Permutation Polynomials over Finite Fields
|
|
- Rodney Nichols
- 5 years ago
- Views:
Transcription
1 Permutation Polynomials over Finite Fields Omar Kihel Brock University
2 1 Finite Fields 2 How to Construct a Finite Field 3 Permutation Polynomials 4 Characterization of PP
3 Finite Fields Let p be a prime. Then Z/pZ is a finite field.
4 Finite Fields The mapping Φ : Z F n n 1 is a homomorphism. For F a finite field, ker Φ.
5 Finite Fields The mapping Φ : Z F n n 1 is a homomorphism. For F a finite field, ker Φ. ker Φ = pz, where p is prime since F is a finite field. F p = Z/pZ F
6 Finite Fields F is an F p -vector space of dimension r. Then the number of elements in F is p r.
7 Finite Fields F is an F p -vector space of dimension r. Then the number of elements in F is p r. For every q = p r, there exists a unique field F with F = q, up to isomorphism.
8 Finite Fields Theorem Let r be a positive integer, p a prime, q = p r. For every x F q, x q = x.
9 Finite Fields Theorem Let r be a positive integer, p a prime, q = p r. For every x F q, x q = x. P roof: F q = {x F q ; x 0} is a multiplicative group of order q 1. Then x q 1 = 1. for every x F q. Hence for every x F q. x q = x
10 Finite Fields Let Ω be an algebraic closure of F p. Let q = p r, r a positive integer. F q is the splitting field of x q x.
11 Finite Fields Let Ω be an algebraic closure of F p. Let q = p r, r a positive integer. F q is the splitting field of x q x. The unicity of the splitting field implies the unicity of F q, up to isomorphism.
12 How to Construct a Finite Field Let q = p n, n a positive integer. Let f(x) be an irreducible polynomial of degree n over F p. Then is a finite field of q elements. F p [x]/ ( f(x) )
13 Polynomials in Finite Fields The number of monic irreducible polynomials in F q [x] of degree n is 1 µ(d)q n/d. n d n
14 Polynomials in Finite Fields The number of monic irreducible polynomials in F q [x] of degree n is 1 µ(d)q n/d. n d n The product of monic irreducible polynomials over F q of degree d n is x qn x.
15 Polynomials in Finite Fields Let h : F q F q be an application. Then there exists a unique polynomial of degree less than q, f(x) = ) (1 (x c) q 1 ) h(c), c F q with f(b) = h(b) for every b F q.
16 Permutation Polynomials A polynomial f F q [x] is a permutation polynomial if the induced application F q F q c f(c) permutes the elements of F q, i.e. f is one-to-one from F q to F q.
17 Permutation Polynomials Recently, permutation polynomials have become of considerable interest in the construction of crytographic systems for the secure transmission of data.
18 Permutation Polynomials Recently, permutation polynomials have become of considerable interest in the construction of crytographic systems for the secure transmission of data. Let M (an element of F q ) be a message which is to be sent securely from A to B. If P (x) is a permutation of F q, then A sends to B the field element N = P (M). Since P (x) is a bijection, B can obtain the original message M by calculating P 1 (N) = P 1( P (M) ) = M.
19 Permutation Polynomials Recently, permutation polynomials have become of considerable interest in the construction of crytographic systems for the secure transmission of data. Let M (an element of F q ) be a message which is to be sent securely from A to B. If P (x) is a permutation of F q, then A sends to B the field element N = P (M). Since P (x) is a bijection, B can obtain the original message M by calculating P 1 (N) = P 1( P (M) ) = M. Permutation polynomials are also useful in several combinatorial applications.
20 Some examples of permutation polynomials Polynomials of degree 0 are not PP over F q ; x i is PP over F q if and only if gcd(i, q 1) = 1; ax 2 + bx + c, (a 0), is PP over F q if and only if b = 0 and char(f q ) = 2; x 4 + 3x and x 5 + 2x 2 permute F 7 ; x 5 ix permutes F 9 = F 3 (i), where i 2 = 1; x 8 + 4x permutes F 29 ; If a F q, then the Dickson polynomial g k (x, a) = [k/2] j=0 k k j ( k j permutes F q if and only if (k, q 2 1) = 1. j ) ( a) j x k 2j
21 Permutation Polynomials Lidl and Mullen in American Math Monthly listed 17 open problems in 2 papers (1988, 1993). P 2. Find new classes of PP. P 13. Find conditions on m, n, q so that ax n + x m permutes F q.
22 Permutation Polynomials Let f(x) = ax n + x m, where gcd(n, m) = 1, a 0, n > m. Let d = gcd(n m, q 1).
23 Permutation Polynomials Let f(x) = ax n + x m, where gcd(n, m) = 1, a 0, n > m. Let d = gcd(n m, q 1). If ( a) (F q ) d, then f(x) = 0 has d + 1 distinct roots in F q. Hence d = 1, implying f is not PP over F q.
