This is a preview of "ANSI X9.80:2005". Click here to purchase the full version from the ANSI store. American National Standard for Financial Services X9.80–2005 Prime Number Generation, Primality Testing, and Primality Certificates Accredited Standards Committee X9, Incorporated Financial Industry Standards Date Approved: August 15, 2005 American National Standards Institute American National Standards, Technical Reports and Guides developed through the Accredited Standards Committee X9, Inc., are copyrighted. Copying these documents for personal or commercial use outside X9 membership agreements is prohibited without express written permission of the Accredited Standards Committee X9, Inc. For additional information please contact ASC X9, Inc., P.O. Box 4035, Annapolis, Maryland 21403. This is a preview of "ANSI X9.80:2005". Click here to purchase the full version from the ANSI store. ANS X9.80–2005 Foreword Approval of an American National Standard requires verification by ANSI that the requirements for due process, consensus, and other criteria for approval have been met by the standards developer. Consensus is established when, in the judgment of the ANSI Board of Standards Review, substantial agreement has been reached by directly and materially affected interests. Substantial agreement means much more than a simple majority, but not necessarily unanimity. Consensus requires that all views and objections be considered, and that a concerted effort be made toward their resolution. The use of American National Standards is completely voluntary; their existence does not in any respect preclude anyone, whether he has approved the standards or not from manufacturing, marketing, purchasing, or using products, processes, or procedures not conforming to the standards. The American National Standards Institute does not develop standards and will in no circumstances give an interpretation of any American National Standard. Moreover, no person shall have the right or authority to issue an interpretation of an American National Standard in the name of the American National Standards Institute. Requests for interpretations should be addressed to the secretariat or sponsor whose name appears on the title page of this standard. CAUTION NOTICE: This American National Standard may be revised or withdrawn at any time. The procedures of the American National Standards Institute require that action be taken to reaffirm, revise, or withdraw this standard no later than five years from the date of approval. Published by Accredited Standards Committee X9, Incorporated Financial Industry Standards P.O. Box 4035 Annapolis, MD 21403 USA X9 Online http://www.x9.org Copyright © 2005 ASC X9, Inc. All rights reserved. No part of this publication may be reproduced in any form, in an electronic retrieval system or otherwise, without prior written permission of the publisher. Printed in the United States of America. ii ©ASC X9, Inc. 2005 – All rights reserved This is a preview of "ANSI X9.80:2005". Click here to purchase the full version from the ANSI store. ANS X9.80–2005 Contents Foreword......................................................................................................................................................................ii Tables...........................................................................................................................................................................v Introduction................................................................................................................................................................vi 1 Scope ..............................................................................................................................................................1 2 Normative references....................................................................................................................................2 3 Terms and definitions ...................................................................................................................................2 4 Symbols and abbreviated terms ..................................................................................................................4 5 Prime Generation Methods...........................................................................................................................5 5.1 General Discussion .......................................................................................................................................5 5.2 Generation of Primes Using Random Integers...........................................................................................7 5.2.1 Generation of Random Primes with Sequential Search ............................................................................7 5.2.2 Generation of Random Primes with Uniform Distribution ........................................................................8 5.2.3 Testing Using Probabilistic Methods ..........................................................................................................8 5.2.4 Testing Using Deterministic Methods .......................................................................................................11 5.3 Constructive Methods.................................................................................................................................16 5.3.1 Shawe-Taylor’s Algorithm..........................................................................................................................16 5.3.2 Maurer’s Algorithm......................................................................................................................................17 5.4 Side Conditions for Generating Primes using Random Integers ...........................................................19 6 Candidate Prime Testing Methods.............................................................................................................20 7 Tables of Parameters ..................................................................................................................................21 7.1 Rounds Required for Miller-Rabin if Followed by Lucas.........................................................................21 7.2 Rounds Required for Frobenius-Grantham ..............................................................................................21 Annex A (normative).................................................................................................................................................23 A.1 Modular Exponentiation..............................................................................................................................23 A.2 Jacobi Symbol .............................................................................................................................................23 A.3 Sieve Procedure...........................................................................................................................................25 A.4 Algorithms for Polynomial Arithmetic.......................................................................................................26 A.5 Lucas Sequence ..........................................................................................................................................28 Annex B (informative) ..............................................................................................................................................30 B.1 Discussion of General Prime Proving Methods .......................................................................................30 B.2 Discussion of the Distribution of Randomly Chosen Primes .................................................................30 Annex C Summary of Changes from ANS X9.80–2001 (informative)..................................................................31 C.1 Introduction..................................................................................................................................................31 C.2 Technical changes.......................................................................................................................................31 C.2.1 Search Range for primes ............................................................................................................................31 C.2.2 Errors in Jacobi symbol algorithm ............................................................................................................31 C.2.3 Range of bases in Miller-Rabin test...........................................................................................................32 C.2.4 Perfect squares in Lucas test.....................................................................................................................32 C.2.5 Discriminants with Jacobi symbol 0 in Lucas test ..................................................................................32 C.2.6 Boundary conditions in Shawe-Taylor’s algorithm .................................................................................32 C.3 Editorial issues ............................................................................................................................................33 C.3.1 Random bit generators ...............................................................................................................................33 ©ASC X9, Inc. 2005 – All rights reserved iii This is a preview of "ANSI X9.80:2005". 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