Cryptography and Computational Number Theory

375 Pages·2001·11.07 MB·English

by  Kwok-Yan Lam, Igor Shparlinski, Huaxiong Wang, Chaoping Xing (eds.)

Progress in Computer Science and Applied Logic
Volume 20
Editor
John C. Cherniavsky, National Science Foundation
Associate Editors
Robert Constable, Cornell University
Jean Gallier, University of Pennsylvania
Richard Platek, Cornell University
Richard Statman, Carnegie-Mellon University
C r y p t o g r a p hy a nd
C o m p u t a t i o n al
N u m b er T h e o ry
Kwok-Yan Lam
Igor Shparlinski
Huaxiong Wang
Chaoping Xing
Editors
Springer Basel AG
Editors:
Kwok-Yan Lam Igor Shparlinski
Department of Computer Science Department of Computing
National University of Singapore Macquarie University
2 Science Drive 2 NSW 2109
Singapore 117543 Australia
e-mail: [email protected] e-mail: [email protected]
Huaxiong Wang Chaoping Xing
Department of Computer Science Department of Mathematics
University of Wollongong National University of Singapore
NSW 2522 2 Science Drive 2
Australia Singapore 117543
e-mail: [email protected] e-mail: [email protected]
2000 Mathematics Subject Classification 11 Yxx, 1 lTxx, 94Axx, 68P25
A CIP catalogue record for this book is available from the Library of Congress,
Washington D.C., USA
Deutsche Bibliothek Cataloging-in-Publication Data
Cryptography and computational number theory : workshop in Singapore 1999 /
Kwok-Yan Lam ed. - Basel; Boston ; Berlin : Birkhäuser, 2001
(Progress in computer science and applied logic ; Vol. 20)
ISBN 978-3-0348-9507-1 ISBN 978-3-0348-8295-8 (eBook)
DOI 10.1007/978-3-0348-8295-8
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, re-use of
illustrations, broadcasting, reproduction on microfilms or in other ways, and storage in
data banks. For any kind of use whatsoever, permission from the copyright owner must
be obtained.
© 2001 Springer Basel AG
Originally published by Birkhäuser Verlag, Basel in 2001
Member of the BertelsmannSpringer Publishing Group
Printed on acid-free paper produced of chlorine-free pulp. TCF «>
98765432 1
Contents
Preface ................................................................... Vll
Computational Number Theory
c. Alonso, J. Gutierrez and R. Rubio
On the Dimension and the Number of Parameters
of a Unirational Variety ............................................. 3
A. Conflitti
On Elements of High Order in Finite Fields .......................... 11
C. Ding, D.R. Kohel and S. Ling
Counting the Number of Points on Affine Diagonal Curves........... 15
J. Friedlander, C. Pomemnce and I.E. Shparlinski
Small Values of the Carmichael Function and
Cryptographic Applications .......................................... 25
J. von zur Gathen and F. Pappalardi
Density Estimates Related to Gauss Periods 33
W. Han
Distribution of the Coefficients of Primitive
Polynomials over Finite Fields ....................................... 43
J. Hoffstein and D. Lieman
The Distribution of the Quadratic Symbol in Function Fields
and a Faster Mathematical Stream Cipher ........................... 59
D. Kohel
Rational Groups of Elliptic Curves Suitable for Cryptography ........ 69
K. Y. Lam and F. Sica
Effective Determination of the Proportion of
Split Primes in Number Fields....................................... 81
P. Mihalescu
Algorithms for Generating, Testing and Proving Primes:
A Survey............................................................ 93
R. Pemlta
Elliptic Curve Factorization Using a
"Partially Oblivious" Function ....................................... 123
A.J. van der Poorten
The Hermite-Serret Algorithm and 122 + 332 ........................ 129
C.P. Xing
Applications of Algebraic Curves to Constructions of Sequences ...... 137
vi Contents
Cryptography
N. Alexandris, M. Burmester, V. Chrissikopoulos and Y. Desmedt
Designated 2-Verifier Proofs and their Application
to Electronic Commerce ............................................. 149
E. Dawson, L. Simpson and J. Golic
Divide and Conquer Attacks on Certain Irregularly Clocked
Stream Ciphers...................................................... 165
A. De Bonis and A. De Santis
New Results on the Randomness of
Visual Cryptography Schemes ....................................... 187
D. Gollmann
Authentication - Myths and Misconceptions ......................... 203
M.1. Gonzales Vasco and M. Naslund
A Survey of Bit-security and Hard Core Functions ................... 227
M.1. Gonzales Vasco and I.E. Shparlinski
On the Security of Diffie-Hellman Bits ............................... 257
J. Hoffstein and J.H. Silverman
Polynomial Rings and Efficient Public Key Authentication II ........ 269
P. Mihalescu
Security of Biased Sources for Cryptographic Keys ................... 287
M. Naslund and A. Russell
Achieving Optimal Fairness from Biased Coinflips .................... 303
P.Q. Nguyen
The Dark Side of the Hidden Number Problem:
Lattice Attacks on DSA ............................................. 321
P.Q. Nguyen, I.E. Shparlinski and J. Stern
Distribution of Modular Sums and the Security of
the Server Aided Exponentiation .................................... 331
R. Safavi-Naini and W. Susilo
A General Construction for Fail-Stop Signatures
