Using Elliptic Curve Parameters | RSA Security 5.2.2 manual

Performing Elliptic Curve Operations

Using Elliptic Curve Parameters

See the section “Generating Elliptic Curve Parameters” on page 260 for the steps you must complete to generate a new curve. You need a properly initialized pseudo- random number generator. Assume that the function InitializeRandomAlgorithm goes through Steps 1 through 4 in the section “Generating Random Numbers” on page 165. Also assume that the function InitializeECParamsObj generates new parameters and places them in ecParamsObj, following the steps in “Using Elliptic Curve Parameters” on page 298:

B_ALGORITHM_OBJ randomAlgorithm = (B_ALGORITHM_OBJ)NULL_PTR; B_ALGORITHM_OBJ ecParamsObj = (B_ALGORITHM_OBJ)NULL_PTR;

if ((status = InitializeRandomAlgorithm (&randomAlgorithm)) != 0) break;

if ((status = InitializeECParamsObj (&ecParamsObj, &randomAlgorithm)) != 0)

break;

You now have a properly initialized random algorithm object, randomAlgorithm, and an algorithm object, ecParamsObj, containing the parameters that describe the elliptic curve that you will use.

Using an EC Key Pair

Before you can encrypt, you need to generate a public/private key pair. As described in “Using an EC Key Pair” on page 298, key generation requires a properly initialized random algorithm and the parameters describing an elliptic curve, both of which you have created in the previous step:

B_KEY_OBJ publicKey = (B_KEY_OBJ)NULL_PTR;

B_KEY_OBJ privateKey = (B_KEY_OBJ)NULL_PTR;

Assume that the steps in “Using an EC Key Pair” have been completed and that publicKey and privateKey are ready to be used.

ECAES Public-Key Encryption

Once you have gone through the preliminary steps of generating your elliptic curve parameters and creating your public/private key pair, you are ready to encrypt your message.

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RSA Security 5.2.2 manual Using Elliptic Curve Parameters, Using an EC Key Pair, Ecaes Public-Key Encryption

