Discrete logarithms in finite fields and their cryptographic significance
A. M. Odlyzko
AT&T Bell Laboratories
Murray Hill, New Jersey 07974
ABSTRACT
Given a primitive element g of a finite field GF(q), the discrete logarithm of a nonzero element
u GF(q)
QUADRATIC RECIPROCITY IN CHARACTERISTIC 2
KEITH CONRAD
1. Introduction
Let F be a finite field. When F has odd characteristic, the quadratic reciprocity law
in F[T ] lets us decide whether or not a quadratic congruence f x2 mod is solvable,
where the modu
TRACE AND NORM
KEITH CONRAD
1. Introduction
Let L/K be a finite extension of fields, with n = [L : K]. We will associate to this
extension two important functions L K, called the trace and the norm. They are related
to the trace and determinant of matrice
0.2.
Problems
2.1 Construct the group under modulo-6 addition.
2.2 Construct the group under modulo-3 multiplication.
2.3 Let in be a positive integer. If in is not a prime, prove that the set cfw_1, 2, , m1 is not a group
under modulo-m multiplication.
2
Chapter 1
Finite Fields
1.1
Introduction
Finite fields are one of the essential building blocks in coding theory and cryptography
and thus appear in many areas in IT security. This section introduces finite fields systematically stating for which orders f
Chapter 5
Automorphisms and bases
10 Automorphisms
In this chapter, we will once again adopt the viewpoint that a finite extension F = Fqm of a finite
field K = Fq is a vector space of dimension m over K.
In Theorem 7.3 we saw that the set of roots of an
Chapter 3
Finite fields
We have seen, in the previous chapters, some examples of finite fields. For example, the residue
class ring Z/pZ (when p is a prime) forms a field with p elements which may be identified with the
Galois field Fp of order p.
The fie
42
5
2013 10
ADVANCES IN MATHEMATICS
Vol.42, No.5
Oct., 2013
A Survey on Normal Bases Over Finite Fields
LIAO Qunying
(Institute of Mathematics and Software Science, Sichuan Normal University, Chengdu, Sichuan,
610066, P. R. China)
Abstract: Normal b
Chapter 1
Introduction
Finite fields is a branch of mathematics which has come to the fore in the last 50 years due to
its numerous applications, from combinatorics to coding theory. In this course, we will study the
properties of finite fields, and gain
EE 387
Algebraic Error-Control Codes
October 28, 2015
Handout #24
Homework #4 Solutions
e be a nonsystematic generator matrix for a linear block code over GF(5).
1. LBC over GF(5). Let G
2 4 2 2 4 4
e = 0 0 3 0 1 1 .
G
3 1 4 0 4 0
2 3 4 1 1 1
Find the sys
9. Finite fields
9.1
9.2
9.3
Uniqueness
Frobenius automorphisms
Counting irreducibles
1. Uniqueness
Among other things, the following result justifies speaking of the field with pn elements (for prime p and
integer n), since, we prove, these parameters co
International Journal of Algebra, Vol. 5, 2011, no. 24, 1175 - 1179
A Note on Trace Map
Dhirendra Singh Yadav
Department of Mathematics
Indian Institute of Technology Delhi
Hauz Khas, New Delhi - 110016, India
ds.yadav.iitd@gmail.com
R. K. Sharma
Departme
EXERCISES N 2, DUALITY
Exercise 1. Let C Fnq be a code. Let I cfw_1, . . . , n. We define the following codes
constructed from C:
The punctured code on I is defined as:
PI (C) := cfw_(ci )iI | c C, F|I|
q .
Roughly speaking, it is the set of codewords o
BIT 21 (1981), 326-334
ON COMPUTING LOGARITHMS OVER GF(2 p)
TORE HERLESTAM and ROLF JOHANNESSON
Abstract.
In this paper we present a new, heuristic method for computing logarithms over GF(2P).
