Digital Certificates, Math | RSA Security 5.2.2 specs

Cryptography Overview

a= gu1 mod p

b= yu2 mod p

v = (a · b mod p) mod q

3.If v = r, the signature is verified. If v ≠ r, the signature is invalid.

The Math

To see that this is indeed the signature, consider the following. We have the values:

y = gx mod p

and

u2 = r · sinv mod q

Make the following algebraic substitutions:

a · b mod p = gu1 · gx·u2 mod p

=gu1 + x·u2 mod p

=gdigest·sinv + x·r·sinv mod p

=gsinv(digest + x·r) mod p

=gk mod p

Recall that:

r= (gk mod p) mod q

This means that:

v = (a · b mod p) mod q

=(gk mod p) mod q

=r

Digital Certificates

Suppose you own an RSA public/private key pair. You must make your public key public so that others can use it to verify your digital signature or to encrypt session keys when creating an RSA envelope. How do you publicize your key?

Probably the best way is to register public keys with a trusted authority. Then, this trusted authority can certify that a particular public key belongs to a particular entity. Currently, such a public key registration infrastructure exists in the form of digital certificates.

C h a p t e r 3 C r y p t o g r a p h y

6 1

Image 83

$RSA Security 5.2.2 manual Digital Certificates, Math$

Contents

Cryptographic Components for C Crypto-C Limit distribution of this document to trusted personnel TrademarksLicense agreement Distribution Contents Cryptography Quick Start N t e n t s Algorithms in Crypto-C Using Crypto-C 101 Saved State Non-Cryptographic Operations 151 Public-Key Operations 213 Symmetric-Key Operations 177 Putting It All Together An X9.31 Example Secret Sharing Operations 305 Glossary 339 Index 349 Appendix a Command-Line Demos 327 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 MultiPrime RSA What’s New in Version 5.2.2?Improved performance Hardware support E f a c e Organization of This ManualAdvanced Encryption Standard AES Organization of This Manual 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 Http//stdsbbs.ieee.org/groups/1363/index.html Ieee Standard Specifications for Public-Key Cryptography on You can get technical support as follows How to Contact RSA SecurityRSA Security Web Site Getting Support and Service How to Contact RSA Security Introduction Algorithms Crypto-C Toolkit Secret Sharing Public-Key AlgorithmsDigital Signatures Elliptic Curve Public-Key Algorithms Nist Approval and Windows 32-bit Platforms Cryptographic Standards and Crypto-CPkcs Standards and Crypto-C Nist Standards and Crypto-C Nist Approval and Windows NT Platforms Pkcs Compared with Nist Ansi X9 Standards and Crypto-C Quick Start Six-Step Sequence Six-Step Sequence Introductory Example Introductory ExampleInclude Files Creating an Algorithm Object #define Nullptr POINTER0 Where Pointer is defined in aglobal.hNullptr is also defined in aglobal.h Typedef unsigned char *POINTER Algorithm information Setting the Algorithm ObjectBalgorithmobj algorithmObject Algorithm object Break Init For our example, we use Setting a Key ObjectCreating a Key Object Key information Int BSetKeyInfo Now we can complete the call to BSetKeyInfo Selecting an Algorithm Chooser We can now complete our call to BEncryptInit UpdateSaving the Object State optional Surrender Context Input data Output data bufferUnsigned int Length of output data Unsigned char *encryptedData = Nullptr Unsigned int outputLenUpdateFor now, we declare Final If status = BEncryptUpdate Asurrenderctx *surrenderContext Surrender context Destroy Pointer block For our example, we use the followingPointer to key object Balgorithmobj * algorithmObject If rc4KeyItem.data != Nullptr Putting It All TogetherFor this example, call Tfree as follows Tfree encryptedData Init If status = BEncryptInit Introductory Example EncryptedData = Nullptr End main If encryptedData != Nullptr First create the algorithm object Decrypting the Introductory ExampleCreating the Key Object Decrypting the Introductory Example Setting the Key Object DecryptedData = Nullptr Always destroy objects when you no longer need them Multiple Updates Multiple Updates Break End while Updatesize Multiple Updates Set Summary of the Six StepsInclude Create Final Page Cryptography Cryptography Overview Cryptography OverviewSymmetric-Key Cryptography Ciphers Crypto-C implements the following block ciphers Block CiphersPadding Ciphers in Crypto-C 2Triple DES Encryption as Implemented in Crypto-C Triple DES RC5 RC6 Four Modes Modes of Operation 3Electronic Codebook ECB Mode Electronic Codebook ECB Mode Cipher Feedback CFB Mode Cipher Block Chaining CBC Mode 5Cipher Feedback