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Data publikacji: 4 czerwca 2021

Data aktualizacji: 5 lutego 2022

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1

Jogia, Danesh Michael Mathematics &amp Statistics Faculty of Science UNSW. "Algebraic aspects of integrability and reversibility in maps." Publisher:University of New South Wales. Mathematics & Statistics, 2008. http://handle.unsw.edu.au/1959.4/40947.

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We study the cause of the signature over finite fields of integrability in two dimensional discrete dynamical systems by using theory from algebraic geometry. In particular the theory of elliptic curves is used to prove the major result of the thesis: that all birational maps that preserve an elliptic curve necessarily act on that elliptic curve as addition under the associated group law. Our result generalises special cases previously given in the literature. We apply this theorem to the specific cases when the ground fields are finite fields of prime order and the function field $mathbb{C}(t

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2

Voloch, J. F. "Curves over finite fields." Thesis, University of Cambridge, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.355283.

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3

Rovi, Carmen. "Algebraic Curves over Finite Fields." Thesis, Linköping University, Department of Mathematics, 2010. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-56761.

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<p>This thesis surveys the issue of finding rational points on algebraic curves over finite fields. Since Goppa's construction of algebraic geometric codes, there has been great interest in finding curves with many rational points. Here we explain the main tools for finding rational points on a curve over a nite eld and provide the necessary background on ring and field theory. Four different articles are analyzed, the first of these articles gives a complete set of table showing the numbers of rational points for curves with genus up to 50. The other articles provide interesting constructions

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4

Lockard, Shannon Renee. "Random vectors over finite fields." Connect to this title online, 2007. http://etd.lib.clemson.edu/documents/1181251515/.

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Giuzzi, Luca. "Hermitian varieties over finite fields." Thesis, University of Sussex, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.326913.

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6

Sharkey, Andrew. "Random polynomials over finite fields." Thesis, University of Glasgow, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.299963.

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7

Park, Jang-Woo. "Discrete dynamics over finite fields." Connect to this title online, 2009. http://etd.lib.clemson.edu/documents/1252937730/.

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8

Cooley, Jenny. "Cubic surfaces over finite fields." Thesis, University of Warwick, 2014. http://wrap.warwick.ac.uk/66304/.

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It is well-known that the set of rational points on an elliptic curve forms an abelian group. When the curve is given as a plane cubic in Weierstrass form the group operation is defined via tangent and secant operations. Let S be a smooth cubic surface over a field K. Again one can define tangent and secant operations on S. These do not give S(K) a group structure, but one can still ask for the size of a minimal generating set. In Chapter 2 of the thesis I show that if S is a smooth cubic surface over a field K with at least 4 elements, and if S contains a skew pair of lines defined over K, th

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9

Lotter, Ernest Christiaan. "On towers of function fields over finite fields." Thesis, Stellenbosch : University of Stellenbosch, 2007. http://hdl.handle.net/10019.1/1283.

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Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2007.<br>Explicit towers of algebraic function fields over finite fields are studied by considering their ramification behaviour and complete splitting. While the majority of towers in the literature are recursively defined by a single defining equation in variable separated form at each step, we consider towers which may have different defining equations at each step and with arbitrary defining polynomials. The ramification and completely splitting loci are analysed by directed graphs with irreducible polynomials as vert

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10

Lötter, Ernest C. "On towers of function fields over finite fields /." Link to the online version, 2007. http://hdl.handle.net/10019.1/1283.

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11

Imran, Muhammad. "Reducibility of Polynomials over Finite Fields." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2012. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-17994.

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Reducibility of certain class of polynomials over Fp, whose degree depends on p, can be deduced by checking the reducibility of a quadratic and cubic polynomial. This thesis explains how can we speeds up the reducibility procedure.

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12

Hua, Jiuzhao Mathematics &amp Statistics Faculty of Science UNSW. "Representations of quivers over finite fields." Awarded by:University of New South Wales. Mathematics & Statistics, 1998. http://handle.unsw.edu.au/1959.4/40405.

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The main purpose of this thesis is to obtain surprising identities by counting the representations of quivers over finite fields. A classical result states that the dimension vectors of the absolutely indecomposable representations of a quiver ?? are in one-to-one correspondence with the positive roots of a root system ??, which is infinite in general. For a given dimension vector ?? ??? ??+, the number A??(??, q), which counts the isomorphism classes of the absolutely indecomposable representations of ?? of dimension ?? over the finite field Fq, turns out to be a polynomial in q with integer

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13

Sze, Christopher. "Certain diagonal equations over finite fields." [Tampa, Fla] : University of South Florida, 2009. http://purl.fcla.edu/usf/dc/et/SFE0003018.

