Dissertations / Theses: 'Maps over finite fields'

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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 $mathbb{C}(t)$. In the former case, this yields an explanation of the aforementioned signature over finite fields of integrability. In the latter case we arrive at an analogue of the Arnol'd-Liouville theorem. Other results that are related to this approach to integrability are also proven and their consequences considered in examples. Of particular importance are two separate items: (i) we define a generalization of integrability called mixing and examine its relation to integrability; and (ii) we use the concept of rotation number to study differences and similarities between birational integrable maps that preserve the same foliation. The final chapter is given over to considering the existence of the signature of reversibility in higher (three and four) dimensional maps. A conjecture regarding the distribution of periodic orbits generated by such maps when considered over finite fields is given along with numerical evidence to support the conjecture.

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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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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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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 of covering curves: covers by the Hemitian curve, Kummer extensions and Artin-Schreier extensions. With these articles the great difficulty of finding explicit equations for curves with many rational points is overcome. With the method given by Arnaldo García in [6] we have been able to nd examples that can be used to define the lower bounds for the corresponding entries in the tables given in http: //wins.uva.nl/~geer, which to the time of writing this Thesis appear as "no information available". In fact, as the curves found are maximal, these entries no longer need a bound, they can be given by a unique entry, since the exact value of N_q(g) is now known.

At the end of the thesis an outline of the construction of Goppa codes is given and the NXL and XNL codes are presented.

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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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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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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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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, then any non-Eckardt K-point on either line generates S(K). This strengthens a result of Siksek [20]. In Chapter 3, I show that if S is a smooth cubic surface over a finite field K = Fq with at least 8 elements, and if S contains at least one K-line, then there is some point P > S(K) that generates S(K). In Chapter 4, I consider cubic surfaces S over finite fields K = Fq that contain no K-lines. I find a lower bound for the proportion of points generated when starting with a non-Eckardt point P > S(K) and show that this lower bound tends to 1/6 as q tends to infinity. In Chapter 5, I define c-invariants of cubic surfaces over a finite field K = Fq with respect to a given K-line contained in S, give several results regarding these c-invariants and relate them to the number of points SS(K)S. In Chapter 6, I consider the problem of enumerating cubic surfaces over a finite field, K = Fq, with a given point, P > S(K), up to an explicit equivalence relation.

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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.
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 vertices. Algorithms are exhibited to construct these graphs in the case of n-step and -finite towers. These techniques are applied to find new tamely ramified n-step towers for 1 n 3. Various new tame towers are found, including a family of towers of cubic extensions for which numerical evidence suggests that it is asymptotically optimal over the finite field with p2 elements for each prime p 5. Families of wildly ramified Artin-Schreier towers over small finite fields which are candidates to be asymptotically good are also considered using our method.

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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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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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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 coefficients, which have been conjectured to be nonnegative by Kac. The main result of this thesis is a multi-variable formal identity which expresses an infinite series as a formal product indexed by ??+ which has the coefficients of various polynomials A??(??, q) as exponents. This identity turns out to be a qanalogue of the remarkable Weyl-Macdonald-Kac denominator identity modulus a conjecture of Kac, which asserts that the multiplicity of ?? is equal to the constant term of A??(??, q). An equivalent form of this conjecture is established and a partial solution is obtained. A new proof of the integrality of A??(??, q) is given. Three Maple programs have been included which enable one to calculate the polynomials A??(??, q) for quivers with at most three nodes. All sample out-prints are consistence with Kac???s conjectures. Another result of this thesis is as follows. Let A be a finite dimensional algebra over a perfect field K, M be a finitely generated indecomposable module over A ???K ??K. Then there exists a unique indecomposable module M??? over A such that M is a direct summand of M??? ???K ??K, and there exists a positive integer s such that Ms = M ??? ?? ?? ?? ??? M (s copies) has a unique minimal field of definition which is isomorphic to the centre of End ??(M???) rad (End ??(M???)). If K is a finite field, then s can be taken to be 1.

