. -
Acceleration of Finite Field Arithmetic with an Application to Reverse Engineering Genetic Networks
ACCELERATION OF FINITE FIELD ARITHMETIC WITH AN APPLICATION TO REVERSE ENGINEERING GENETIC NETWORKS by Edgar Ferrer Moreno A thesis submitted in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY in COMPUTING AND INFORMATION SCIENCES AND ENGINEERING UNIVERSITY OF PUERTO RICO MAYAGÜEZ CAMPUS 2008 Approved by: ________________________________ __________________ Omar Colón, PhD Date Member, Graduate Committee ________________________________ __________________ Oscar Moreno, PhD Date Member, Graduate Committee ________________________________ __________________ Nayda Santiago, PhD Date Member, Graduate Committee ________________________________ __________________ Dorothy Bollman, PhD Date President, Graduate Committee ________________________________ __________________ Edusmildo Orozco, PhD Date Representative of Graduate Studies ________________________________ __________________ Néstor Rodríguez, PhD Date Chairperson of the Department Abstract Finite field arithmetic plays an important role in a wide range of applications. This research is originally motivated by an application of computational biology where genetic networks are modeled by means of finite fields. Nonetheless, this work has application in various research fields including digital signal processing, error correct- ing codes, Reed-Solomon encoders/decoders, elliptic curve cryptosystems, or compu- tational and algorithmic aspects of commutative algebra. We present a set of efficient algorithms for finite field arithmetic over GF (2m), which are implemented -
Vector Space Theory
Vector Space Theory A course for second year students by Robert Howlett typesetting by TEX Contents Copyright University of Sydney Chapter 1: Preliminaries 1 §1a Logic and commonSchool sense of Mathematics and Statistics 1 §1b Sets and functions 3 §1c Relations 7 §1d Fields 10 Chapter 2: Matrices, row vectors and column vectors 18 §2a Matrix operations 18 §2b Simultaneous equations 24 §2c Partial pivoting 29 §2d Elementary matrices 32 §2e Determinants 35 §2f Introduction to eigenvalues 38 Chapter 3: Introduction to vector spaces 49 §3a Linearity 49 §3b Vector axioms 52 §3c Trivial consequences of the axioms 61 §3d Subspaces 63 §3e Linear combinations 71 Chapter 4: The structure of abstract vector spaces 81 §4a Preliminary lemmas 81 §4b Basis theorems 85 §4c The Replacement Lemma 86 §4d Two properties of linear transformations 91 §4e Coordinates relative to a basis 93 Chapter 5: Inner Product Spaces 99 §5a The inner product axioms 99 §5b Orthogonal projection 106 §5c Orthogonal and unitary transformations 116 §5d Quadratic forms 121 iii Chapter 6: Relationships between spaces 129 §6a Isomorphism 129 §6b Direct sums Copyright University of Sydney 134 §6c Quotient spaces 139 §6d The dual spaceSchool of Mathematics and Statistics 142 Chapter 7: Matrices and Linear Transformations 148 §7a The matrix of a linear transformation 148 §7b Multiplication of transformations and matrices 153 §7c The Main Theorem on Linear Transformations 157 §7d Rank and nullity of matrices 161 Chapter 8: Permutations and determinants 171 §8a Permutations 171 §8b Determinants -
C*-Algebraic Schur Product Theorem, P\'{O} Lya-Szeg\H {O}-Rudin Question and Novak's Conjecture
C*-ALGEBRAIC SCHUR PRODUCT THEOREM, POLYA-SZEG´ O-RUDIN˝ QUESTION AND NOVAK’S CONJECTURE K. MAHESH KRISHNA Department of Humanities and Basic Sciences Aditya College of Engineering and Technology Surampalem, East-Godavari Andhra Pradesh 533 437 India Email: [email protected] August 17, 2021 Abstract: Striking result of Vyb´ıral [Adv. Math. 2020] says that Schur product of positive matrices is bounded below by the size of the matrix and the row sums of Schur product. Vyb´ıral used this result to prove the Novak’s conjecture. In this paper, we define Schur product of matrices over arbitrary C*- algebras and derive the results of Schur and Vyb´ıral. As an application, we state C*-algebraic version of Novak’s conjecture and solve it for commutative unital C*-algebras. We formulate P´olya-Szeg˝o-Rudin question for the C*-algebraic Schur product of positive matrices. Keywords: Schur/Hadamard product, Positive matrix, Hilbert C*-module, C*-algebra, Schur product theorem, P´olya-Szeg˝o-Rudin question, Novak’s conjecture. Mathematics Subject Classification (2020): 15B48, 46L05, 46L08. Contents 1. Introduction 1 2. C*-algebraic Schur product, Schur product theorem and P´olya-Szeg˝o-Rudin question 3 3. Lower bounds for C*-algebraic Schur product 7 4. C*-algebraic Novak’s conjecture 9 5. Acknowledgements 11 References 11 1. Introduction Given matrices A := [aj,k]1≤j,k≤n and B := [bj,k]1≤j,k≤n in the matrix ring Mn(K), where K = R or C, the Schur/Hadamard/pointwise product of A and B is defined as (1) A ◦ B := aj,kbj,k . -
Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF(2 M)
Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF (2m) using polynomial bases by Fan Zhou B.Eng., Zhejiang University, 2013 A Report Submitted in Partial Fulfillment of the Requirements for the Degree of MASTER OF ENGINEERING in the Department of Electrical and Computer Engineering c Fan Zhou, 2018 University of Victoria All rights reserved. This report may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Study of Extended Euclidean and Itoh-Tsujii Algorithms in GF (2m) using polynomial bases by Fan Zhou B.Eng., Zhejiang University, 2013 Supervisory Committee Dr. Fayez Gebali, Supervisor (Department of Electrical and Computer Engineering) Dr. Watheq El-Kharashi, Departmental Member (Department of Electrical and Computer Engineering) iii ABSTRACT Finite field arithmetic is important for the field of information security. The inversion operation consumes most of the time and resources among all finite field arithmetic operations. In this report, two main classes of algorithms for inversion are studied. The first class of inverters is Extended Euclidean based inverters. Extended Euclidean Algorithm is an extension of Euclidean algorithm that computes the greatest common divisor. The other class of inverters is based on Fermat's little theorem. This class of inverters is also called multiplicative based inverters, because, in these algorithms, the inversion is performed by a sequence of multiplication and squaring. This report represents a literature review of inversion algorithm and implements a multiplicative based inverter and an Extended Euclidean based inverter in MATLAB. The experi- mental results show that inverters based on Extended Euclidean Algorithm are more efficient than inverters based on Fermat's little theorem. -
Matrix Theory” Involves the Use of Matrices and Operations Defined on Them As Conceptual Data Structures for Understanding Various Ideas in Algebra and Combinatorics
Matrix Canonical Forms notational skills and proof techniques S. Gill Williamson arXiv:1407.5130v1 [math.HO] 18 Jul 2014 ©S. Gill Williamson 2012. All rights reserved. Preface This material is a rewriting of notes handed out by me to beginning graduate students in seminars in combinatorial mathematics (Department of Mathematics, University of California San Diego). Topics covered in this seminar were in algebraic and algorithmic combinatorics. Solid skills in linear and multilinear algebra were required of students in these seminars - especially in algebraic combinatorics. I developed these notes to review the students' undergraduate linear algebra and improve their proof skills. We focused on a careful development of the general matrix canonical forms as a training ground. I would like to thank Dr. Tony Trojanowski for a careful reading of this material and numerous corrections and helpful suggestions. I would also like to thank Professor Mike Sharpe, UCSD Department of Mathematics, for considerable LaTeX typesetting assistance and for his Linux Libertine font options to the newtxmath package. S. Gill Williamson, 2012 http://cseweb.ucsd.edu/∼gill 3 x 4 CONTENTS Chapter 1. Functions and Permutations7 Algebraic terminology7 Sets, lists, multisets and functions 14 Permutations 19 Exercises 24 Chapter 2. Matrices and Vector Spaces 27 Review 27 Exercises: subspaces 30 Exercises: spanning sets and dimension 30 Matrices { basic stuff 33 Chapter 3. Determinants 39 Elementary properties of determinants 41 Laplace expansion theorem 49 Cauchy-Binet theorem 54 Exercises: Cauchy Binet and Laplace 59 Chapter 4. Hermite/echelon forms 61 Row equivalence 61 Hermite form, canonical forms and uniqueness 67 Stabilizers of GL(n; K); column Hermite forms 73 Finite dimensional vector spaces and Hermite forms 76 Diagonal canonical forms { Smith form 81 Chapter 5. -
3 Finite Field Arithmetic