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14

Liu, Xiaoyu Wilson R. M. "On divisible codes over finite fields /." Diss., Pasadena, Calif. : Caltech, 2006. http://resolver.caltech.edu/CaltechETD:etd-05252006-010331.

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15

Giangreco, Maidana Alejandro José. "Cyclic abelian varieties over finite fields." Thesis, Aix-Marseille, 2019. http://www.theses.fr/2019AIXM0316.

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L'ensemble A(k) des points rationnels d'une variété abélienne A définie sur un corps fini k forme un groupe abélien fini. Ce groupe convient pour des multiples applications, et sa structure est très importante. Connaître les possibles structures de groupe des A(k) et quelques statistiques est donc fondamental. Dans cette thèse, on s'intéresse aux "variétés cycliques", i.e. variétés abéliennes définies sur des corps finis avec groupe des points rationnels cyclique.Les isogénies nous donnent une classification plus grossière que celle donnée par les classes d'isomorphisme des variétés abéliennes

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16

Colon-Reyes, Omar. "Monomial Dynamical Systems over Finite Fields." Diss., Virginia Tech, 2005. http://hdl.handle.net/10919/27415.

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Linking the structure of a system with its dynamics is an important problem in the theory of finite dynamical systems. For monomial dynamical systems, that is, a system that can be described by monomials, information about the limit cycles can be obtained from the monomials themselves. In particular, this work contains sufficient and necessary conditions for a monomial dynamical system to have only fixed points as limit cycles.<br>Ph. D.

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17

Hanif, Sajid, and Muhammad Imran. "Factorization Algorithms for Polynomials over Finite Fields." Thesis, Linnéuniversitetet, Institutionen för datavetenskap, fysik och matematik, DFM, 2011. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-11553.

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Integer factorization is a dicult task. Some cryptosystem such asRSA (which stands for Rivest, Shamir and Adleman ) are in fact designedaround the diculty of integer factorization.For factorization of polynomials in a given nite eld Fp we can useBerlekamp's and Zassenhaus algorithms. In this project we will see howBerlekamp's and Zassenhaus algorithms work for factorization of polyno-mials in a nite eld Fp. This project is aimed toward those with interestsin computational algebra, nite elds, and linear algebra.

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18

Spencer, Andrew. "A study of matrices over finite fields." Thesis, University of Oxford, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.365392.

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19

Hammarhjelm, Gustav. "Construction of Irreducible Polynomials over Finite Fields." Thesis, Uppsala universitet, Algebra och geometri, 2014. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-224900.

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20

Grout, Jason Nicholas. "The Minimum Rank Problem Over Finite Fields." Diss., CLICK HERE for online access, 2007. http://contentdm.lib.byu.edu/ETD/image/etd1995.pdf.

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21

Stones, Brendan. "Aspects of harmonic analysis over finite fields." Thesis, University of Edinburgh, 2005. http://hdl.handle.net/1842/14492.

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In this thesis we study three topics in Harmonic Analysis in the finite field setting. The methods used are purely combinatorial in nature. We prove a sharp result for the maximal operator associated to dilations of quadric surfaces. We use Christ’s method ([Christ, Convolution, Curvature and Combinatorics. A case study, International Math. Research Notices 19 (1998)]), for L<sup>p</sup>→ L<sup>q</sup> estimates for convolution with the twisted n-bic curve in the European setting, to give L<sup>p</sup> → L<sup>q</sup> estimates for convolution with k-dimensional surfaces in the finite field se

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22

Cenk, Murat. "Results On Complexity Of Multiplication Over Finite Fields." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/3/12610363/index.pdf.

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Let n and l be positive integers and f (x) be an irreducible polynomial over Fq such that ldeg( f (x)) &lt<br>2n - 1, where q is 2 or 3. We obtain an effective upper bound for the multiplication complexity of n-term polynomials modulo f (x)^l. This upper bound allows a better selection of the moduli when Chinese Remainder Theorem is used for polynomial multiplication over Fq. We give improved formulae to multiply polynomials of small degree over Fq. In particular we improve the best known multiplication complexities over Fq in the literature in some cases. Moreover, we present a method for mul

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23

Prešern, Mateja. "Existence problems of primitive polynomials over finite fields." Connect to e-thesis. Move to record for print version, 2007. http://theses.gla.ac.uk/50/.

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Thesis (Ph.D.) - University of Glasgow, 2007.<br>Ph.D. thesis submitted to the Department of Mathematics, Faculty of Information and Mathematical Sciences, University of Glasgow, 2007. Includes bibliographical references.