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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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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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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, mais elles offrent un outil très puissant en géométrie algébrique. Chaque classe d'isogénie est déterminée par son polynôme de Weil. On donne un critère pour caractériser les "classes d'isogénies cycliques", i.e. classes d'isogénies de variétés abéliennes définies sur des corps finis qui contiennent seulement des variétés cycliques. Ce critère est basé sur le polynôme de Weil de la classe d'isogénie.À partir de cela, on donne des bornes de la proportion de classes d'isogénies cycliques parmi certaines familles de classes d'isogénies paramétrées par ses polynômes de Weil.On donne aussi la proportion de classes d'isogénies cycliques "locaux" parmi les classes d'isogénie définies sur des corps finis mathbb{F}_q avec q éléments, quand q tend à l'infini
The set A(k) of rational points of an abelian variety A defined over a finite field k forms a finite abelian group. This group is suitable for multiple applications, and its structure is very important. Knowing the possible group structures of A(k) and some statistics is then fundamental. In this thesis, we focus our interest in "cyclic varieties", i.e. abelian varieties defined over finite fields with cyclic group of rational points. Isogenies give us a coarser classification than that given by the isomorphism classes of abelian varieties, but they provide a powerful tool in algebraic geometry. Every isogeny class is determined by its Weil polynomial. We give a criterion to characterize "cyclic isogeny classes", i.e. isogeny classes of abelian varieties defined over finite fields containing only cyclic varieties. This criterion is based on the Weil polynomial of the isogeny class.From this, we give bounds on the fractions of cyclic isogeny classes among certain families of isogeny classes parameterized by their Weil polynomials.Also we give the proportion of "local"-cyclic isogeny classes among the isogeny classes defined over the finite field mathbb{F}_q with q elements, when q tends to infinity

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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.
Ph. D.

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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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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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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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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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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^p→ L^q estimates for convolution with the twisted n-bic curve in the European setting, to give L^p → L^q estimates for convolution with k-dimensional surfaces in the finite field setting. We give solution to the k-plane Radon transform problem and embark on a study of a generalisation of this problem.

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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)) <
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 multiplication in finite fields improving finite field multiplication complexity muq(n) for certain values of q and n. We use local expansions, the lengths of which are further parameters that can be used to optimize the bounds on the bilinear complexity, instead of evaluation into residue class field. We show that we obtain improved bounds for multiplication in Fq^n for certain values of q and n where 2 <
= n <
=18 and q = 2, 3, 4.

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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.
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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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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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 plenteous source of permutation polynomials. We also describe a piecewise construction of permutation polynomials over a finite field Fq which uses a subgroup of Fq*, a “selection” function, and several “case” functions. Chapter 5 of this dissertation is devoted to this piecewise construction which generalizes several recently discovered families of permutation polynomials.

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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 Function Fields, it is shown that if d < ⁴√q and C(f) < 2q/d then f(x) - f(y) factors into at least d/2 absolutely irreducible factors and f(x) has one of the following forms: (UNFORMATTED TABLE FOLLOWS) f(x - λ) = D(d,a)(x) + c, d|(q² - 1), f(x - λ) = D(r,a)(∙ ∙ ∙ ((x²+b₁)²+b₂)²+ ∙ ∙ ∙ +b(m)), d|(q² - 1), d=2ᵐ∙r, and (2,r) = 1 f(x - λ) = (x² + a)ᵈ/² + b, d/2|(q - 1), f(x - λ) = (∙ ∙ ∙((x²+b₁)²+b₂)² + ∙ ∙ ∙ +b(m))ʳ+c, d|(q - 1), d=2ᵐ∙r, f(x - λ) = xᵈ + a, d|(q - 1), f(x - λ) = x(x³ + ax + b) + c, f(x - λ) = x(x³ + ax + b) (x² + a) + e, f(x - λ) = D₃,ₐ(x² + c), c² ≠ 4a, f(x - λ) = (x³ + a)ⁱ + b, i = 1, 2, 3, or 4, f(x - λ) = x³(x³ + a)³ +b, f(x - λ) = x⁴(x⁴ + a)² +b or f(x - λ) = (x⁴ + a) ⁱ + b, i = 1,2 or 3, where D(d,a)(x) denotes the Dickson’s polynomial of degree d. Finally to show other polynomials with small value set, the following equation is obtained C((fᵐ + b)ⁿ) = αq/d + O(√q) where α = (1 – (1 – 1/m)ⁿ)m and the constant implied in O(√q) is independent of q.

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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 schemes include bit parallel ones using optimized BCH codes that are useful in applications where power and area are not prime concerns. The scheme is also extended to dynamically correcting scheme to reduce decoder delay. Other method that suits low power and area applications such as RFIDs and smart cards using cross parity codes is also proposed. The experimental evaluation shows that the proposed techniques can mitigate single and multiple bit errors with wider error coverage compared to existing methods with lesser area and power consumption. The proposed scheme is used to mask the errors appearing at the output of the circuit irrespective of their cause. This thesis also investigates the error mitigation schemes in emerging technologies (QCA, CNTFET)to compare area, power and delay with existing CMOS equivalent. Though the proposed novel multiple error correcting techniques can not ensure 100% error mitigation, inclusion of these techniques to actual design can improve the reliability of the circuits or increase the dif culty in hacking crypto-devices. Proposed schemes can also be extended to non GF digital circuits.