Contents

Crypto-C Cryptographic Components for C Trademarks License agreementDistribution Limit distribution of this document to trusted personnel Contents Quick Start Cryptography N t e n t s Using Crypto-C 101 Algorithms in Crypto-C Non-Cryptographic Operations 151 Saved State Symmetric-Key Operations 177 Public-Key Operations 213 Secret Sharing Operations 305 Putting It All Together An X9.31 Example Appendix a Command-Line Demos 327 Glossary 339 Index 349 List of Figures A B S a F E C r y p t o C D e v e l o p e r ’s G u i d e List of Tables A B S a F E C r y p t o C D e v e l o p e r ’s G u i d e Preface What’s New in Version 5.2.2? Improved performanceHardware support MultiPrime RSA Organization of This Manual Advanced Encryption Standard AESOrganization of This Manual E f a c e Conventions Used in This Manual Crypto-C procedures to use with algorithm objectConventions Used in This Manual Terms and Abbreviations Terms and Abbreviations A B S a F E C r y p t o C D e v e l o p e r ’s G u i d e Related DocumentsRelated Documents Ieee Standard Specifications for Public-Key Cryptography on Http//stdsbbs.ieee.org/groups/1363/index.html How to Contact RSA Security RSA Security Web Site Getting Support and ServiceHow to Contact RSA Security You can get technical support as follows Introduction Crypto-C Toolkit Algorithms Public-Key Algorithms Digital SignaturesElliptic Curve Public-Key Algorithms Secret Sharing Cryptographic Standards and Crypto-C Pkcs Standards and Crypto-CNist Standards and Crypto-C Nist Approval and Windows 32-bit Platforms Pkcs Compared with Nist Nist Approval and Windows NT Platforms Ansi X9 Standards and Crypto-C Quick Start Six-Step Sequence Six-Step Sequence Introductory Example Include FilesCreating an Algorithm Object Introductory Example Where Pointer is defined in aglobal.h Nullptr is also defined in aglobal.hTypedef unsigned char *POINTER #define Nullptr POINTER0 Algorithm information Setting the Algorithm ObjectBalgorithmobj algorithmObject Algorithm object Init Break For our example, we use Setting a Key ObjectCreating a Key Object Int BSetKeyInfo Key information Selecting an Algorithm Chooser Now we can complete the call to BSetKeyInfo Update Saving the Object State optionalSurrender Context We can now complete our call to BEncryptInit Output data buffer Unsigned intLength of output data Input data Unsigned char *encryptedData = Nullptr Unsigned int outputLenUpdateFor now, we declare If status = BEncryptUpdate Final Destroy Asurrenderctx *surrenderContext Surrender context For our example, we use the following Pointer to key objectBalgorithmobj * algorithmObject Pointer block Putting It All Together For this example, call Tfree as followsTfree encryptedData If rc4KeyItem.data != Nullptr Init If status = BEncryptInit Introductory Example If encryptedData != Nullptr EncryptedData = Nullptr End main Decrypting the Introductory Example Creating the Key ObjectDecrypting the Introductory Example First create the algorithm object Setting the Key Object Always destroy objects when you no longer need them DecryptedData = Nullptr Multiple Updates Multiple Updates Updatesize Break End while Multiple Updates Summary of the Six Steps IncludeCreate Set Final Page Cryptography Cryptography Overview Symmetric-Key CryptographyCiphers Cryptography Overview Block Ciphers PaddingCiphers in Crypto-C Crypto-C implements the following block ciphers Triple DES 2Triple DES Encryption as Implemented in Crypto-C RC5 RC6 Modes of Operation Four Modes Electronic Codebook ECB Mode 3Electronic Codebook ECB Mode Cipher Block Chaining CBC Mode Cipher Feedback CFB Mode 5Cipher Feedback CFB Mode Output Feedback OFB Mode 6Output Feedback Mode OFB Stream Ciphers Message Digests RC4 algorithm with MAC Message Digests and Pseudo-Random Numbers Password-Based Encryption Hash-Based Message Authentication Codes Hmac 8DES Key and IV Generation for Password Based Encryption Public-Key Cryptography RSA Algorithm 9Public-Key Cryptography Modular Math Prime NumbersMultiPrime Numbers RSA