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The efficient generation of public-key parameters is a prerequisite in public-key systems. A specific example is the requirement of a prime number p to define a finite field Zp for use in the Diffie-Hellman key agreement protocol and its derivatives (x12.6). In this case, an element of high order in Z p is also required. Another example is the requirement of primes p and q for an RSA modulus n = pq (x8.2). | This is a Chapter from the Handbook of Applied Cryptography by A. Menezes P. van Oorschot and S. Vanstone CRC Press 1996. For further information see www.cacr.math.uwaterloo.ca hac CRC Press has granted the following specihc permissions for the electronic version of this book Permission is granted to retrieve print and store a single copy of this chapter for personal use. This permission does not extend to binding multiple chapters of the book photocopying or producing copies for other than personal use of the person creating the copy or making electronic copies available for retrieval by others without prior permission in writing from CRC Press. Except where over-ridden by the specihc permission above the standard copyright notice from CRC Press applies to this electronic version Neither this book nor any part may be reproduced or transmitted in any form or by any means electronic or mechanical including photocopying microhlming and recording or by any information storage or retrieval system without prior permission in writing from the publisher. The consent of CRC Press does not extend to copying for general distribution for promotion for creating new works or for resale. Specihc permission must be obtained in writing from CRC Press for such copying. @1997 by CRC Press Inc. Chapter Public-Key Parameters Contents in Brief 4.1 Introduction.133 4.2 Probabilistic primality tests.135 4.3 True Primality tests.142 4.4 Prime number generation.145 4.5 Irreducible polynomials over Zp.154 4.6 Generators and elements of high order.160 4.7 Notes and further references.165 4.1 Introduction The efficient generation of public-key parameters is a prerequisite in public-key systems. A specific example is the requirement of a prime number p to define a finite field Zp for use in the Diffie-Hellman key agreement protocol and its derivatives 12.6 . In this case an element of high order in Zp is also required. Another example is the requirement of primes p and q for an RSA modulus n .