24 Permutation Polynomials Let f(x) = ax n + x m, where gcd(n, m) = 1, a 0, n > m. Let d = gcd(n m, q 1). If ( a) (F q ) d, then f(x) = 0 has d + 1 distinct roots in F q. Hence d = 1, implying f is not PP over F q. f(x) being PP over F q implies f(x) = 0 has a unique root in F q (not neccesarily simple).
25 Permutation Polynomials Let f(x) = ax n + x m, where gcd(n, m) = 1, a 0, n > m. Let d = gcd(n m, q 1). If ( a) (F q ) d, then f(x) = 0 has d + 1 distinct roots in F q. Hence d = 1, implying f is not PP over F q. f(x) being PP over F q implies f(x) = 0 has a unique root in F q (not neccesarily simple). The reciprocal is not true in general: x i is PP if and only if gcd(i, q 1) = 1.
26 Permutation Polynomials Proposition Let f(x) = i 0 a i x pi F q [x]. f is a permutation polynomial if and only if f has exactly one unique root in F q.
27 Permutation Polynomials Proposition Let f(x) = i 0 a i x pi F q [x]. f is a permutation polynomial if and only if f has exactly one unique root in F q. P roof: f(x ± y) = f(x) ± f(y).
28 Permutation Polynomials A new family of PP (Ayad and K). ( ) f(x) = x u x (q 1)/2 + x (q 1)/4 + 1, (i) (u, q 1) = 1; (ii) q 1 (mod 8); (iii) 3 (q 1)/4 1 (mod p).
29 Permutation Polynomials Conjecture (Conjecture of Carlitz) If n is an even integer greater than 0, there exists a constant c(n) such that if q > c(n), then there is no permutation polynomial of degree n over F q.
30 Permutation Polynomials Conjecture (Conjecture of Carlitz) If n is an even integer greater than 0, there exists a constant c(n) such that if q > c(n), then there is no permutation polynomial of degree n over F q. Fried, Guralnick, and Saxl proved the Conjecture of Carlitz.
31 Characterization of PP Theorem (Hermite-Dickson) Let p be a prime, q = p r, and g(x) F q [x]. Then g(x) is a permutation polynomial over F q if and only if (i) g(x) = 0 has a unique root in F q ; (ii) for every l {1, 2,..., q 2}, ( ) deg g l (x) (mod x q x) q 2.
32 Characterization of PP Theorem (Hermite-Dickson) Let p be a prime, q = p r, and g(x) F q [x]. Then g(x) is a permutation polynomial over F q if and only if (i) g(x) = 0 has a unique root in F q ; (ii) for every l {1, 2,..., q 2}, ( ) deg g l (x) (mod x q x) q 2. Corollary If deg g(x) = d and d (q 1), then g(x) is not a permutation polynomial over F q.
33 Characterization of PP Theorem (Ayad-Belghaba-K) Let f(x) = ax n + x m F q [x], with a 0, gcd(m, n) = 1. Let d = gcd(n m, q 1), and suppose that d > 2. Then f(x) is a permutation polynomial over F q if and only if (i) f(x) = 0 has a unique root in F q (ii) for every l {1,..., q 2} such that d l, we have ( ) deg f l (x) q 2.
34 Characterization of PP To give the idea of the proof of the Theorem of Hermite-Dickson, we require the following lemma. Lemma a 1, a 2,..., a q F q are distinct if and only if q a t i = 0 for 1 t q 2 i=1 and q a q 1 i = 1 i=1
35 Characterization of PP The reduction of ( f(x)) t modulo (x q x) has the form q 1 α i,t x i. i=0
36 Characterization of PP The reduction of ( f(x)) t modulo (x q x) has the form q 1 α i,t x i. i=0 We have f(x) = c F q (1 (x c) q 1) f(c), then α q 1,t = c F q f(c) t.
37 Characterization of PP f has a unique root in F q, which implies that c F q f(c) q 1 = 1.
38 Characterization of PP f has a unique root in F q, which implies that c F q f(c) q 1 = 1. ( ) deg f t (x) q 2, 1 t q 2 implies that the coefficient of x q 1 is 0. Hence α q 1,t = 0. Then α q 1,t = c F q (f(c)) t = 0, for 1 t q 2.
39 Characterization of PP For the reciprocal, f being PP implies f has a unique root in F q. f is PP, then f(c) are distinct. Then c F q ( f(c)) t = 0, 1 t q 2 implies ( ) ( ) t deg f(x) q 2 for 1 t q 2.
40 Characterization of PP Ayad-Belghaba-K proved the following: Theorem If f(x) = ax n + x m permutes F p, where n > m > 0 and a F p, then p 1 (d 1)d. We showed that this bound in terms of d only is sharp.
41 Characterization of PP Corollary If f(x) = x n + ax m permutes F p, where 1 m < n < p and a F p, then gcd(n m, p 1) > 4, except if d = 3, p = 7, and f(x) is one of the following: (i) f(x) = x 4 + 3x, (ii) f(x) = x 4 3x, (iii) f(x) = x 5 + 2x 2, (iv) f(x) = x 5 2x 2.
