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Finite Fields with Applications to Coding Theory, Cryptography and Related Areas: Proceedings of the Sixth International Conference on Finite Fields and Applications, held at Oaxaca, México, May 21–25, 2001 PDF

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Finite Fields with Applications to Coding Theory, Cryptography and Related Areas Springer Berlin Heidelberg New York Hong Kong London Milan Paris Tokyo Gary L. Mullen Henning Stichtenoth Horacio Tapia-Recillas Editors Finite Fields with Applications to Coding Theory, Cryptography and Related Areas Proceedings of the Sixth International Conference on Finite Fields and Applications, held at Oaxaca, Mexico, May 21-25, 2001 Springer Editors Gary L. Mullen The Pennsylvania State University Department of Mathematics 16802 University Park, PA, USA e-mail: [email protected] Henning Stichtenoth Universitat Essen FB6 Mathematik und Informatik 45117 Essen, Deutschland e-mail: [email protected] Horacio Tapia-Recillas Universidad Aut6noma Metropolitana, Mexico Departamento de Matematicas Iztapalapa 09340 Av. San Rafael Atlixco 186 09340 Mexico, D.E, Mexico e-mail: [email protected] Cataloging-in-Publication Data applied for Die Deutsche Bibliothek -CIP-Einheitsaufnahme Finite fields with applications to coding theory, cryptography and related areas: proceedings of the Sixth Inter national Conference on Finite Fields and Applications, held at Oaxaca, Mexico, May 21-25, 20011 Gary 1. Mullen ... ed. -Berlin; Heidelberg; New York; Hong Kong; London; Milan; Paris; Singapore; Tokyo: Springer, 2002 ISBN-13:978-3-642-63976-0 e-ISBN-13:978-3-642-59435-9 DOl: 10.1007/978-3-642-59435-9 Mathematics Subject Classification (2000): primary: nTxx; secondary: 05-XX, 51Exx, 94Axx, 94Bxx ISBN -13 :978-3-642-63976-0 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permit ted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg New York a member of BertelsmannSpringer Science+Business Media GmbH http://www.springer.de © Springer-Verlag Berlin Heidelberg 2002 Softcover reprint of the hardcover 1st edition 2002 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. production: PRO EDIT GmbH, Heidelberg Typeset by the authors using a Springer TEX macro package Cover design: design & production GmbH, Heidelberg Printed on acid-free paper SPIN 10881880 46/3142hs - 543210 Preface This volume represents the refereed proceedings of the "Sixth International Conference on Finite Fields and Applications (Fq6)" held in the city of Oaxaca, Mexico, between 22-26 May 200l. The conference was hosted by the Departmento do Matermiticas of the U niversidad Aut6noma Metropolitana Iztapalapa, Nlexico. This event continued a series of biennial international conferences on Finite Fields and Applications, following earlier meetings at the University of Nevada at Las Vegas (USA) in August 1991 and August 1993, the University of Glasgow (Scotland) in July 1995, the University of Waterloo (Canada) in August 1997, and at the University of Augsburg (Ger many) in August 1999. The Organizing Committee of Fq6 consisted of Dieter Jungnickel (University of Augsburg, Germany), Neal Koblitz (University of Washington, USA), Alfred }.lenezes (University of Waterloo, Canada), Gary Mullen (The Pennsylvania State University, USA), Harald Niederreiter (Na tional University of Singapore, Singapore), Vera Pless (University of Illinois, USA), Carlos Renteria (lPN, Mexico). Henning Stichtenoth (Essen Univer sity, Germany). and Horacia Tapia-Recillas, Chair (Universidad Aut6noma l'vIetropolitan-Iztapalapa. Mexico). The program of the conference consisted of four full days and one half day of sessions, with 7 invited plenary talks, close to 60 contributed talks, basic courses in finite fields. cryptography and coding theory and a series of lectures at local educational institutions. Finite fields have an inherently fascinating structure and they are im portant tools in discrete mathematics. Their applications range from com binatorial design tlwory, finite geollletries, and algebraic geometry to coding theory, cryptology. and scientific