using Authentication Codes .......................................... 343
R. Safavi-Naini and H. Wang
Robust Additive Secret Sharing Schemes over Zm .................... 357
R.D. Silverman
RSA Public Key Validation .......................................... 369
Preface
This volume contains the refereed proceedings of the Workshop on Cryptography
and Computational Number Theory, CCNT'99, which has been held in Singapore
during the week of November 22-26, 1999.
The workshop was organized by the Centre for Systems Security of the Na
tional University of Singapore. We gratefully acknowledge the financial support
from the Singapore National Science and Technology Board under the grant num
ber RP960668/M.
The idea for this workshop grew out of the recognition of the recent, rapid
development in various areas of cryptography and computational number the
ory. The event followed the concept of the research programs at such well-known
research institutions as the Newton Institute (UK), Oberwolfach and Dagstuhl
(Germany), and Luminy (France). Accordingly, there were only invited lectures at
the workshop with plenty of time for informal discussions.
It was hoped and successfully achieved that the meeting would encourage and
stimulate further research in information and computer security as well as in the
design and implementation of number theoretic cryptosystems and other related
areas. Another goal of the meeting was to stimulate collaboration and more active
interaction between mathematicians, computer scientists, practical cryptographers
and engineers in academia, industry and government.
Talks concerning many different aspects of cryptography and computational
number theory such as theory, techniques, applications and practical experiences
were given. Some other related areas of number theory and computer science were
covered as well. These include but are not limited to talks devoted to
o new cryptographic systems and protocols;
o new attacks on the existing cryptosystems;
o new cryptographic paradigms such as visual and audio cryptography;
o pseudorandom number generator and stream cipher;
o primality proving and integer factorization;
o fast algorithms;
o cryptographic aspects of the theory of elliptic and higher genus curves;
o polynomials over finite fields;
o analytical number theory.
Some of the talks, and their associated papers, were surveys giving comprehensive
state-of-the-art outlines of some number theoretic research areas of significance to
cryptography. Some were descriptions of new original results and ideas for which
this workshop provided the first forum where they were publicly presented.
viii Preface
The contents of this volume reflects the whole variety of the topics which
have been considered at the workshop. We believe it will provide a valuable con
tribution to and stimulate further progress of cryptography and computational
number theory.
The Organizers:
Kwok-Yan Lam (National University of Singapore)
Igor Shparlinski (Macquarie University, Australia)
Huaxiong Wang (University of Wollongong, Australia)
Chaoping Xing (National University of Singapore)
Computational
N umber Theory
Progress in Computer Science and Applied Logic, Vol. 20
© 2001 Birkhauser Verlag Basel/Switzerland
On the Dimension and the Number
of Parameters of a Unirational Variety
Cesar Alonso, Jaime Gutierrez, and Rosario Rubio
Abstract. In this paper we study the relation between the dimension of a
parametric variety and the number of parameters. We present an algorithm
to reparameterize a variety in order to obtain a parameterization where the
number of parameters equals the dimension of the variety.
1. Introduction
Let OC be an algebraically closed field of characteristic zero and T = (T1, ... , T m)
indeterminates over IK. Following the notation of [5], given n rational functions in
OC(T)
FlN(T) FnN(T)
F1(T) = FID(T)"" , Fn(T) = FnD(T);
we consider W = Ul<i<n V(FiD(T)) c OCm, where V(FiD(T)) is the zero set of
the polynomial FiD(T) Eo OC[T], and the map rp: (OCm - W) ---+ ocn such that:
rp(h, ... ,tm) = (F1(tl,'" ,tm), ... ,Fn(h, ... ,tm)).
An affine variety V <;;; OCn is a parametric or unirational variety if there exists a
collection of rational functions F = {F1( T), ... , Fn (T)} such that V is the Zariski
closure of rp(OCm - W). The set F is a parameterization of V.
Most affine varieties cannot be parameterized in the sense described above.
In general, it is difficult to tell whether a given variety is unirational or not. It is
well known that any parametric variety is irreducible; in fact, the coordinate ring
of V is isomorphic to OC[ Fl (T), ... , Fn (T)], that is,
OC[Xl,'" , xnl/I(V) ~ OC[Fl (T), ... , Fn(T)],
where I(V) is the polynomial ideal of V. So, the dimension of V is the transcen
dence degree of OC(Fl (T), ... , Fn(T)) over OC, that is,
d = dim(V) = trans.degoc (OC( Fl (T), ... , Fn (T)).
A parameterization F is called faithful if OC( Fl (T), ... , Fn (T)]) = OC(T).
In geometric terms, it means there exist a one to one map from points of the
variety to values of the parameters T, except an algebraic set of dimension d - 1.
If the parameterization is not faithful, naturally we would ask whether we can