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1 
New methods for finite field arithmeticYanik, Tu��rul 21 November 2001 (has links)
We describe novel methods for obtaining fast software implementations of the
arithmetic operations in the finite field GF(p) and GF(p[superscript k]). In GF(p) we realize
an extensive speedup in modular addition and subtraction routines and some small
speedup in the modular multiplication routine with an arbitrary prime modulus p
which is of arbitrary length. The most important feature of the method is that it
avoids bitlevel operations which are slow on microprocessors and performs wordlevel
operations which are significantly faster. The proposed method has applications in
publickey cryptographic algorithms defined over the finite field GF(p), most notably
the elliptic curve digital signature algorithm. The new method provides up to 13% speedup in the execution of the ECDSA algorithm over the field GF(p) for the
length of p in the range 161���k���256.
In the finite extension field GF(p[superscript k]) we describe two new methods for obtaining
fast software implementations of the modular multiplication operation with an
arbitrary prime modulus p, which has less bitlength than the wordlength of a microprocessor
and an arbitrary generator polynomial. The second algorithm is a significant
improvement over the first algorithm by using the same concepts introduced
in GF(p) arithmetic. / Graduation date: 2002

2 
POLYNOMIALS WITH SMALL VALUE SET OVER FINITE FIELDS.GOMEZCALDERON, JAVIER. January 1986 (has links)
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.

3 
Existence problems of primitive polynomials over finite fieldsPrešern, Mateja. January 2007 (has links)
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.

4 
THE ANALYSIS OF INTERCONNECTIONS OF SEQUENTIAL MACHINES BY POLYNOMIAL FUNCTIONHunt, Bobby Ray, 1941 January 1967 (has links)
No description available.

5 
The minimum rank problem over finite fields /Grout, Jason Nicholas, January 2007 (has links) (PDF)
Thesis (Ph. D.)Brigham Young University. Dept. of Mathematics, 2007. / Includes bibliographical references (p. 8183).

6 
A survey on CalabiYau manifolds over finite fields.January 2008 (has links)
Mak, Kit Ho. / Thesis (M.Phil.)Chinese University of Hong Kong, 2008. / Includes bibliographical references (leaves 7881). / Abstracts in English and Chinese. / Chapter 1  Introduction  p.7 / Chapter 2  Preliminaries on Number Theory  p.10 / Chapter 2.1  Finite Fields  p.10 / Chapter 2.2  padic Numbers  p.13 / Chapter 2.3  The Teichmuller Representatives  p.16 / Chapter 2.4  Character Theory  p.18 / Chapter 3  Basic CalabiYau Geometry  p.26 / Chapter 3.1  Definition and Basic Properties of CalabiYau Manifolds  p.26 / Chapter 3.2  CalabiYau Manifolds of Low Dimensions  p.29 / Chapter 3.3  Constructions of CalabiYau Manifolds  p.32 / Chapter 3.4  Importance of CalabiYau Manifolds in Physics  p.35 / Chapter 4  Number of Points on CalabiYau Manifolds over Finite Fields  p.39 / Chapter 4.1  The General Method  p.39 / Chapter 4.2  The Number of Points on a Family of CalabiYau Varieties over Finite Fields  p.45 / Chapter 4.2.1  The Case ψ = 0  p.45 / Chapter 4.2.2  The Case ψ ß 0  p.50 / Chapter 4.3  The Number of Points on the Affine Mirrors over Finite Fields  p.55 / Chapter 4.3.1  The Case ψ = 0  p.55 / Chapter 4.3.2  The Case ψ ß 0  p.56 / Chapter 4.4  The Number of points on the Projective Mirror over Finite Fields  p.59 / Chapter 4.5  Summary of the Results and Related Conjectures  p.61 / Chapter 5  The Relation Between Periods and the Number of Points over Finite Fields modulo q  p.67 / Chapter 5.1  Periods of CalabiYau Manifolds  p.67 / Chapter 5.2  The Case for Elliptic Curves  p.69 / Chapter 5.2.1  The Periods of Elliptic Curves  p.69 / Chapter 5.2.2  The Number of Fgpoints on Elliptic Curves Modulo q  p.70 / Chapter 5.3  The Case for a Family of Quintic Threefolds  p.73 / Chapter 5.3.1  The Periods of Xψ  p.73 / Chapter 5.3.2  The Number of F9points on Quintic Three folds Modulo q  p.75 / Bibliography  p.78

7 
Finite projective planes and related combinatorial systemsGlynn, David G. January 1978 (has links) (PDF)
Includes bibliography.

8 
Efficient algorithms for finite fields, with applications in elliptic curve cryptographyBaktir, Selcuk. January 2003 (has links)
Thesis (M.S.)Worcester Polytechnic Institute. / Keywords: multiplication; OTF; optimal extension fields; finite fields; optimal tower fields; cryptography; OEF; inversion; finite field arithmetic; elliptic curve cryptography. Includes bibliographical references (p. 5052).

9 
On the undecidability of certain finite theoriesGarfunkel, Solomon A., January 1967 (has links)
Thesis (Ph. D.)University of Wisconsin, 1967. / Typescript. Vita. eContent providerneutral record in process. Description based on print version record. Includes bibliographical references.

10 
Low complexity normal bases /Thomson, David January 1900 (has links)
Thesis (M.SC.)  Carleton University, 2007. / Includes bibliographical references (p. 108109). Also available in electronic format on the Internet.

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