Algorithm Cryptography Overview Summary SecurityCalculation is shown in Table 1Calculation of 827 mod Digital Envelopes Optimal Asymmetric Encryption Padding Oaep 10Digital Envelope Authentication and Digital Signatures Cryptography Overview 11RSA Digital Signature Digital Signature Algorithm DSA Digital Certificates Math Diffie-Hellman Public Key Agreement Algorithm Parameter Generation PhasePhase Math Elliptic Curve Cryptography Multiple-Party Key Agreement Elliptic Curve Parameters Finite Field Fields of Even Characteristic Odd Prime Fields Coefficients Over an Odd Prime Field = 0·I ≡ 2·2m-1·Imod2m Coefficients Over a Field of Even Characteristic Point P and its OrderPoints of an Elliptic Curve Elliptic curve parameters Order of an Elliptic Curve Order of a PointPoint of Prime Order Is written EFq Summary of Elliptic Curve Terminology CofactorOrder n of P Is sometimes called the base point 2Elliptic Curve Parameters Representing Fields of Even Characteristic Elliptic Curve Key Pair Generation Signing a Message Ecdsa Signature SchemeCreating the Key Pair Verifying a Signature Math Now recall that s = k-1e+dr mod n, so Elliptic Curve Authenticated Encryption Scheme EcaesRecall that Q = dP, so Encrypting a Message Using the Public Key Decrypting a Message Using the Private Key Elliptic Curve Diffie-Hellman Key Agreement Phase 13Elliptic Curve Diffie-Hellman Key Agreement Secret Sharing First coordinate of S, xS, is their agreed-upon secret key Working with Keys Key Generation Key Management Key Escrow Applications of Cryptography Ascii Encoding and DecodingLocal Applications Applications of Cryptography Point-to-Point Applications Client/Server Applications Peer-to-Peer Applications Choosing Algorithms Public-Key vs. Symmetric-Key CryptographyStream vs. Block Symmetric-Key Algorithms Choosing Algorithms Block Symmetric-Key Algorithms Key Agreement vs. Digital Envelopes Secret Sharing and Key Escrow Elliptic Curve Algorithms Interoperability Security Considerations Handling Private KeysElliptic Curve Standards Security Considerations Temporary Buffers Pseudo-Random Numbers and Seed Generation Choosing Passwords 3DES Weak and Semi-Weak Keys Initialization Vectors and SaltsDES Weak Keys Stream Ciphers Timing Attacks and Blinding Pick a random value r and compute = mre mod n Choosing Key Sizes MD5p padP MD5q padQ m RSA Keys 4Summary of Recommended Key Sizes Diffie-Hellman Parameters and DSA Keys RC2 Effective Key BitsRC4 Key Bits RC5 Key Bits and Rounds Elliptic Curve Keys Using Crypto-C Algorithms in Crypto-C Information Formats Provided by Crypto-C Basic Algorithm Info TypesBER-Based Algorithm Info Types Algorithms in Crypto-C Summary of AIs PEM-Based Algorithm Info TypesBSAFE1 Algorithm Info Types 1Message Digests Algorithms in Crypto-C 2Message Authentication 3ASCII Encoding4Pseudo-Random Number Generation 5Symmetric Stream Ciphers Algorithms in Crypto-C 5Symmetric Stream Ciphers 6Symmetric Block Ciphers Algorithms in Crypto-C 6Symmetric Block Ciphers RC2 7RSA Public-Key Cryptography Key Generation Algorithms in Crypto-C 7RSA Public-Key Cryptography Oaep 8DSA Public-Key Cryptography Digital Signatures Algorithms in Crypto-C 9Diffie-Hellman Key Agreement 10Elliptic Curve Public-Key Cryptography 12Hardware Interface 10 Elliptic Curve Public-Key Cryptography11Bloom-Shamir Secret Sharing Algorithms in Crypto-C 13Advanced Encryption Standard AES Algorithm Info Type Keys In Crypto-C Summary of KIsKeys In Crypto-C 15Block Cipher Keys 17DSA Public and Private Keys Keys In Crypto-C 15Block Cipher Keys16RSA Public and Private Keys Keys In Crypto-C 18Elliptic Curve Keys System Considerations In Crypto-C Algorithm ChoosersAn Encryption Algorithm Chooser System Considerations In Crypto-C An RSA Algorithm Chooser Surrender Context Description of AIX962RandomV0 instead of AISHA1Random Sample Surrender Function Also gives the form that a surrender function must haveReserved for future use Int *Surrender Pointer handle Saving State When to Allocate Memory Memory-Management Routines Memory-Management