42 Characterization of PP Niederreiter and Robinson proved the following: Theorem Given a positive integer n, there exists a constant c(n) such that for q > c(n), no polynomial of the form ax n + bx m + c F q with n > m > 1, gcd(n, m) = 1, and ab 0 permutes F q. The constant c(n) is not explicit.
43 Characterization of PP Turnwald proved the following: Theorem If f(x) = ax n + x m permutes F q, where n > m > 0 and a F q, then either q (n 2) 4 + 4n 4 or n = mp i.
44 Characterization of PP Turnwald proved the following: Theorem If f(x) = ax n + x m permutes F q, where n > m > 0 and a F q, then either q (n 2) 4 + 4n 4 or n = mp i. The proof uses Weil lower bound for the number of the points on the curve f(x) f(y) x y over F q.
45 Characterization of PP When q = p, Turnwald proved the following: Theorem If f(x) = ax n + x m permutes F p, where n > m > 0 and a F p, then p < n max{m, n m}.
46 Characterization of PP In 2009, Masuda and Zieve proved the following: Theorem If f(x) = ax n + x m permutes F p, where n > m > 0 and a F p, then p 1 (d + 1)d.
47 Thank you for your time.
ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE
ON PERMUTATION POLYNOMIALS OF PRESCRIBED SHAPE AMIR AKBARY, DRAGOS GHIOCA, AND QIANG WANG Abstract. We count permutation polynomials of F q which are sums of m + 2 monomials of prescribed degrees. This
More informationCryptography and Schur s Conjecture UM Bozeman, November 19, 2004
Cryptography and Schur s Conjecture UM Bozeman, November 19, 2004 Advertisement for our seminar at MSU-Billings. WHAT-DO-YOU-KNOW? MATHEMATICS COLLOQUIUM: We plan talks this year on historical topics with
More informationA Weil bound free proof of Schur s conjecture
A Weil bound free proof of Schur s conjecture Peter Müller Department of Mathematics University of Florida Gainesville, FL 32611 E-mail: pfm@math.ufl.edu Abstract Let f be a polynomial with coefficients
More informationOn the degree of local permutation polynomials
On the degree of local permutation polynomials Wiebke S. Diestelkamp Department of Mathematics University of Dayton Dayton, OH 45469-2316 wiebke@udayton.edu Stephen G. Hartke Department of Mathematics
More informationOn Permutation Polynomials over Local Finite Commutative Rings
International Journal of Algebra, Vol. 12, 2018, no. 7, 285-295 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8935 On Permutation Polynomials over Local Finite Commutative Rings Javier
More informationDICKSON POLYNOMIALS OVER FINITE FIELDS. n n i. i ( a) i x n 2i. y, a = yn+1 a n+1 /y n+1
DICKSON POLYNOMIALS OVER FINITE FIELDS QIANG WANG AND JOSEPH L. YUCAS Abstract. In this paper we introduce the notion of Dickson polynomials of the k + 1)-th kind over finite fields F p m and study basic
More informationSome fundamental contributions of Gary Mullen to finite fields
Some fundamental contributions of Gary Mullen to finite fields School of Mathematics and Statistics, Carleton University daniel@math.carleton.ca Carleton Finite Fields Day 2017 September 29, 2017 Overview
More informationPERFECT POLYNOMIALS OVER F p WITH p + 1 IRREDUCIBLE DIVISORS. 1. Introduction. Let p be a prime number. For a monic polynomial A F p [x] let d
PERFECT POLYNOMIALS OVER F p WITH p + 1 IRREDUCIBLE DIVISORS L. H. GALLARDO and O. RAHAVANDRAINY Abstract. We consider, for a fixed prime number p, monic polynomials in one variable over the finite field
More informationOn generalized Lucas sequences
Contemporary Mathematics On generalized Lucas sequences Qiang Wang This paper is dedicated to Professor G. B. Khosrovshahi on the occasion of his 70th birthday and to IPM on its 0th birthday. Abstract.