computing. A particularly fruitful aspect is the interplay between theory and applications which has led to many new perspectives in research on finite fields. This interplay has been a dominant theme in earlier FC] conferences and was very much in evidence at Fq6. Ap plied or applications-oriented topics accounted for a significant part of the program. These proceedings reflect the wide variety of topics represented at the con ference. Most invited talks and a good proportion of the contributed talks are on permanent record here. All contributed talks were screened before the conference and all full papers were carefully refereed. We would like to take this opportunity to thank the members of the Organizing Committee and all referees who helped in these tasks. These colleagues contributed enor mously to the quality of the conference presentations and to guaranteeing high standards for these proceedings. We greatly appreciate the generons financial support received for the con ference. A fair portion of the funds were provided by a grant from the Consejo VI Preface Nacional de Ciencia y Tecnologfa (CONACYT), Mexico and from various of fices of the host institution. We also thank Universidad Benito Juarez de Oaxaca, Instituto Tecnol6gico de Oaxaca, Direcci6n General de Servicios do C6mputo Academico-UNAM, Instituto Politecnico Nacional, Sociedad Matematica Mexicana, Certicom Corp., Institute of Combinatorics and Ap plications, and Red de Criptologfa (CONACYT-UAM) for diverse kinds of support. We are grateful to various offices of the state of Oaxaca who helped with additional funds and organizational issues. Thanks are also due to the Gov ernor of the state of Oaxaca, who gave a reception for the participants in the splendid setting of the Centro Cultural Santo Domingo in the city of Oax aca. Last but not least, the highly efficient and friendly manner in which the conference took place would not have been possible without the enthusiasm and hard work by the assistants, secretaries and students who saw to many details involved in such a major event; we are grateful to all of them. Regarding the present proceedings, we thank Dr. Martin Peters of Sprin ger-Verlag who gave us the opportunity to edit this volume with a top publisher and in an attractive form. Working with him and all the staff at Springer-Verlag is always a pleasure. Finally, we are pleased to confirm that the Fq series will continue with Fq7 in Toulouse, France in May 2003. We expect another lively and stim ulating meeting there, which should, like the previous conferences, serve as an important meeting place for theoretical as well as applied aspects of finite fields. We hope to see you there! May 2002 Horacio Tapia-Recillas Gary Mullen Henning Stichtenoth Contents Commutative Semifields of Rank 2 Over Their Middle Nucleus 1 Simeon Ball and Michel Lavrauw A Rao-Nam like Cryptosystem with Product Codes . . . . . . . . . . 22 Angela I. Barbero and Juan G. Tena Pseudorandom Sequences from Elliptic Curves ............ 37 P.H. T. Beelen and J.M. Doumen On Cryptographic Complexity of Boolean Functions ......... 53 Claude Carlet On Divisibility of Exponential Sums over Finite Fields of Characteristic 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 F.N. Castro and O. Moreno Value Sets of Polynomials over Finite Fields ............. 80 Pinaki Das and Gary L. Mullen Bounds for Completely Decomposable Jacobians ........... 86 Iwan Duursma and Jean-Yves Enjalbert Twin Irreducible Polynomials over Finite Fields . . . . . . .. 94 G. Effinger, Kenneth H. Hicks, and Gary L. Mullen Invariants of Finite Groups over Finite Fields: Recent Progress and New Conjectures ............. 112 Peter Fleischmann The Group Law on Elliptic Curves on Hesse form . . . . . . . . . . . 123 H ege Reithe Frium On Curves with Many Rational Points over Finite Fields " . . . . . 152 Arnalda Garcia VHDL Specification of a FPGA to Divide and Multiply in GF(2m) .. 164 Mario Alberto Garcia-Martinez and Guillermo Morales-Luna Distribution of Irreducible Polynomials over F2 .. . . . . . . . . . . 177 Kenneth H. Hicks, Gary L. Mullen, and Ikuro Sat a VIII Contents Arithmetic on a Family of Picard Curves .............. 