Routines and Standard C LibrariesCrypto-C uses the following memory-management routines Tmalloc Trealloc Tfree Tmemset Tmemcpy Tmemmove Tmemcmp Binary Data BER/DER EncodingMemory Allocation Typedef struct unsigned char *data unsigned int len System Considerations In Crypto-C Input and Output Symmetric Block AlgorithmsInput constraints Output considerations 20Input Limits for RSA Pkcs Encryption General Considerations Public Key Size Key SizeDES Keys Private Key Size C2 58 60 B1 00 C2 58 60 B1 Interfacing with a Bhapi Implementation Using Cryptographic HardwareUsing Cryptographic Hardware 2Algorithm Object in a Hardware Implementation Pkcs #11 Support Using a Pkcs #11 Device with Crypto-C Using Cryptographic Hardware Balgorithmmethod *RSASIGNHWCHOOSER = &AMMD5 Using Cryptographic Hardware KeypairGenParams.privateKeyAttributes.tokenFlag = Tfprivate Now generate Using Cryptographic Hardware Using Cryptographic Hardware Return Behardware Pkcs #11 Support for DSA Key Pair Generation Balgorithmmethod *DSAKEYGENCHOOSER = &AMDSAKEYGEN P11KeyGenParams.privateKeyAttributes.keyUsage = POINTER&pubKeyData-params Advanced Pkcs #11 For an RSA private key, it would be thisDigest 00 00 00 66 a9 47 2d 80 5a Hardware Issues Random NumbersCkrv rv Rv = CKBYTEPTRseedBuffer, CKULONGseedLen Using Cryptographic Hardware Page Non-Cryptographic Operations Message Digests Creating a DigestMessage Digests Example in this section corresponds to the file mdigest.c If status = BDigestInit #define Digestlen If status = BDigestUpdateYour call will be the following Remember to destroy all objects when you are done with them BER-Encoding the Digest Following example BER-encodes the preceeding sample digest Saving the State of a Digest Algorithm ObjectSaved State Item stateInfo = Null Message Digests 1Code Sample DigestDataSavedState Status = Rsademoealloc break Message Digests Hash-Based Message Authentication Code Hmac Hash-Based Message Authentication Code Create the key object Hash-Based Message Authentication Code HmacGenerate a random 24-byte key using KI24Byte RandomAlgorithm, keyData, KEYSIZE, Asurrenderctx *NULLPTR != Unsigned char *digestedData unsigned int digestedDataLen Generating Random Numbers Generating Random NumbersGenerating Random Numbers with SHA1 Setting The Algorithm Object Random Seed Puts Enter a random seed if status = Unsigned char *gets char *randomSeed != 0 break Generate If status = BGenerateRandomBytes Generating Independent Streams of Randomness This step is identical to the previous exampleTypedef struct Additional seeding Item randomSeed AX931RANDOMPARAMS x931Params Steps 4, 5These steps are identical to the previous example Converting Data Between Binary and Ascii Converting Data Between BinaryEncoding Binary Data To Ascii If status = BEncodeInit asciiEncoder != 0 break Decoding ASCII-Encoded Data BDestroyAlgorithmObject &asciiEncoder Tfree asciiEncoding If status = BDecodeInit asciiDecoder != 0 break If status = binaryDecoding == Nullptr != 0 break BDestroyAlgorithmObject &asciiDecoder Tfree binaryDecoding Symmetric-Key Operations Block Ciphers DES with CBCBlock Ciphers Example in this section corresponds to the file descbc.c Examples include des, rc5 RC5 parametersPoints at init vector Item For new padding schemes Now set up the Bblkcipherwfeedbackparams structure Call BGenerateRandomBytes Depends on cipher type, as follows Format of info supplied to BSetKeyInfo Unsigned int outputBufferSize Decrypting RC2 Cipher Now you can set your algorithm object as follows Init Use a random number generator to come up with 24 bytes If rc2KeyItem.data != Nullptr Update EncryptedData = Nullptr If rc2KeyItem.data != Nullptr RC5 Cipher Initialization vector 0x10 Unsigned int255 Unsigned int Unsigned char initVector8 ARC5CBCPARAMS rc5Params If rc5KeyItem.data != Nullptr Use a random number generator to create 10 bytesItem rc5KeyItem Update Final RC6 Cipher EncryptedData = Nullptr #define Blocksize Unsigned char initVectorBLOCKSIZE Setting the Key Data If rc6KeyItem.data != Nullptr