More informationHomework 8 Solutions to Selected Problems
Homework 8 Solutions to Selected Problems June 7, 01 1 Chapter 17, Problem Let f(x D[x] and suppose f(x is reducible in D[x]. That is, there exist polynomials g(x and h(x in D[x] such that g(x and h(x
More informationOn the Hansen-Mullen Conjecture for Self-Reciprocal Irreducible Polynomials
On the Hansen-Mullen Conjecture for Self-Reciprocal Irreducible Polynomials Giorgos N. Kapetanakis (joint work with T. Garefalakis) University of Crete Fq10 July 14, 2011 Giorgos Kapetanakis (Crete) On
More informationPermutation polynomials
8 Permutation polynomials 8.1 One variable......................................... 208 Introduction Criteria Enumeration and distribution of PPs Constructions of PPs PPs from permutations of multiplicative
More informationSalem numbers of trace 2 and a conjecture of Estes and Guralnick
Salem numbers of trace 2 and a conjecture of Estes and Guralnick James McKee, Pavlo Yatsyna (Royal Holloway) November 2015 Properties of minimal polynomials Let A be an integer symmetric matrix, n n. Properties
More informationGalois Theory, summary
Galois Theory, summary Chapter 11 11.1. UFD, definition. Any two elements have gcd 11.2 PID. Every PID is a UFD. There are UFD s which are not PID s (example F [x, y]). 11.3 ED. Every ED is a PID (and
More informationMATH 361: NUMBER THEORY TENTH LECTURE
MATH 361: NUMBER THEORY TENTH LECTURE The subject of this lecture is finite fields. 1. Root Fields Let k be any field, and let f(x) k[x] be irreducible and have positive degree. We want to construct a
More informationQuasi-reducible Polynomials
Quasi-reducible Polynomials Jacques Willekens 06-Dec-2008 Abstract In this article, we investigate polynomials that are irreducible over Q, but are reducible modulo any prime number. 1 Introduction Let
More informationClassification of Finite Fields
Classification of Finite Fields In these notes we use the properties of the polynomial x pd x to classify finite fields. The importance of this polynomial is explained by the following basic proposition.
More informationFIELD THEORY. Contents
FIELD THEORY MATH 552 Contents 1. Algebraic Extensions 1 1.1. Finite and Algebraic Extensions 1 1.2. Algebraic Closure 5 1.3. Splitting Fields 7 1.4. Separable Extensions 8 1.5. Inseparable Extensions
More informationMath 201C Homework. Edward Burkard. g 1 (u) v + f 2(u) g 2 (u) v2 + + f n(u) a 2,k u k v a 1,k u k v + k=0. k=0 d
Math 201C Homework Edward Burkard 5.1. Field Extensions. 5. Fields and Galois Theory Exercise 5.1.7. If v is algebraic over K(u) for some u F and v is transcendental over K, then u is algebraic over K(v).
More informationMath 120 HW 9 Solutions
Math 120 HW 9 Solutions June 8, 2018 Question 1 Write down a ring homomorphism (no proof required) f from R = Z[ 11] = {a + b 11 a, b Z} to S = Z/35Z. The main difficulty is to find an element x Z/35Z
More informationFrom now on we assume that K = K.
Divisors From now on we assume that K = K. Definition The (additively written) free abelian group generated by P F is denoted by D F and is called the divisor group of F/K. The elements of D F are called
More informationTC10 / 3. Finite fields S. Xambó
TC10 / 3. Finite fields S. Xambó The ring Construction of finite fields The Frobenius automorphism Splitting field of a polynomial Structure of the multiplicative group of a finite field Structure of the
More informationarxiv: v1 [cs.it] 12 Jun 2016
New Permutation Trinomials From Niho Exponents over Finite Fields with Even Characteristic arxiv:606.03768v [cs.it] 2 Jun 206 Nian Li and Tor Helleseth Abstract In this paper, a class of permutation trinomials
More informationSelected exercises from Abstract Algebra by Dummit and Foote (3rd edition).
Selected exercises from Abstract Algebra by Dummit and Foote (3rd edition). Bryan Félix Abril 12, 2017 Section 14.2 Exercise 3. Determine the Galois group of (x 2 2)(x 2 3)(x 2 5). Determine all the subfields
More informationMath 121 Homework 2 Solutions
Math 121 Homework 2 Solutions Problem 13.2 #16. Let K/F be an algebraic extension and let R be a ring contained in K that contains F. Prove that R is a subfield of K containing F. We will give two proofs.
More informationbut no smaller power is equal to one. polynomial is defined to be
13. Radical and Cyclic Extensions The main purpose of this section is to look at the Galois groups of x n a. The first case to consider is a = 1. Definition 13.1. Let K be a field. An element ω K is said
More informationField Theory Qual Review
Field Theory Qual Review Robert Won Prof. Rogalski 1 (Some) qual problems ˆ (Fall 2007, 5) Let F be a field of characteristic p and f F [x] a polynomial f(x) = i f ix i. Give necessary and sufficient conditions
More informationFinite Fields. Mike Reiter
1 Finite Fields Mike Reiter reiter@cs.unc.edu Based on Chapter 4 of: W. Stallings. Cryptography and Network Security, Principles and Practices. 3 rd Edition, 2003. Groups 2 A group G, is a set G of elements
More informationPrescribing coefficients of invariant irreducible polynomials
Prescribing coefficients of invariant irreducible polynomials Giorgos Kapetanakis 1 Faculty of Engineering and Natural Sciences, Sabancı Üniversitesi. Ortha Mahalle, Tuzla 34956, İstanbul, Turkey Abstract
More informationOn Computably Enumerable Sets over Function Fields
On Computably Enumerable Sets over Function Fields Alexandra Shlapentokh East Carolina University Joint Meetings 2017 Atlanta January 2017 Some History Outline 1 Some History A Question and the Answer
More informationFinite fields Michel Waldschmidt
Finite fields Michel Waldschmidt http://www.imj-prg.fr/~michel.waldschmidt//pdf/finitefields.pdf Updated: 03/07/2018 Contents 1 Background: Arithmetic 1.1 Cyclic groups If G is a finite multiplicative
More informationMATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION
MATH 431 PART 2: POLYNOMIAL RINGS AND FACTORIZATION 1. Polynomial rings (review) Definition 1. A polynomial f(x) with coefficients in a ring R is n f(x) = a i x i = a 0 + a 1 x + a 2 x 2 + + a n x n i=0
More informationSection 33 Finite fields
Section 33 Finite fields Instructor: Yifan Yang Spring 2007 Review Corollary (23.6) Let G be a finite subgroup of the multiplicative group of nonzero elements in a field F, then G is cyclic. Theorem (27.19)
More informationAlgebra Review 2. 1 Fields. A field is an extension of the concept of a group.