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24

Mohseni, Rajaei Sedighe. "Rational invariants of orthogonal groups over finite fields." Thesis, Queen Mary, University of London, 1998. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.301116.

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25

Fernando, Neranga. "A Study of Permutation Polynomials over Finite Fields." Scholar Commons, 2013. http://scholarcommons.usf.edu/etd/4484.

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Let p be a prime and q = pk. The polynomial gn,q isin Fp[x] defined by the functional equation Sigmaa isin Fq (x+a)n = gn,q(xq- x) gives rise to many permutation polynomials over finite fields. We are interested in triples (n,e;q) for which gn,q is a permutation polynomial of Fqe. In Chapters 2, 3, and 4 of this dissertation, we present many new families of permutation polynomials in the form of gn,q. The permutation behavior of gn,q is becoming increasingly more interesting and challenging. As we further explore the permutation behavior of gn,q, there is a clear indication that gn,q is a p

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26

GOMEZ-CALDERON, JAVIER. "POLYNOMIALS WITH SMALL VALUE SET OVER FINITE FIELDS." Diss., The University of Arizona, 1986. http://hdl.handle.net/10150/183933.

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Let K(q) be the finite field with q elements and characteristic p. Let f(x) be a monic polynomial of degree d with coefficients in K(q). Let C(f) denote the number of distinct values of f(x) as x ranges over K(q). It is easy to show that C(f) ≤ [|(q - 1)/d|] + 1. Now, there is a characterization of polynomials of degree d < √q for which C(f) = [|(q - 1)/d|] +1. The main object of this work is to give a characterization for polynomials of degree d < ⁴√q for which C(f) < 2q/d. Using two well known theorems: Hurwitz genus formula and Andre Weil's theorem, the Riemann Hypothesis for Algebraic Fun

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27

Poolakkaparambil, Mahesh. "Multiple bit error correcting architectures over finite fields." Thesis, Oxford Brookes University, 2012. https://radar.brookes.ac.uk/radar/items/f9340342-9c82-415c-99aa-7d58e931e640/1/.

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This thesis proposes techniques to mitigate multiple bit errors in GF arithmetic circuits. As GF arithmetic circuits such as multipliers constitute the complex and important functional unit of a crypto-processor, making them fault tolerant will improve the reliability of circuits that are employed in safety applications and the errors may cause catastrophe if not mitigated. Firstly, a thorough literature review has been carried out. The merits of efcient schemes are carefully analyzed to study the space for improvement in error correction, area and power consumption. Proposed error correction

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28

Roy, Sankhadip. "Trace Forms Over Finite Fields of Characteristic Two." OpenSIUC, 2011. https://opensiuc.lib.siu.edu/dissertations/345.

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This dissertation provides elementary facts about trace forms over finite fields of characteristic two in order to generalize some results concerning Gold functions. Then determines new set of maximal Artin-Schreier curves with possible pair of invariants, finds possible trace forms with specific co-dimemntion and provides a new family of binary sequences having low correlation values.

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29

Berardini, Elena. "Algebraic geometry codes from surfaces over finite fields." Thesis, Aix-Marseille, 2020. http://www.theses.fr/2020AIXM0170.

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Nous proposons, dans cette thèse, une étude théorique des codes géométriques algébriques construits à partir de surfaces définies sur les corps finis. Nous prouvons des bornes inférieures pour la distance minimale des codes sur des surfaces dont le diviseur canonique est soit nef soit anti-strictement nef et sur des surfaces sans courbes irréductibles de petit genre. Nous améliorons ces bornes inférieures dans le cas des surfaces dont le nombre de Picard arithmétique est égal à un, des surfaces sans courbes de petite auto-intersection et des surfaces fibrées. Ensuite, nous appliquons ces borne

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30

Schram, Erin Jay. "T-designs on vector spaces over finite fields /." The Ohio State University, 1989. http://rave.ohiolink.edu/etdc/view?acc_num=osu1487675687175974.

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31

Presern, Mateja. "Existence problems of primitive polynomials over finite fields." Thesis, University of Glasgow, 2007. http://theses.gla.ac.uk/50/.

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This thesis concerns existence of primitive polynomials over finite fields with one coefficient arbitrarily prescribed. It completes the proof of a fundamental conjecture of Hansen and Mullen (1992), which asserts that, with some explicable general exceptions, there always exists a primitive polynomial of any degree n over any finite field with an arbitrary coefficient prescribed. This has been proved whenever n is greater than or equal to 9 or n is less than or equal to 3, but was unestablished for n = 4, 5, 6 and 8. In this work, we efficiently prove the remaining cases of the conjecture in

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32

Lappano, Stephen. "Some Results Concerning Permutation Polynomials over Finite Fields." Scholar Commons, 2016. http://scholarcommons.usf.edu/etd/6293.