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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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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 bornes aux surfaces plongées dans P3. Une attention particulière est accordée aux codes construits à partir des surfaces abéliennes. Dans ce contexte, nous donnons une borne générale sur la distance minimale et nous démontrons que cette estimation peut être améliorée en supposant que la surface abélienne ne contient pas de courbes absolument irréductibles de petit genre. Dans cette optique nous caractérisons toutes les surfaces abéliennes qui ne contiennent pas de courbes absolument irréductibles de genre inférieur ou égal à 2. Cette approche nous conduit naturellement à considérer les restrictions de Weil de courbes elliptiques et les surfaces abéliennes qui n'admettent pas de polarisation principale
In this thesis we provide a theoretical study of algebraic geometry codes from surfaces defined over finite fields. We prove lower bounds for the minimum distance of codes over surfaces whose canonical divisor is either nef or anti-strictly nef and over surfaces without irreducible curves of small genus. We sharpen these lower bounds for surfaces whose arithmetic Picard number equals one, surfaces without curves with small self-intersection and fibered surfaces. Then we apply these bounds to surfaces embedded in P3. A special attention is given to codes constructed from abelian surfaces. In this context we give a general bound on the minimum distance and we prove that this estimation can be sharpened under the assumption that the abelian surface does not contain absolutely irreducible curves of small genus. In this perspective we characterize all abelian surfaces which do not contain absolutely irreducible curves of genus up to 2. This approach naturally leads us to consider Weil restrictions of elliptic curves and abelian surfaces which do not admit a principal polarization

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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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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 a selfcontained way and with little computation; this is achieved by separately considering the polynomials with second, third or fourth coefficient prescribed, and in each case developing methods involving the use of character sums and sieving techniques. When the characteristic of the field is 2 or 3, we also use p-adic analysis. In addition to proving the previously unestablished cases of the conjecture, we also offer shorter and self-contained proof of the conjecture when the first coefficient of the polynomial is prescibed, and of some other cases where the proof involved a large amount of computation. For degrees n = 6, 7 and 8 and selected values of m, we also prove the existence of primitive polynomials with two coefficients prescribed (the constant term and any other coefficient).

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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 version of the reversed Dickson polynomial, when the integer n is of the form n = qa - qb - 1. This leads to the third chapter where we consider binomials and trinomials taking special forms. In this case we are able to give explicit conditions that guarantee the given binomial or trinomial is a permutation polynomial. In the fourth chapter we are concerned with permutation polynomials of 𝔽q, where q is even, that can be represented as the sum of a power function and a linearized polynomial. These types of permutation polynomials have applications in cryptography. Lastly, chapter five is concerned with a conjecture on monomial graphs that can be formulated in terms of polynomials over finite fields.

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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 we produce concrete computations in dimension 2, 3 and 4.

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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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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.

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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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37

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 Dimension drei mit isolierten Singularitäten stets Defekt besitzt. Das zweite Kapitel der Dissertation behandelt Bertini-Sätze über endlichen Körpern, aufbauend auf Poonens Formel für die Dichte glatter Hyperflächenschnitte in einer glatten Umgebungsvarietät. Diese wird auf quasiglatte Hyperflächen in simpliziellen torischen Varietäten verallgemeinert. Die Hauptanwendung ist zu zeigen, dass Hyperflächen mit einem in Relation zum Grad großen singulären Ort die Dichte null haben. Weiterhin enthält das Kapitel einen Bertini-Irreduzibilitätssatz, der auf einer Arbeit von Charles und Poonen beruht. Im dritten Kapitel werden ebenfalls Dichten über endlichen Körpern untersucht. Zunächst werden gewisse Faserungen über glatten projektiven Basisvarietäten in einem gewichteten projektiven Raum betrachtet. Das erste Resultat ist ein Bertini-Satz für glatte Faserungen, der Poonens Formel über glatte Hyperflächen impliziert. Der letzte Abschnitt behandelt elliptische Kurven über einem Funktionskörper einer Varietät der Dimension mindestens zwei. Die zuvor entwickelten Techniken ermöglichen es, eine untere Schranke für die Dichte solcher Kurven mit Mordell-Weil-Rang null anzugeben. Dies verbessert ein Ergebnis von Kloosterman.
The first topic of this dissertation is the defect of projective hypersurfaces. It is indicated that hypersurfaces with defect have a rather large singular locus. In the first chapter of this thesis, this will be made precise and proven for hypersurfaces with arbitrary isolated singularities over a field of characteristic zero, and for certain classes of hypersurfaces in positive characteristic. Moreover, over a finite field, an estimate on the density of hypersurfaces without defect is given. Finally, it is shown that a non-factorial threefold hypersurface with isolated singularities always has defect. The second chapter of this dissertation deals with Bertini theorems over finite fields building upon Poonen’s formula for the density of smooth hypersurface sections in a smooth ambient variety. This will be extended to quasismooth hypersurfaces in simplicial toric varieties. The main application is to show that hypersurfaces admitting a large singular locus compared to their degree have density zero. Furthermore, the chapter contains a Bertini irreducibility theorem for simplicial toric varieties generalizing work of Charles and Poonen. The third chapter continues with density questions over finite fields. In the beginning, certain fibrations over smooth projective bases living in a weighted projective space are considered. The first result is a Bertini-type theorem for smooth fibrations, giving back Poonen’s formula on smooth hypersurfaces. The final section deals with elliptic curves over a function field of a variety of dimension at least two. The techniques developed in the first two sections allow to produce a lower bound on the density of such curves with Mordell-Weil rank zero, improving an estimate of Kloosterman.