Algebra Review 2 1 Fields A field is an extension of the concept of a group. Definition 1. A field (F, +,, 0 F, 1 F ) is a set F together with two binary operations (+, ) on F such that the following conditions
More informationExplicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields
Southern Illinois University Carbondale OpenSIUC Articles and Preprints Department of Mathematics 2007 Explicit Factorizations of Cyclotomic and Dickson Polynomials over Finite Fields Robert W. Fitzgerald
More informationFinite Fields. Saravanan Vijayakumaran Department of Electrical Engineering Indian Institute of Technology Bombay
1 / 25 Finite Fields Saravanan Vijayakumaran sarva@ee.iitb.ac.in Department of Electrical Engineering Indian Institute of Technology Bombay September 25, 2014 2 / 25 Fields Definition A set F together
More informationSelf-reciprocal Polynomials Over Finite Fields
Self-reciprocal Polynomials Over Finite Fields by Helmut Meyn 1 and Werner Götz 1 Abstract. The reciprocal f (x) of a polynomial f(x) of degree n is defined by f (x) = x n f(1/x). A polynomial is called
More informationPolynomials. Chapter 4
Chapter 4 Polynomials In this Chapter we shall see that everything we did with integers in the last Chapter we can also do with polynomials. Fix a field F (e.g. F = Q, R, C or Z/(p) for a prime p). Notation
More informationSchool of Mathematics and Statistics. MT5836 Galois Theory. Handout 0: Course Information
MRQ 2017 School of Mathematics and Statistics MT5836 Galois Theory Handout 0: Course Information Lecturer: Martyn Quick, Room 326. Prerequisite: MT3505 (or MT4517) Rings & Fields Lectures: Tutorials: Mon
More informationHonors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35
Honors Algebra 4, MATH 371 Winter 2010 Assignment 4 Due Wednesday, February 17 at 08:35 1. Let R be a commutative ring with 1 0. (a) Prove that the nilradical of R is equal to the intersection of the prime
More informationA Mass Formula for Cyclic Self-Orthogonal Codes
A Mass Formula for Cyclic Self-Orthogonal Codes Chekad Sarami Department of Mathematics & Computer Science Fayettevle State University Fayettevle, North Carolina, U.S.A. Abstract - We give an algorithm
More informationChapter 8. P-adic numbers. 8.1 Absolute values
Chapter 8 P-adic numbers Literature: N. Koblitz, p-adic Numbers, p-adic Analysis, and Zeta-Functions, 2nd edition, Graduate Texts in Mathematics 58, Springer Verlag 1984, corrected 2nd printing 1996, Chap.
More informationDICKSON POLYNOMIALS THAT ARE PERMUTATIONS. Mihai Cipu
Serdica Math. J. 30 (004, 177 194 DICKSON POLYNOMIALS THAT ARE PERMUTATIONS Mihai Cipu Communicated by P. Pragacz Abstract. A theorem of S.D. Cohen gives a characterization for Dickson polynomials of the
More informationMath 429/581 (Advanced) Group Theory. Summary of Definitions, Examples, and Theorems by Stefan Gille
Math 429/581 (Advanced) Group Theory Summary of Definitions, Examples, and Theorems by Stefan Gille 1 2 0. Group Operations 0.1. Definition. Let G be a group and X a set. A (left) operation of G on X is
More informationRINGS: SUMMARY OF MATERIAL
RINGS: SUMMARY OF MATERIAL BRIAN OSSERMAN This is a summary of terms used and main results proved in the subject of rings, from Chapters 11-13 of Artin. Definitions not included here may be considered
More informationFields and Galois Theory. Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory.