187 Rolf-Peter Holzapfel and Florin Nicolae New Quantum Error-Correcting Codes from Hermitian Self-Orthogonal Codes over G F ( 4) .................. 209 Jon-Lark Kim A Note on the Counter-Example of Patterson-Wiedemann ...... 214 Philippe Langevin and Jean-Pierre Zanotti Continued Fractions for Certain Algebraic Power Series over a Finite Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 Alain Lasjaunias Linear Complexity and Polynomial Degree of a Function Over a Finite Field .......................... 229 Wilfried Meidl and Arne Winterhof Primitive Roots in Cubic Extensions of Finite Fields ......... 239 Donald Mills and Gavin McNay On Polynomial Families in n Indeterminates over Finite Prime Fields Coming from Planar Functions .................... 251 Nobuo Nakagawa Cryptanalysis of the Sakazaki-Okamoto-Mambo ID-based Key Distribution System over Elliptic Curves . . . . . . . . . . 263 Minghua Qu, Doug Stinson, and Scott Vanstone Differential and Linear Distributions of Substitution Boxes for Symmetric-Key Cryptosystems .................. 270 Peter Roelse Exponential Sums and Lattice Reduction: Applications to Cryptography . . . . . . . . . . . . . . . . . . . . . 286 Igor E. Shparlinski An Alternate Construction of the Berlekamp Subalgebra ....... 299 Greg Stein On the Fp-Linearity of the Generalized Gray Map Image of a Zpk+l-Linear Code ................... 306 H. Tapia-Recillas and G. Vega Contents IX Construction of Modular Curves and Computation of Their Cardinality over lFp ...................... 313 Cedric Tavernier' Asymptotie Properties of Global Fields ................ 328 M. A. Tsfasrnan Author Index .......................... 335 Commutative Semifields of Rank 2 Over Their Middle Nucleus Simeon BallI * and Michel Lavrauw2 Queen Mary, University of London, London, El 4NS, United Kingdom 1 2 Eindhoven University of Technology, Eindhoven, 5600MB, The Netherlands Abstract. This article is about finite commutative semifields that are of rank 2 over their middle nucleus, the largest subset of elements that is a finite field. These semifields have a direct correspondence to certain flocks of the quadratic cone in PG(3, q) and to certain ovoids of the parabolic space Q( 4, q). We shall consider these links, the known examples and non-existence results. 1 Semifields A finite semifield S is a finite algebraic system that possesses two binary operations, addition and multiplication, which satisfy the following axioms. o. (S1) Addition is a group with identity (S2) a(b+c) = ab+ac and (a+ b)c = ac+be for all a, b, e E S. oJ (S3) There exists an element 1 0 such that 1a = a = a1 for all a E S. (84) If ab = 0 then either a = 0 or b = O. Throughout this article the term semifield will refer to a finite semifield. The additive group of a semifield must be commutative. By (S2), (ae + ad) + (be + bd) = (a + b)(e + d) = (ae + be) + (ad + bd). Hence, ad + be = be + ad and any elements that can be written as products commute under additioll. By (84) and finiteness, any element of S can be written as a product and so it follows that the additive group is abelian. Moreover it is not difficult to show that the group is elementary abelian. oJ Let a 0, and let p be the additive order of a. If p is not prime then we can write p = rs for rand s integers not equal to 1, and by observing that o = (pa)a = (rsa)a = (ra)(sa) we get a contradiction from (S4). The fact that every nonzero element has prime order suffices to show that the group is elementary abelian. and that all nonzero elements have the same prime order p. This number p is the characteristic of the semifield. An elementary abelian group can be viewed as a vector space over a finite field. In particular S has pn elements where n is the dimension of S over the field GF(p). There are * The author acknowledges the support of an EPSRC (UK) Advanced Research Fellowship AF /990 480. G. L. Mullen et al. (eds.), Finite Fields with Applications to Coding Theory, Cryptography and Related Areas © Springer-Verlag Berlin Heidelberg 2002

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