ISO/IEC INTERNATIONAL STANDARD 15946-1 Third edition 2016-07-01 Information 1technologySecurity techniques Cryptographic techniques based on elliptic curves Part 1: General Technologies de I'information-Techniques desecurité- Techniques cryptographiques basees sur lescourbeselliptiques- Partie l:Généralites Reference number ISO/IEC 15946-1:2016(E) IEC Piowded by IHS unde lone wimisO No reprodicton or netwoaing ge tom Hs OISO/IEC2016 ISO/IEC15946-1:2016(E) COPYRIGHTPROTECTEDDOCUMENT O ISO/IEC 2016,Published in Switzerland Allrights reserved.Unless otherwise specified,no part of this publication may be reproduced or utilized otherwise in any form or by any means,electronic or mechanical,including photocopying,or posting on the internet or an intranet,without prior written permission.Permission can be requested from either ISo at the address below or ISO's member body in the country of the requester ISO copyright office Ch.de Blandonnet 8-CP 401 CH-1214 Vernier,Geneva,Switzerland Tel.+41227490111 Fax+41227490947 [email protected] www.iso.org henatonbrontzatn terSandadlaton Noreproducionor netvorkrng pemiedwthut foense trom IHS ISO/IEC 15946-1:2016(E) Contents Page Foreword Introduction 1 Scope 2 Normative references 3 Terms and definitions 4 Symbols 5 Conventions for fields 5.1 Finite prime fields F(p.) 5.2 Finite fields F (pm).. 6 Conventions for elliptic curves 6.1 Definitions of elliptic curves 6.1.1 Elliptic curves over F (pm) 6.1.2 Elliptic curves over F(2m) 6.1.3 Elliptic curves over F(3m) 6.2 Group law onelliptic curves 6.3 Generation ofelliptic curves 6.4 Cryptographic bilinear map .. 7 Conversion functions 7.1 Octet string /bit stringconversion:OS2BSPandBS2OSP 7.2 Bit string /integer conversion : BS2IPand I2BSP. 6 7.3 Octet string /string conversion :OS2IP and 120SP 7.4 Finite field element /integer conversion :FE2IPp 7.5 Octet string /finite field element conversion :OS2FEPpand FE2OSPr 7.6 Elliptic curve point / octet string conversion : EC2OSPe and OS2ECPE 7.6.1 Compressed ellipticcurvepoints 7.6.2 Point decompression algorithms 7.6.3 Conversion functions 7.7 Integer/elliptic curveconversion:I2ECP 8 Ellipticcurvedomainparametersandpublickey 8.1 Elliptic curve domain parameters over F(q) 8.2 Elliptic curve key generation Annex A(informative)Background informationonfinite fields AnnexB(informative)Backgroundinformationonellipticcurves Annex C(informative)Background informationon ellipticcurve cryptosystems Annex D (informative)Summary of coordinate systems Bibliography 31 InternatonalOroanizatonQsISQ/IEC2016-Allrightsreserved ii pSou ters tomMs