Fields and Galois Theory Below are some results dealing with fields, up to and including the fundamental theorem of Galois theory. This should be a reasonably logical ordering, so that a result here should
More informationMATH 8253 ALGEBRAIC GEOMETRY WEEK 12
MATH 8253 ALGEBRAIC GEOMETRY WEEK 2 CİHAN BAHRAN 3.2.. Let Y be a Noetherian scheme. Show that any Y -scheme X of finite type is Noetherian. Moreover, if Y is of finite dimension, then so is X. Write f
More informationMTH Abstract Algebra II S17. Review for the Final Exam. Part I
MTH 411-1 Abstract Algebra II S17 Review for the Final Exam Part I You will be allowed to use the textbook (Hungerford) and a print-out of my online lecture notes during the exam. Nevertheless, I recommend
More informationOn transitive polynomials modulo integers
Notes on Number Theory and Discrete Mathematics Print ISSN 1310 5132, Online ISSN 2367 8275 Vol. 22, 2016, No. 2, 23 35 On transitive polynomials modulo integers Mohammad Javaheri 1 and Gili Rusak 2 1
More informationEXPLICIT EVALUATIONS OF SOME WEIL SUMS. 1. Introduction In this article we will explicitly evaluate exponential sums of the form
EXPLICIT EVALUATIONS OF SOME WEIL SUMS ROBERT S. COULTER 1. Introduction In this article we will explicitly evaluate exponential sums of the form χax p α +1 ) where χ is a non-trivial additive character
More informationNOTES ON FINITE FIELDS
NOTES ON FINITE FIELDS AARON LANDESMAN CONTENTS 1. Introduction to finite fields 2 2. Definition and constructions of fields 3 2.1. The definition of a field 3 2.2. Constructing field extensions by adjoining
More informationOn the Number of Trace-One Elements in Polynomial Bases for F 2
On the Number of Trace-One Elements in Polynomial Bases for F 2 n Omran Ahmadi and Alfred Menezes Department of Combinatorics & Optimization University of Waterloo, Canada {oahmadid,ajmeneze}@uwaterloo.ca
More information1. Group Theory Permutations.
1.1. Permutations. 1. Group Theory Problem 1.1. Let G be a subgroup of S n of index 2. Show that G = A n. Problem 1.2. Find two elements of S 7 that have the same order but are not conjugate. Let π S 7
More informationFinite Fields and Their Applications
Finite Fields and Their Applications 1 01 14 31 Contents lists available at SciVerse ScienceDirect Finite Fields and Their Applications www.elsevier.com/locate/ffa Dickson polynomials over finite fields
More informationLagrange s polynomial
Lagrange s polynomial Nguyen Trung Tuan November 16, 2016 Abstract In this article, I will use Lagrange polynomial to solve some problems from Mathematical Olympiads. Contents 1 Lagrange s interpolation
More informationA New Approach to Permutation Polynomials over Finite Fields
A New Approach to Permutation Polynomials over Finite Fields Joint work with Dr. Xiang-dong Hou and Stephen Lappano Department of Mathematics and Statistics University of South Florida Discrete Seminar
More informationComplete permutation polynomials of monomial type
Complete permutation polynomials of monomial type Giovanni Zini (joint works with D. Bartoli, M. Giulietti and L. Quoos) (based on the work of thesis of E. Franzè) Università di Perugia Workshop BunnyTN
More informationCarmen s Core Concepts (Math 135)
Carmen s Core Concepts (Math 135) Carmen Bruni University of Waterloo Week 8 1 The following are equivalent (TFAE) 2 Inverses 3 More on Multiplicative Inverses 4 Linear Congruence Theorem 2 [LCT2] 5 Fermat
More informationAbstract Algebra, Second Edition, by John A. Beachy and William D. Blair. Corrections and clarifications
1 Abstract Algebra, Second Edition, by John A. Beachy and William D. Blair Corrections and clarifications Note: Some corrections were made after the first printing of the text. page 9, line 8 For of the
More informationMath 581 Problem Set 6 Solutions
Math 581 Problem Set 6 Solutions 1. Let F K be a finite field extension. Prove that if [K : F ] = 1, then K = F. Proof: Let v K be a basis of K over F. Let c be any element of K. There exists α c F so
More informationFinite Fields Appl. 8 (2002),
On the permutation behaviour of Dickson polynomials of the second kind Finite Fields Appl. 8 (00), 519-530 Robert S. Coulter 1 Information Security Research Centre, Queensland University of Technology,
More informationLecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman
Lecture Notes Math 371: Algebra (Fall 2006) by Nathanael Leedom Ackerman October 31, 2006 TALK SLOWLY AND WRITE NEATLY!! 1 0.1 Symbolic Adjunction of Roots When dealing with subfields of C it is easy to
More information9. Finite fields. 1. Uniqueness
9. Finite fields 9.1 Uniqueness 9.2 Frobenius automorphisms 9.3 Counting irreducibles 1. Uniqueness Among other things, the following result justifies speaking of the field with p n elements (for prime
More informationMath 4310 Solutions to homework 7 Due 10/27/16
Math 4310 Solutions to homework 7 Due 10/27/16 1. Find the gcd of x 3 + x 2 + x + 1 and x 5 + 2x 3 + x 2 + x + 1 in Rx. Use the Euclidean algorithm: x 5 + 2x 3 + x 2 + x + 1 = (x 3 + x 2 + x + 1)(x 2 x
More informationLagrange s polynomial
Lagrange s polynomial Nguyen Trung Tuan November 13, 2016 Abstract...In numerical analysis, Lagrange polynomials are used for polynomial interpolation. For a given set of distinct points x j and numbers
More informationDickson Polynomials that are Involutions
Dickson Polynomials that are Involutions Pascale Charpin Sihem Mesnager Sumanta Sarkar May 6, 2015 Abstract Dickson polynomials which are permutations are interesting combinatorial objects and well studied.