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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) by an involution W, then the number of rational points of the quotient curve WX_{0} (n) is not less than half of the original number. On the other hand, if this involution has many fixed points, then by the Hurwitz-Genus formula the genus of the curve WX_{0} (n) is much less than half of the g (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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In this thesis we look at simple abelian varieties defined over a finite field k = [special characters omitted]_pn with End_k( A) commutative. We derive a formula that connects the p -rank r(A) with the splitting behavior of p in E = [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 p in [special characters omitted]_E, and we do so for abelian varieties of dimension less than or equal to four defined over [special characters omitted]_p. We then look for when p divides [[special characters omitted]_E : [special characters omitted][π, π]]. This allows us to prove that the endomorphism ring of an absolutely simple abelian surface is maximal at p when p ≥ 3. We also derive a condition that guarantees that p divides [[special characters omitted]_E: [special characters omitted][π, π]]. Last, we explicitly describe the structure of some intermediate subrings of p-power index between [special characters omitted][π, π] and [special characters omitted]_E when A is an abelian 3-fold with r(A) = 1.

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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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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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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.
Includes bibliography.
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 prove a modified version of the Latimer-MacDuffee theorem which holds for proper equivalence classes of matrices.
AFRIKAANSE OPSOMMING: Latimer & MacDuffee het in 1933 bewys dat daar 'n een-tot-een korrespondensie is tussen ekwivalensieklasse van matrikse met 'n gegewe minimumpolinoom en ekwivalensieklasse van ideale van 'n sekere ring. In die geval waar die matrikse heeltallige inskrywings het, het Behn en Van der Merwe in 2002 'n algoritme ontwikkel om verteenwoordigers in elke ekwivalensieklas voort te bring. Ons brei hierdie algoritme uit na die geval van matrikse met inskrywings in die ring Fq[T] van polinome oor 'n eindige liggaam en ons bewys 'n gewysigde weergawe van die Latimer-MacDuffee stelling wat geld vir klasse van streng ekwivalente matrikse.

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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.
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 densité des matrices de multiplication et d’évaluer telle faible densité pour les systèmes de polynômes génériques. Algorithmes originaux sont présentés pour la décomposition des ensembles de polynômes en ensembles triangulaires simples sur les corps finis. Nous définissons également décomposition sans carré et factorisation des polynômes sur produits non mélangés d’extensions des corps et proposons des lgorithmes pour les calculer. L’efficacité et l’efficience de ces algorithmes ont été vérifiées par des expériences avec nos implémentations. Méthodes de résolution de systèmes polynomiaux sur les corps finis sont également appliquées pour résoudre les problèmes pratiques posés par la Biologie et la Théorie du Codage
Polynomial system solving over finite fields is of particular interest because of its applications in Cryptography, Coding Theory, and other areas of information science and technologies. In this thesis we study several important theoretical and computational aspects for solving polynomial systems over finite fields, in particular on the two widely used tools Gröbner bases and triangular sets. We propose efficient algorithms for change of ordering of Gröbner bases of zero-dimensional ideals by using the sparsity of multiplication matrices and evaluate such sparsity for generic polynomial systems. Original algorithms are presented for decomposing polynomial sets into simple triangular sets over finite fields. We also define squarefree decomposition and factorization of polynomials over unmixed products of field extensions and propose algorithms for computing them. The effectiveness and efficiency of these algorithms have been verified by experiments with our implementations. Methods for polynomial system solving over finite fields are also applied to solve practical problems arising from Biology and Coding Theory

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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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50

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 Technology are listed and their properties are discussed.

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Dissertations / Theses on the topic 'Maps over finite fields'

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