CFB Mode Output Feedback OFB Mode Stream Ciphers 6Output Feedback Mode OFB RC4 algorithm with MAC Message Digests Message Digests and Pseudo-Random Numbers Hash-Based Message Authentication Codes Hmac Password-Based Encryption Public-Key Cryptography 8DES Key and IV Generation for Password Based Encryption 9Public-Key Cryptography RSA Algorithm RSA Algorithm Modular MathPrime Numbers MultiPrime Numbers Cryptography Overview 1Calculation of 827 mod SummarySecurity Calculation is shown in Table Optimal Asymmetric Encryption Padding Oaep Digital Envelopes 10Digital Envelope Authentication and Digital Signatures Cryptography Overview 11RSA Digital Signature Digital Signature Algorithm DSA Math Digital Certificates Algorithm Diffie-Hellman Public Key Agreement Parameter Generation PhasePhase Math Multiple-Party Key Agreement Elliptic Curve Cryptography Finite Field Elliptic Curve Parameters Odd Prime Fields Fields of Even Characteristic = 0·I ≡ 2·2m-1·Imod2m Coefficients Over an Odd Prime Field Elliptic curve parameters Coefficients Over a Field of Even CharacteristicPoint P and its Order Points of an Elliptic Curve Is written EFq Order of an Elliptic CurveOrder of a Point Point of Prime Order 2Elliptic Curve Parameters Summary of Elliptic Curve TerminologyCofactor Order n of P Is sometimes called the base point Elliptic Curve Key Pair Generation Representing Fields of Even Characteristic Signing a Message Ecdsa Signature SchemeCreating the Key Pair Math Verifying a Signature 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 Elliptic Curve Diffie-Hellman Key Agreement Decrypting a Message Using the Private Key Phase 13Elliptic Curve Diffie-Hellman Key Agreement First coordinate of S, xS, is their agreed-upon secret key Secret Sharing Key Generation Working with Keys Key Escrow Key Management Applications of Cryptography Applications of CryptographyAscii Encoding and Decoding Local Applications Point-to-Point Applications Client/Server Applications Peer-to-Peer Applications Choosing Algorithms Choosing AlgorithmsPublic-Key vs. Symmetric-Key Cryptography Stream vs. Block Symmetric-Key Algorithms Key Agreement vs. Digital Envelopes Block Symmetric-Key Algorithms Elliptic Curve Algorithms Secret Sharing and Key Escrow Interoperability Security Considerations Security ConsiderationsHandling Private Keys Elliptic Curve Standards Pseudo-Random Numbers and Seed Generation Temporary Buffers Choosing Passwords 3DES Weak and Semi-Weak Keys Initialization Vectors and SaltsDES Weak Keys Timing Attacks and Blinding Stream Ciphers = mre mod n Pick a random value r and compute MD5p padP MD5q padQ m Choosing Key Sizes 4Summary of Recommended Key Sizes RSA Keys RC5 Key Bits and Rounds Diffie-Hellman Parameters and DSA KeysRC2 Effective Key Bits RC4 Key Bits Elliptic Curve Keys Algorithms in Crypto-C Using Crypto-C Algorithms in Crypto-C Information Formats Provided by Crypto-CBasic Algorithm Info Types BER-Based Algorithm Info Types 1Message Digests Summary of AIsPEM-Based Algorithm Info Types BSAFE1 Algorithm Info Types 5Symmetric Stream Ciphers Algorithms in Crypto-C 2Message Authentication3ASCII Encoding 4Pseudo-Random Number Generation 6Symmetric Block Ciphers Algorithms in Crypto-C 5Symmetric Stream Ciphers RC2 Algorithms in Crypto-C 6Symmetric Block Ciphers Key Generation 7RSA Public-Key Cryptography Oaep Algorithms in Crypto-C 7RSA Public-Key Cryptography Digital Signatures 8DSA Public-Key Cryptography 10Elliptic Curve Public-Key Cryptography Algorithms in Crypto-C 9Diffie-Hellman Key Agreement 12Hardware Interface 10 Elliptic Curve Public-Key Cryptography11Bloom-Shamir Secret Sharing Algorithm Info Type Algorithms in Crypto-C 13Advanced Encryption Standard AES 15Block Cipher Keys Keys In Crypto-CSummary of KIs Keys In Crypto-C 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 System Considerations In Crypto-CAlgorithm Choosers An Encryption Algorithm Chooser An RSA Algorithm Chooser Description of AIX962RandomV0 instead of AISHA1Random Surrender Context Int *Surrender Pointer handle Sample Surrender FunctionAlso gives the form that a surrender function must have Reserved for future use Saving State When to Allocate Memory Tmalloc Trealloc Tfree Tmemset Tmemcpy Tmemmove Tmemcmp Memory-Management RoutinesMemory-Management Routines and Standard C Libraries Crypto-C uses the following memory-management routines Binary Data BER/DER EncodingMemory Allocation Typedef struct unsigned char *data unsigned int len System Considerations In Crypto-C Output considerations Input and OutputSymmetric Block Algorithms Input constraints 20Input Limits for