More informationFour classes of permutation polynomials of F 2 m
Finite Fields and Their Applications 1 2007) 869 876 http://www.elsevier.com/locate/ffa Four classes of permutation polynomials of F 2 m Jin Yuan,1, Cunsheng Ding 1 Department of Computer Science, The
More informationMath 547, Exam 2 Information.
Math 547, Exam 2 Information. 3/19/10, LC 303B, 10:10-11:00. Exam 2 will be based on: Homework and textbook sections covered by lectures 2/3-3/5. (see http://www.math.sc.edu/ boylan/sccourses/547sp10/547.html)
More informationFUNCTIONAL DECOMPOSITION OF A CLASS OF WILD POLYNOMIALS
FUNCTIONAL DECOMPOSITION OF A CLASS OF WILD POLYNOMIALS ROBERT S. COULTER, GEORGE HAVAS AND MARIE HENDERSON Dedicated to Professor Anne Penfold Street. Abstract. No general algorithm is known for the functional
More informationThe Galois group of a polynomial f(x) K[x] is the Galois group of E over K where E is a splitting field for f(x) over K.
The third exam will be on Monday, April 9, 013. The syllabus for Exam III is sections 1 3 of Chapter 10. Some of the main examples and facts from this material are listed below. If F is an extension field
More informationModern Computer Algebra
Modern Computer Algebra Exercises to Chapter 25: Fundamental concepts 11 May 1999 JOACHIM VON ZUR GATHEN and JÜRGEN GERHARD Universität Paderborn 25.1 Show that any subgroup of a group G contains the neutral
More informationMath 121 Homework 3 Solutions
Math 121 Homework 3 Solutions Problem 13.4 #6. Let K 1 and K 2 be finite extensions of F in the field K, and assume that both are splitting fields over F. (a) Prove that their composite K 1 K 2 is a splitting
More informationThe Number of Irreducible Polynomials of Even Degree over F 2 with the First Four Coefficients Given
The Number of Irreducible Polynomials of Even Degree over F 2 with the First Four Coefficients Given B. Omidi Koma School of Mathematics and Statistics Carleton University bomidi@math.carleton.ca July
More informationDivision of Trinomials by Pentanomials and Orthogonal Arrays
Division of Trinomials by Pentanomials and Orthogonal Arrays School of Mathematics and Statistics Carleton University daniel@math.carleton.ca Joint work with M. Dewar, L. Moura, B. Stevens and Q. Wang
More informationMath 581 Problem Set 3 Solutions
Math 581 Problem Set 3 Solutions 1. Prove that complex conjugation is a isomorphism from C to C. Proof: First we prove that it is a homomorphism. Define : C C by (z) = z. Note that (1) = 1. The other properties
More informationAn Approach to Hensel s Lemma
Irish Math. Soc. Bulletin 47 (2001), 15 21 15 An Approach to Hensel s Lemma gary mcguire Abstract. Hensel s Lemma is an important tool in many ways. One application is in factoring polynomials over Z.
More informationPage Points Possible Points. Total 200
Instructions: 1. The point value of each exercise occurs adjacent to the problem. 2. No books or notes or calculators are allowed. Page Points Possible Points 2 20 3 20 4 18 5 18 6 24 7 18 8 24 9 20 10
More informationFine-grain decomposition of F q
Fine-grain decomposition of F q n David Thomson with Colin Weir Army Cyber Institute at USMA West Point David.Thomson@usma.edu WCNT 2015 D. Thomson (ACI at West Point) Linear algebra over finite fields
More informationRings and Fields Theorems
Rings and Fields Theorems Rajesh Kumar PMATH 334 Intro to Rings and Fields Fall 2009 October 25, 2009 12 Rings and Fields 12.1 Definition Groups and Abelian Groups Let R be a non-empty set. Let + and (multiplication)
More informationOn one class of permutation polynomials over finite fields of characteristic two *
On one class of permutation polynomials over finite fields of characteristic two * Leonid Bassalygo, Victor A. Zinoviev To cite this version: Leonid Bassalygo, Victor A. Zinoviev. On one class of permutation
More informationwith Good Cross Correlation for Communications and Cryptography
m-sequences with Good Cross Correlation for Communications and Cryptography Tor Helleseth and Alexander Kholosha 9th Central European Conference on Cryptography: Trebíc, June 26, 2009 1/25 Outline m-sequences
More informationp-adic fields Chapter 7
Chapter 7 p-adic fields In this chapter, we study completions of number fields, and their ramification (in particular in the Galois case). We then look at extensions of the p-adic numbers Q p and classify
More informationThe Chinese Remainder Theorem
Chapter 5 The Chinese Remainder Theorem 5.1 Coprime moduli Theorem 5.1. Suppose m, n N, and gcd(m, n) = 1. Given any remainders r mod m and s mod n we can find N such that N r mod m and N s mod n. Moreover,
