SEC 1: Elliptic Curve Cryptography

1 STANDARDS FOREFFICIENTCRYPTOGRAPHYSEC 1: Elliptic Curve CryptographyCerticom 20, 2000Ve r s i o n 1 . 0c 2000 Certicom to copy this document is granted providedit is identified as Standards for Efficient Cryptography (SEC) ,in all material mentioning or referencing 1 - ContentsPage Overview .. Aim .. Compliance .. DocumentEvolution .. Finite Finite curves curves DataTypesandConversions .. Conversion .. Conversion .. FieldElement-to-OctetStringConversion.

2 OctetString-to-FieldElementConversion .. 143 Cryptographic Curve Domain Parameters .. Curve Domain Parameters Curve Domain Parameters 19 Page iiSEC 1: Elliptic Curve Cryptography Ver. Curve Key Pairs .. Curve Key Pair Generation Primitive .. of Elliptic Curve Public Keys .. Validation of Elliptic Curve Public Keys .. Curve Diffie-Hellman Primitives .. Curve Diffie-Hellman Primitive .. Curve Cofactor Diffie-Hellman Primitive .. Curve MQV Primitive.

3 HashFunctions .. KeyDeployment .. TaggingOperation .. TagCheckingOperation .. SymmetricEncryptionSchemes .. KeyDeployment .. 344 Signature Curve Digital Signature KeyDeployment .. SigningOperation .. VerifyingOperation .. 385 Encryption Curve Integrated Encryption Scheme .. 41 sec 1 - ContentsPage KeyDeployment .. 436 Key Agreement Curve Diffie-Hellman Scheme.

4 KeyDeployment .. Agreement Operation .. Curve MQV KeyDeployment .. Agreement Operation .. 49A Terms .. 56B Commentary on Section 2 - Mathematical Foundations .. Commentary on Section 3 - Cryptographic Components .. on Elliptic Curve Domain Parameters .. on Elliptic Curve Key on Elliptic Curve Diffie-Hellman Primitives .. on the Elliptic Curve MQV Primitive .. CommentaryonSection4-SignatureSchemes .. on the Elliptic Curve Digital Signature Algorithm.

5 CommentaryonSection5-EncryptionSchemes .. on the Elliptic Curve Integrated Encryption Scheme .. Commentary on Section 6 - Key Agreement Schemes .. on the Elliptic Curve Diffie-Hellman Scheme .. 69 Page ivSEC 1: Elliptic Curve Cryptography Ver. on the Elliptic Curve MQV Scheme .. AlignmentwithOtherStandards .. 72C for Elliptic Curve Syntax for Elliptic Curve Domain Parameters .. Syntax for Elliptic Curve Public Keys .. Syntax for Elliptic Curve Private Keys.

6 84D References85 SEC1-ListofFiguresPagevList of Tables1 Representations .. 603 Comparable key sizes .. 624 AlignmentwithothercoreECCstandards .. 73 List of Figures1 ConvertingbetweenDataTypes ..9 sec 1 - 1 IntroductionPage OverviewThis document specifies public-key cryptographic schemes based on Elliptic Curve Cryptography (ECC).In particular, it specifies: signature schemes; encryption schemes; and key agreement also describes cryptographic primitives which are used to construct the schemes, and syntax foridentifying the schemes are intended for general application within computer and communications AimThe aim of this document is to facilitate deployment of ECC by completely specifying efficient, well-established, and well-understood public-key cryptographic schemes based on to encourage deployment of interoperable implementations of ECC by profiling existing stan-dards like ANSI [3]

7 , IEEE P1363 [40], and WAP WTLS [87], and draft standards like [4] and IEEE P1363A [41], but restricting the options allowed in these standards to increase thelikelihood of interoperability and to ensure conformance with all standards to help ensure ongoing detailed analysis of ECC by cryptographers by clearly, completely, andpublicly specifying baseline ComplianceImplementations may claim compliance with the cryptographic schemes specified in this document pro-vided the external interface (input and output)

8 To the schemes is identical to the interface specified computations may be performed as specified here, or may be performed via an equivalent se-quence of that this compliance definition implies that conformant implementations must perform all the cryp-tographic checks included in the scheme specifications in this document. This is important because thechecks are essential to the prevention of subtle is intended to make a validation system available so that implementors can check compliance with thisdocument - see the SECG website, , for further 2 sec 1 : Elliptic Curve Cryptography Ver.

9 Document EvolutionThis document will be reviewed every five years to ensure it remains up to date with cryptographicadvances. The next scheduled review will therefore take place in September intermittent reviews may also be performed from time-to-time as deemed necessary by theStandards for Efficient Cryptography Intellectual PropertyThe reader s attention is called to the possibility that compliance with this document may require use ofan invention covered by patent rights. By publication of this document, no position is taken with respectto the validity of this claim or of any patent rights in connection therewith.

10 The patent holder(s) mayhave filed with the SECG a statement of willingness to grant a license under these rights on reasonableand nondiscriminatory terms and conditions to applicants desiring to obtain such a license. Additionaldetails may be obtained from the patent holder and from the SECG website, OrganizationThis document is organized as main body of the document focuses on the specification of public-key cryptographic schemes basedon ECC. Section 2 describes the mathematical foundations fundamental to the operation of all theschemes.