18.783 Elliptic Curves Spring 2021 Lecture #3 02/24/2021 3 Finite field arithmetic In order to perform explicit computations with elliptic curves over finite fields, we first need to understand how to compute in finite fields. In many of the applications we will consider, the finite fields involved will be quite large, so it is important to understand the computational complexity of finite field operations. This is a huge topic, one to which an entire course could be devoted, but we will spend just one or two lectures on this topic, with the goal of understanding the most commonly used algorithms and analyzing their asymptotic complexity. This will force us to omit many details, but references to the relevant literature will be provided for those who want to learn more. Our first step is to fix an explicit representation of finite field elements. This might seem like a technical detail, but it is actually quite crucial; questions of computational complexity are meaningless otherwise. Example 3.1. By Theorem 3.12 below, the multiplicative group of a finite field Fq is cyclic. One way to represent the nonzero elements of a finite field is as explicit powers of a fixed generator, in which case it is enough to know the exponent, an integer in [0; q − 1]. With this representation multiplication and division are easy, solving the discrete logarithm problem is trivial, but addition is costly (not even polynomial time). We will instead choose a representation that makes addition (and subtraction) very easy, multiplication slightly harder but still easy, division slightly harder than multiplication but still easy (all these operations take quasi-linear time). -
Finite Field Arithmetic (Galois Field)
ASS.PROF.DR Thamer Information Theory 4th Class in Communication Finite Field Arithmetic (Galois field) Introduction: A finite field is also often known as a Galois field, after the French mathematician Pierre Galois. A Galois field in which the elements can take q different values is referred to as GF(q). The formal properties of a finite field are: (a) There are two defined operations, namely addition and multiplication. (b) The result of adding or multiplying two elements from the field is always an element in the field. (c) One element of the field is the element zero, such that a + 0 = a for any element a in the field. (d) One element of the field is unity, such that a • 1 = a for any element a in the field. (e) For every element a in the field, there is an additive inverse element -a, such that a + ( - a) = 0. This allows the operation of subtraction to be defined as addition of the inverse. (f) For every non-zero element b in the field there is a multiplicative inverse element b-1 such that b b-1= 1. This allows the operation of division to be defined as multiplication by the inverse. (g) The associative [a + (b + c) = (a + b) + c, a • (b • c) = [(a • b) • c], commutative [a + b = b + a, a • b = b • a], and distributive [a • (b + c) = a • b + a • c] laws apply. 1 ASS.PROF.DR Thamer Information Theory 4th Class in Communication These properties cannot be satisfied for all possible field sizes. They can, however, be satisfied if the field size is any prime number or any integer power of a prime. -
On a New Method for Defining the Norm of Fourier-Stieltjes Algebras
Pacific Journal of Mathematics ON A NEW METHOD FOR DEFINING THE NORM OF FOURIER-STIELTJES ALGEBRAS MARTIN E. WALTER Vol. 137, No. 1 January 1989 PACIFIC JOURNAL OF MATHEMATICS Vol. 137, No. 1, 1989 ON A NEW METHOD FOR DEFINING THE NORM OF FOURIER-STIELTJES ALGEBRAS MARTIN E. WALTER This paper is dedicated to the memory of Henry Abel Dye P. Eymard equipped B(G), the Fourier-Stieltjes algebra of a locally compact group G, with a norm by considering it the dual Banach space of bounded linear functional on another Banach space, namely the universal C*-algebra, C*(G). We show that B(G) can be given the exact same norm if it is considered as a Banach subalgebra of 3f(C*(G))9 the Banach algebra of completely bounded maps of C* (G) into itself equipped with the completely bounded norm. We show here how the latter approach leads to a duality theory for finite (and, more generally, discrete) groups which is not available if one restricts attention to the "linear functional" [as opposed to the "completely bounded map99] approach. 1. Preliminaries. For most of this paper G will be a finite (discrete) group, although we consider countably infinite discrete groups in the last section. We now summarize certain aspects of duality theory that we will need. The discussion is for a general locally compact group G whenever no simplification is achieved by assuming G to be finite. The reader familiar with Fourier and Fourier-Stieltjes algebras may proceed to §2 and refer back to this section as might be necessary.