Break OutputLenTotal = outputLenUpdate + outputLenFinal AES Cipher Been allocated Unsigned char *aesParams Creating a Key Object If aesKeyItem.data != Nullptr Final Password-Based Encryption Unsigned char * salt Iteration count Init Tmemset pbeKeyItem.data, 0, Maxpwlen Update Final Page Public-Key Operations Performing RSA Operations Performing RSA OperationsGenerating a Key Pair Where Item is There is no in generating a key pair Destroy What is MultiPrime? MultiPrimeMultiPrime Milliseconds Private non-CRT How Many Primes? Two-Prime RSA 48.8 17.5 Three-Prime RSA 10.9 Sample BGenerateInit RsaGenBGenerateKeypair BGetKeyInfo Generating an RSA MultiPrime Key Set the Algorithm Object Distributing an RSA Public Key If status = BGenerateInit keypairGenerator If you looked at the elements of the struct Crypto-C FormatBER/DER Encoding Format of info returned by BGetKeyInfo If status = myPublicKeyBER.data == Nullptr != Tfree myPublicKeyBER.data RSA Public-Key EncryptionSend it off Remember to free any memory you allocated Reference Manual entry on AIPKCSRSAPublic states Input constraints #define Blocksize RSA Private-Key Decryption If status = BDecryptInit Optimal Asymetric Encryption Padding Oaep Raw RSA Encryption and Decryption MultiPrime RSA Digital Signatures Computing a Digital Signature Setting The Algorithm Object Entry for the AI in use Verifying a Digital Signature Surrender context outlined in The Surrender Context on If status = BVerifyInit Update Performing DSA Operations Performing DSA OperationsGenerating DSA Parameters There is no in generating DSA parameters Bdsaparamgenparams dsaParams DsaParams.primeBits = Generate Generating a DSA Key Pair DSA Signatures Computing a Digital Signature Creating An Algorithm Object Properly cast Nullptr for the surrender context #define Maxsiglen To verify the signature created here, use the same AI Data and you know its length, your call is the following Generating Diffie-Hellman Parameters Performing Diffie-Hellman Key Agreement Adhparamgenparams There is no in generating Diffie-Hellman parameters Adhparamgenparams dhParams Destroy Distributing Diffie-Hellman Parameters Base generator BER Format If you look at the elements of the struct A p t e r 7 P u b l i c K e y O p e r a t i o n s Diffie-Hellman Key Agreement Item dhParametersBER If status = BKeyAgreeInit Phase Saving the Object State Other party should send their public value and its length Performing Elliptic Curve Operations Performing Elliptic Curve OperationsGenerating Elliptic Curve Parameters Input of 0 defaults to fieldElementBits Input of 0 defaults toFormat of info supplied to BSetAlgorithmInfo Implementation version Performing Elliptic Curve Operations No Update step is necessary for parameter generation Retrieving Elliptic Curve Parameters GeneralFlag = Indicates type of base field It is the prime numberCase that fieldType = Ftfp Elliptic curve coefficient If status = output-base.data == Nullptr != 0 break FreeECParamInfo&ecParamInfo If status = output-order.data == Nullptr != 0 breakAecparams ecParamInfo Generating an Elliptic Curve Key Pair Sample code, FreeECParamInfo is implemented as follows Becparams paramInfo There is no Update step for key generation Initialize Retrieving an Elliptic Curve Key KIECPublic gives a pointer to an Aecpublickey structurePublic component Aecparams curveParams Performing Elliptic Curve Operations End FreeECPubKeyInfo Generating Acceleration TablesGenerating a Generic Acceleration Table Aecparams *cryptocECParamInfo Aecparams ecParamInfo Retrieve the elliptic curve parametersFormat the information There is no Update step for building acceleration tables For odd prime field Item accelTableItem Allocate memoryBuild the acceleration table Generating a Public-Key Acceleration Table Retrieve the public key information Put the information retrieved in the proper format FreeECPubKeyInfo&publicKeyInfo Item pubKeyAccelTableItem unsigned int maxTableLen Performing EC Diffie-Hellman Key Agreement Build the public-key acceleration tableItem