When 2P-1 is a Mersenne prime < 2 3 1 - 1 it works in very sho
Finite Fields
Semestre dautomne 2013
Prof. Elisa Gorla, Maike Massierer
Universite de Neuchatel Mathematiques
www.math.unibas.ch/massierer/finitefields-hs13
Exercise sheet 5
Date: 18 October 2013
Exercise 1
Norm
Let Fqm |Fq be an extension of finite field
ACTA ARITHMETICA
XLVII (1986) =
A polynomial representation for logarithms in GF(q)
by . '
GARY L. MULLEN (University Park, Pa.) and DAVlD WHITE (Logan, Ut.)
1. Introduction. Let F4 =GFitI) denote the nite eld of order q n p"
where p is prime and n 21. Th
5. Linear algebra I: dimension
5.1
5.2
5.3
Some simple results
Bases and dimension
Homomorphisms and dimension
1. Some simple results
Several observations should be made. Once stated explicitly, the proofs are easy.
[1]
The intersection of a (non-empty)
15-859V: Introduction to coding theory
Carnegie Mellon University
Spring 2010
Venkatesan Guruswami
P ROBLEM S ET 2
Due by Friday, March 19
I NSTRUCTIONS
This problem sets can be turned in groups of two people; i.e., a single write-up for each two person
Row Rank of a Matrix equals its Column Rank
S. Kumaresan
Dept. of Math. & Stat.
University of Hyderabad
Hyderabad 500 046
kumaresa@gmail.com
Let A = (aij ) be an m n matrix over a field F . We denote by Ai , the i-th row of A:
Ai := (ai1 , . . . , ain ).
Coding Theory: Assignment 1
1. Let p(x) = x6 + x + 1 Z2 [x]. Show that F := Z2 [x]/(p(x) describes a finite field. What is the
size of F?
2. Find all the irreducible polynomials of degree 5 over GF (2).
3. Let be a primitive element in GF (24 ). Solve the
Information and Coding Theory
Autumn 2014
Homework 3
Due: December 4, 2014
Note: You may discuss these problems in groups. However, you must write up your own solutions
and mention the names of the people in your group. Also, please do mention any books,
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 3, MARCH 2005
415
Transactions Papers
A Decoding Algorithm for Finite-Geometry
LDPC Codes
Zhenyu Liu, Student Member, IEEE, and Dimitris A. Pados, Member, IEEE
AbstractIn this paper, we develop a new low-c
PROBLEMS
X 2.1 Construct the group under modulo6 addition.
X 2.2 Construct the group under modulo3 multiplication.
x 2.3 Let in be a positive integer. If m is not a prime. prove that the set cfw_1. 2. - ~ . m 1
15 not a group under modulo-m multiplication
Group Theory
July 14, 2012
Denitions and Examples
Denition of a Group
A set S is a group under an operation if:
1. For every two elements s, t S, s t S. This is called closure under the group operation.
2. For every three elements s, t, r S, (s t) r = s (
Homework 3
Due: Tuesday, February 14
EE 451
Spring 2012
T. R. Fischer
1. Let V5 be the space of binary 5-tuples over GF(2). Let S be the subspace of V5 spanned
by cfw_(0, 1, 1, 0, 0), (1, 0, 1, 0, 1).
a. List all vectors in S. What is the dimension of S?
Assignment 2
Information Theory and Coding II (Error Control Coding)
Science and Research Branch in Azad University
Semester 1, 90-91
1. Show that in a binary group block codes either all the code words have even weight or half
have even weight, half odd.
Assignment 3
Information Theory and Coding II (Error Control Coding)
Science and Research Branch in Azad University
Semester 1, 90-91
1. Erasure correcting ability. The erasure correcting ability of a block code is the largest number
such that all patter
Assignment 1
Information Theory and Coding II (Error Control Coding)
Science and Research Branch in Azad University
Semester 1, 90-91
1.
2.
3. Which of the following sentences is correct? Justify your answers.
a) The set of all non-negative integers is a