More informationS11MTH 3175 Group Theory (Prof.Todorov) Final (Practice Some Solutions) 2 BASIC PROPERTIES
S11MTH 3175 Group Theory (Prof.Todorov) Final (Practice Some Solutions) 2 BASIC PROPERTIES 1 Some Definitions For your convenience, we recall some of the definitions: A group G is called simple if it has
More information: Coding Theory. Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, upattane
2301532 : Coding Theory Notes by Assoc. Prof. Dr. Patanee Udomkavanich October 30, 2006 http://pioneer.chula.ac.th/ upattane Chapter 1 Error detection, correction and decoding 1.1 Basic definitions and
More informationOn complete permutation polynomials 1
Fourteenth International Workshop on Algebraic and Combinatorial Coding Theory September 7 13, 2014, Svetlogorsk (Kaliningrad region), Russia pp. 57 62 On complete permutation polynomials 1 L. A. Bassalygo
More informationCDM. Finite Fields. Klaus Sutner Carnegie Mellon University. Fall 2018
CDM Finite Fields Klaus Sutner Carnegie Mellon University Fall 2018 1 Ideals The Structure theorem Where Are We? 3 We know that every finite field carries two apparently separate structures: additive and
More informationSection V.6. Separability
V.6. Separability 1 Section V.6. Separability Note. Recall that in Definition V.3.10, an extension field F is a separable extension of K if every element of F is algebraic over K and every root of the
More informationUniversity of Ottawa
University of Ottawa Department of Mathematics and Statistics MAT3143: Ring Theory Professor: Hadi Salmasian Final Exam April 21, 2015 Surname First Name Instructions: (a) You have 3 hours to complete
More informationH. W. Lenstra, Jr. and M. Zieve
A family of exceptional polynomials in characteristic three H. W. Lenstra, Jr. and M. Zieve Abstract. We present a family of indecornposable polynomials of non prime-power degree over the finite fleld
More informationIntroduction to finite fields
Chapter 7 Introduction to finite fields This chapter provides an introduction to several kinds of abstract algebraic structures, particularly groups, fields, and polynomials. Our primary interest is in
More informationarxiv: v1 [math.nt] 11 May 2016
arxiv:1605.03375v1 [math.nt] 11 May 2016 On Some Permutation Binomials and Trinomials Over F 2 n Srimanta Bhattacharya Centre of Excellence in Cryptology, Indian Statistical Institute, Kolkata. E-mail:
More informationI216e Discrete Math (for Review)
I216e Discrete Math (for Review) Nov 22nd, 2017 To check your understanding. Proofs of do not appear in the exam. 1 Monoid Let (G, ) be a monoid. Proposition 1 Uniquness of Identity An idenity e is unique,
More informationSPLITTING FIELDS AND PERIODS OF FIBONACCI SEQUENCES MODULO PRIMES
SPLITTING FIELDS AND PERIODS OF FIBONACCI SEQUENCES MODULO PRIMES SANJAI GUPTA, PAROUSIA ROCKSTROH, AND FRANCIS EDWARD SU 1. Introduction The Fibonacci sequence defined by F 0 = 0, F 1 = 1, F n+1 = F n
More informationThe Kummer Pairing. Alexander J. Barrios Purdue University. 12 September 2013
The Kummer Pairing Alexander J. Barrios Purdue University 12 September 2013 Preliminaries Theorem 1 (Artin. Let ψ 1, ψ 2,..., ψ n be distinct group homomorphisms from a group G into K, where K is a field.
More informationHow Are Irreducible and Primitive Polynomials Distributed overjuly Finite 21, Fields? / 28
How Are Irreducible and Primitive Polynomials Distributed over Finite Fields? Gary L. Mullen Penn State University mullen@math.psu.edu July 21, 2010 How Are Irreducible and Primitive Polynomials Distributed
More information2 ALGEBRA II. Contents
ALGEBRA II 1 2 ALGEBRA II Contents 1. Results from elementary number theory 3 2. Groups 4 2.1. Denition, Subgroup, Order of an element 4 2.2. Equivalence relation, Lagrange's theorem, Cyclic group 9 2.3.
More informationFURTHER EVALUATIONS OF WEIL SUMS
FURTHER EVALUATIONS OF WEIL SUMS ROBERT S. COULTER 1. Introduction Weil sums are exponential sums whose summation runs over the evaluation mapping of a particular function. Explicitly they have the form
More information1 Rings 1 RINGS 1. Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism
1 RINGS 1 1 Rings Theorem 1.1 (Substitution Principle). Let ϕ : R R be a ring homomorphism (a) Given an element α R there is a unique homomorphism Φ : R[x] R which agrees with the map ϕ on constant polynomials
More informationChapter 4 Finite Fields
Chapter 4 Finite Fields Introduction will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, IDEA, Public Key concern operations on numbers what constitutes a number
More information