pubKeyAccelTableItem GeneralFlag = If status = BBuildTableFinal Create Optional Set Acceleration Table Info If status = alicePublicValue == Nullptr != 0 break Performing Ecdsa in Compliance with Ansi If status = aliceSecretValue == Nullptr != 0 break Generating EC Parameters Generating an EC Key Pair Computing a Digital Signature Create Build an algorithm chooser with the appropriate AMs Item aTableItem If status = signature == Nullptr != 0 break Now, using BSignUpdate, pass in the data to be signedAecparams *ecParamInfo Initialized random algorithm in BSignFinal Destroy all objects that are no longer neededUse the same AI and digestInfo as you did for signing Unsigned int signatureLen Optional Set Public Key Acceleration Table Info Performing Ecdsa with X9.62-Compliant BER Item stockECParamsBER ECParamsBER154 = 0x30 0x810x52 0xd1 0x7a StockECParamsBER.data = ECParamsBER stockECParamsBER.len = DataToSign = 0x53 0x680x73 0x79 Aecparams *ecParamsInfo Use the same AI as you did for signing Using Ecaes Tfreesignature BDestoryAlgorithmObject &ecDSAVerify Ecaes Public-Key Encryption Using Elliptic Curve ParametersUsing an EC Key Pair Item accelerationTableItem Optional Acceleration TableNow, pass this information into your algorithm object Break FieldElementLen = ecParamInfo-fieldElementBits + 7 Final Create an algorithm object Ecaes Private-Key DecryptionSteps for decryption are similar to those for encryption BDecryptUpdate OutputLenUpdateMaxDecryptedDataLen = outputLenTotal = TmallocmaxDecryptedDataLenPage Secret Sharing Generating Shares Secret Sharing Example in this section corresponds to the file scrtshar.c #define Secretsize 16 #define Totalshares If status = secretSharecount == Nullptr != 0 break Break If status != 0 breakUnsigned int outputLenFinal If status = BEncryptFinal Tfree secretSharecount Reconstructing the Secret Use the same AI, AIBSSecretSharing Asurrenderctx *NULLPTR != 0 break If status != 0 break BDestroyAlgorithmObject &secretReconstructer Page Putting It All Together An X9.31 Example Bkeyobj privateKey = Bkeyobjnullptr Item randomSeed X9.31 Sample ProgramX9.31 Sample Program Generating Random Bytes Static unsigned char f4Data = 0x01, 0x00Unsigned int status Printf ================================================\n To create a random algorithm object and set the parameters Providing the Seed Generating a Key Pair Break RsaVerifyX931. For signing, use rsaSignX931 Computing a Digital SignaturePrintf ============================= \n AX931PARAMS Break Init Verifying the Signature If status != Surrendering Control Return End GeneralSurrenderFunction Printing the Buffer Contents Overview of the Demos Appendix a Command-Line Demo User’s Guide Command Line modeInput Redirection mode Command-Line Demo User’s Guide Sign a File Using BdemoSpecifying User Keys Open a File Envelope Create a File EnvelopeVerify a Signed File Generate a Key Pair Enter any arbitrary string of printable characters Using BdemodsaRunning Bdemodsa Sign a File Using Bdemoec Running Bdemoec Table A-1Demo Program Source Files File ReferenceFile Reference BSLite BSLite Table A-1Demo Program Source Files BSLite Page Glossary Advanced Encryption Standard Series of steps used to complete a taskAmerican National Standards Institute Application Programming Interface O s s a r y Generic security service application program interface Electronic business Data InterchangeSee ECC See XOR Process of having a third party hold onto encryption keys Process that creates a larger key from the original keyAct of creating a key An algorithm that generates the subkeys in a block cipher Private key in the RSA public-key cryptosystem Extra bits concatenated with a key, password, or plaintextData to be encrypted Number extracted from a pseudo- random sequence Secure Multipurpose Internet Mail Extensions Simple Mail Transfer Protocol Secure Wide Area NetworkSee secret key Trusted Computer System Evaluation Criteria XOR Page Index Ecdsa D e A B S a F E C r y p t o C D e v e l o p e r ’s G u i d e D e See also RC4 subprime