J. W. P. Hirschfeld

Bio: J. W. P. Hirschfeld is an academic researcher from University of Sussex. The author has contributed to research in topics: Finite geometry & Projective space. The author has an hindex of 23, co-authored 56 publications receiving 4582 citations. Previous affiliations of J. W. P. Hirschfeld include University of Brighton.

Cubic Curves, Finite Geometry and Cryptography

Abstract: Some geometry on non-singular cubic curves, mainly over finite fields, is surveyed. Such a curve has 9,3,1 or 0 points of inflexion, and cubic curves are classified accordingly. The group structure and the possible numbers of rational points are also surveyed. A possible strengthening of the security of elliptic curve cryptography is proposed using a `shared secret' related to the group law. Cubic curves are also used in a new way to construct sets of points having various combinatorial and geometric properties that are of particular interest in finite Desarguesian planes.

Geometry, combinatorial designs and related structures : proceedings of the First Pythagorean Conference

Abstract: Preface Introduction 1. On maximum size anti-Pasch sets of triples 2. Some simple 7-designs 3. Inscribed bundles, Veronese surfaces and caps 4. Embedding partial geometries in Steiner designs 5. Finite geometry after Aschbacher's theorem: PGL(n,q) from a Kleinian viewpoint 6. The Hermitian function field arising from a cyclic arc in a Galois plane 7. Intercalates everywhere 8. Difference sets: an update 9. Computational results for the known biplanes of order 9 10. A survey of small embeddings for partial cycle systems 11. Rosa triple systems 12. Searching for spreads and packings 13. A note of Buekenhout-Metz unitals 14. Elation generalized quadrangles of order (q^2, q) 15. Uniform parallelisms of PG(3,3) 16. Double-fives and partial spreads in PG(5,2) 17. Rank three geometries with simplicial residues 18. Generalized quadrangles and the Axiom of Veblen Talks Participants.

Classification of cubic surfaces with twenty-seven lines over the finite field of order thirteen

Abstract: In Hirschfeld (J Austral Math Soc 4(1):83–89, 1964), the existence of the cubic surface which arises from a double-six over the finite field of order four was considered. In Hirschfeld (Rend Mat Appl 26:115–152, 1967), the existence and the properties of the cubic surfaces over the finite fields of odd and even order was discussed and classified over the fields of order seven, eight, nine. In this paper, cubic surfaces with twenty-seven lines over the finite field of thirteen elements are classified.

On the Number of Solutions of an Equation Over a Finite Field

Abstract: The Stohr–Voloch approach is used to obtain a new bound for the number of solutions in ( F q ) 2 of an equation f ( X , Y ) = 0, where f ( X , Y ) is an absolutely irreducible polynomial with coefficients in a finite field F q .

Analytic Number Theory

Abstract: Introduction Arithmetic functions Elementary theory of prime numbers Characters Summation formulas Classical analytic theory of $L$-functions Elementary sieve methods Bilinear forms and the large sieve Exponential sums The Dirichlet polynomials Zero-density estimates Sums over finite fields Character sums Sums over primes Holomorphic modular forms Spectral theory of automorphic forms Sums of Kloosterman sums Primes in arithmetic progressions The least prime in an arithmetic progression The Goldbach problem The circle method Equidistribution Imaginary quadratic fields Effective bounds for the class number The critical zeros of the Riemann zeta function The spacing of zeros of the Riemann zeta-function Central values of $L$-functions Bibliography Index.

Introduction to finite fields and their applications

Abstract: The first part of this book presents an introduction to the theory of finite fields, with emphasis on those aspects that are relevant for applications. The second part is devoted to a discussion of the most important applications of finite fields especially information theory, algebraic coding theory and cryptology (including some very recent material that has never before appeared in book form). There is also a chapter on applications within mathematics, such as finite geometries. combinatorics. and pseudorandom sequences. Worked-out examples and list of exercises found throughout the book make it useful as a textbook.

A course in combinatorics

Abstract: This is the second edition of a popular book on combinatorics, a subject dealing with ways of arranging and distributing objects, and which involves ideas from geometry, algebra and analysis. The breadth of the theory is matched by that of its applications, which include topics as diverse as codes, circuit design and algorithm complexity. It has thus become essential for workers in many scientific fields to have some familiarity with the subject. The authors have tried to be as comprehensive as possible, dealing in a unified manner with, for example, graph theory, extremal problems, designs, colorings and codes. The depth and breadth of the coverage make the book a unique guide to the whole of the subject. The book is ideal for courses on combinatorical mathematics at the advanced undergraduate or beginning graduate level. Working mathematicians and scientists will also find it a valuable introduction and reference.

The Geometry of Two-Weight Codes

Abstract: On etudie les relations entre les codes [n,k] lineaires a deux poids, les ensembles (n,k,h 1 h 2 ) projectifs et certains graphes fortement reguliers