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Let p be a prime, p a power of p and 𝔽q the finite field with q elements. Any function φ: 𝔽q → 𝔽q can be unqiuely represented by a polynomial, 𝔽φ of degree < q. If the map x ↦ Fφ(x) induces a permutation on the underlying field we say Fφ is a permutation polynomial. Permutation polynomials have applications in many diverse fields of mathematics. In this dissertation we are generally concerned with the following question: Given a polynomial f, when does the map x ↦ F(x) induce a permutation on 𝔽q. In the second chapter we are concerned the permutation behavior of the polynomial gn,q, a q-ary ve

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33

Marseglia, Stefano. "Isomorphism classes of abelian varieties over finite fields." Licentiate thesis, Stockholms universitet, Matematiska institutionen, 2016. http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-130316.

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Deligne and Howe described polarized abelian varieties over finite fields in terms of finitely generated free Z-modules satisfying a list of easy to state axioms. In this thesis we address the problem of developing an effective algorithm to compute isomorphism classes of (principally) polarized abelian varieties over a finite field, together with their automorphism groups. Performing such computations requires the knowledge of the ideal classes (both invertible and non-invertible) of certain orders in number fields. Hence we describe a method to compute the ideal class monoid of an order and w

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34

Witte, Malte. "Noncommutative Iwasawa Main Conjectures for Varieties over Finite Fields." Doctoral thesis, Universitätsbibliothek Leipzig, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:15-20090610-144827-5.

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We state and prove an analogue for varieties over finite fields of T. Fukaya's and K. Kato's version of the noncommutative Iwasawa main conjecture. Moreover, we explain how this statement can be reinterpreted in terms of Waldhausen K-theory.

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35

Thuen, Øystein Øvreås. "Constructing elliptic curves over finite fields using complex multiplication." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2006. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9434.

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<p>We study and improve the CM-method for the creation of elliptic curves with specified group order over finite fields. We include a thorough review of the mathematical theory needed to understand this method. The ability to construct elliptic curves with very special group order is important in pairing-based cryptography.</p>

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36

Panario, Rodriguez Daniel Nelson. "Combinatorial and algebraic aspects of polynomials over finite fields." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1997. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape16/PQDD_0016/NQ28297.pdf.

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Brydon, Duncan. "Exterior squares over finite fields : polynomials, matrices and probabilities." Thesis, University of Oxford, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.394013.

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38

Britnell, John R. "Cycle index methods for matrix groups over finite fields." Thesis, University of Oxford, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.275602.

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39

Lindqvist, Anders. "On four-dimensional unital division algebras over finite fields." Thesis, Uppsala universitet, Algebra och geometri, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-254656.

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40

Lindner, Niels. "Hypersurfaces with defect and their densities over finite fields." Doctoral thesis, Humboldt-Universität zu Berlin, Mathematisch-Naturwissenschaftliche Fakultät, 2017. http://dx.doi.org/10.18452/17704.

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Das erste Thema dieser Dissertation ist der Defekt projektiver Hyperflächen. Es scheint, dass Hyperflächen mit Defekt einen verhältnismäßig großen singulären Ort besitzen. Diese Aussage wird im ersten Kapitel der Dissertation präzisiert und für Hyperflächen mit beliebigen isolierten Singularitäten über einem Körper der Charakteristik null, sowie für gewisse Klassen von Hyperflächen in positiver Charakteristik bewiesen. Darüber hinaus lässt sich die Dichte von Hyperflächen ohne Defekt über einem endlichen Körper abschätzen. Schließlich wird gezeigt, dass eine nicht-faktorielle Hyperfläche der D

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41

Cam, Vural. "Drinfeld Modular Curves With Many Rational Points Over Finite Fields." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613118/index.pdf.

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In our study Fq denotes the finite field with q elements. It is interesting to construct curves of given genus over Fq with many Fq -rational points. Drinfeld modular curves can be used to construct that kind of curves over Fq . In this study we will use reductions of the Drinfeld modular curves X_{0} (n) to obtain curves over finite fields with many rational points. The main idea is to divide the Drinfeld modular curves by an Atkin-Lehner involution which has many fixed points to obtain a quotient with a better #{rational points} /genus ratio. If we divide the Drinfeld modular curve X_{0} (n)

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42

Bradford, Jeremy. "Commutative endomorphism rings of simple abelian varieties over finite fields." Thesis, University of Maryland, College Park, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=3557641.

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<p>In this thesis we look at simple abelian varieties defined over a finite field <i>k</i> = [special characters omitted]<i><sub>p</sub>n</i> with End<i><sub>k</sub></i>(<i> A</i>) commutative. We derive a formula that connects the <i>p </i>-rank <i>r(A)</i> with the splitting behavior of <i> p</i> in <i>E</i> = [special characters omitted](π), where π is a root of the characteristic polynomial of the Frobenius endomorphism. We show how this formula can be used to explicitly list all possible splitting behaviors of <i>p</i> in [special characters omitted]<i><sub>E</sub></i>, and we do

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43

Hines, Peter Anthony. "The linear complexity of de Bruijn sequences over finite fields." Thesis, Royal Holloway, University of London, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.313736.

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44

Abu, Salem Fatima Khaled. "Factorisation algorithms for univariate and bivariate polynomials over finite fields." Thesis, University of Oxford, 2004. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.403928.

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45

Van, Zyl Jacobus Visser. "On the Latimer-MacDuffee theorem for polynomials over finite fields." Thesis, Stellenbosch : University of Stellenbosch, 2011. http://hdl.handle.net/10019.1/6581.

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Thesis (PhD (Mathematical Sciences))--University of Stellenbosch, 2011.<br>Includes bibliography.<br>ENGLISH ABSTRACT: Latimer & MacDuffee showed in 1933 that there is a one-to-one correspondence between equivalence classes of matrices with a given minimum polynomial and equivalence classes of ideals of a certain ring. In the case where the matrices are taken over the integers, Behn and Van der Merwe developed an algorithm in 2002 to produce a representative in each equivalence class. We extend this algorithm to matrices taken over the ring Fq[T] of polynomials over a finite field and pr

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46

Hart, Derrick. "Explorations of geometric combinatorics in vector spaces over finite fields." Diss., Columbia, Mo. : University of Missouri-Columbia, 2008. http://hdl.handle.net/10355/5585.

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Thesis (Ph. D.)--University of Missouri-Columbia, 2008.<br>The entire dissertation/thesis text is included in the research.pdf file; the official abstract appears in the short.pdf file (which also appears in the research.pdf); a non-technical general description, or public abstract, appears in the public.pdf file. Title from title screen of research.pdf file (viewed on June 8, 2009) Vita. Includes bibliographical references.

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47

Karaoglu, Fatma. "The cubic surfaces with twenty-seven lines over finite fields." Thesis, University of Sussex, 2018. http://sro.sussex.ac.uk/id/eprint/78533/.

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In this thesis, we classify the cubic surfaces with twenty-seven lines in three dimensional projective space over small finite fields. We use the Clebsch map to construct cubic surfaces with twenty-seven lines in PG(3; q) from 6-arcs not on a conic in PG(2; q). We introduce computational and geometrical procedures for the classification of cubic surfaces over the finite field Fq. The performance of the algorithms is illustrated by the example of cubic surfaces over F13, F17 and F19.

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48

Mou, Chenqi. "Solving polynomial systems over finite fields : Algorithms, Implementations and applications." Paris 6, 2013. http://www.theses.fr/2013PA066805.

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Résolution de systèmes polynomiaux sur les corps finis est d’un intérêt particulier en raison de ses applications en Cryptographie, Théorie du Codage, et d’autres domaines de la science de l’information. Dans cette thèse, nous étudions plusieurs aspects importants théoriques et informatiques pour résolution de systèmes polynomiaux sur les corps finis, en particulier sur les deux outils largement utiliss: bases de Gröbner et ensembles triangulaires. Nous proposons des algorithmes efficaces pour le changement de l’ordre des bases de Gröbner d’idéaux de dimension zéro en utilisant le faible densi

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Lester, Jeremy W. "The Elliptic Curve Group Over Finite Fields: Applications in Cryptography." Youngstown State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=ysu1348847698.

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Kirlar, Baris Bulent. "Isomorphism Classes Of Elliptic Curves Over Finite Fields Of Characteristic Two." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/2/12606489/index.pdf.

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In this thesis, the work of Menezes on the isomorphism classes of elliptic curves over finite fields of characteristic two is studied. Basic definitions and some facts of the elliptic curves required in this context are reviewed and group structure of elliptic curves are constructed. A fairly detailed investigation is made for the isomorphism classes of elliptic curves due to Menezes and Schoof. This work plays an important role in Elliptic Curve Digital Signature Algorithm. In this context, those isomorphism classes of elliptic curves recommended by National Institute of Standards and Tech

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