RSA Pkcs Encryption General Considerations Public Key Size Key SizeDES Keys Private Key Size 00 C2 58 60 B1 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 Tfprivate KeypairGenParams.privateKeyAttributes.tokenFlag = Now generate Using Cryptographic Hardware Using Cryptographic Hardware Return Behardware Pkcs #11 Support for DSA Key Pair Generation Balgorithmmethod *DSAKEYGENCHOOSER = &AMDSAKEYGEN POINTER&pubKeyData-params P11KeyGenParams.privateKeyAttributes.keyUsage = 00 00 00 66 a9 47 2d 80 5a Advanced Pkcs #11For an RSA private key, it would be this Digest CKBYTEPTRseedBuffer, CKULONGseedLen Hardware IssuesRandom Numbers Ckrv rv Rv = Using Cryptographic Hardware Page Non-Cryptographic Operations Example in this section corresponds to the file mdigest.c Message DigestsCreating a Digest Message Digests If status = BDigestInit #define Digestlen If status = BDigestUpdateYour call will be the following BER-Encoding the Digest Remember to destroy all objects when you are done with them Following example BER-encodes the preceeding sample digest Saving the State of a Digest Algorithm ObjectSaved State Item stateInfo = Null Message Digests Status = Rsademoealloc break 1Code Sample DigestDataSavedState Message Digests Hash-Based Message Authentication Code Hash-Based Message Authentication Code Hmac 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 Unsigned char *gets char *randomSeed != 0 break Puts Enter a random seed if status = If status = BGenerateRandomBytes Generate Additional seeding Generating Independent Streams of RandomnessThis step is identical to the previous example Typedef struct 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 BDestroyAlgorithmObject &asciiEncoder Tfree asciiEncoding Decoding ASCII-Encoded Data If status = binaryDecoding == Nullptr != 0 break If status = BDecodeInit asciiDecoder != 0 break BDestroyAlgorithmObject &asciiDecoder Tfree binaryDecoding Symmetric-Key Operations Example in this section corresponds to the file descbc.c Block CiphersDES with CBC Block Ciphers For new padding schemes Examples include des, rc5RC5 parameters Points at init vector Item Call BGenerateRandomBytes Now set up the Bblkcipherwfeedbackparams structure Format of info supplied to BSetKeyInfo Depends on cipher type, as follows Unsigned int outputBufferSize Decrypting RC2 Cipher Now you can set your algorithm object as follows Init If rc2KeyItem.data != Nullptr Use a random number generator to come up with 24 bytes 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 EncryptedData = Nullptr RC6 Cipher #define Blocksize Unsigned char initVectorBLOCKSIZE Setting the Key Data If rc6KeyItem.data != Nullptr Break OutputLenTotal = outputLenUpdate + outputLenFinal Been allocated AES Cipher Unsigned char *aesParams Creating a Key Object If aesKeyItem.data != Nullptr Final Password-Based Encryption Iteration count Unsigned char * salt 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 Milliseconds Private non-CRT What is MultiPrime?MultiPrime MultiPrime Two-Prime RSA 48.8 17.5 Three-Prime RSA 10.9 How Many Primes? Sample BGetKeyInfo BGenerateInitRsaGen BGenerateKeypair Set the Algorithm Object Generating an RSA MultiPrime Key If status = BGenerateInit keypairGenerator Distributing an RSA Public Key If you looked at the elements of the struct Crypto-C FormatBER/DER Encoding If status = myPublicKeyBER.data == Nullptr != Format of info returned by BGetKeyInfo Tfree myPublicKeyBER.data RSA Public-Key EncryptionSend it off Remember to free any memory you allocated Input constraints Reference Manual entry on AIPKCSRSAPublic states #define Blocksize RSA Private-Key Decryption If status = BDecryptInit Raw RSA Encryption and Decryption Optimal Asymetric Encryption Padding Oaep MultiPrime Computing a Digital Signature RSA Digital Signatures Setting The Algorithm Object Entry for the AI in use Surrender context outlined in The Surrender Context on Verifying a Digital Signature If status = BVerifyInit Update Performing DSA Operations Performing DSA OperationsGenerating DSA Parameters Bdsaparamgenparams dsaParams DsaParams.primeBits = There is no in generating DSA parameters 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 Performing Diffie-Hellman Key Agreement Generating Diffie-Hellman Parameters Adhparamgenparams Adhparamgenparams dhParams There is no in generating Diffie-Hellman parameters Destroy Base generator Distributing Diffie-Hellman Parameters If you look at the elements of the struct BER Format 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 Item dhParametersBER Diffie-Hellman Key Agreement If status = BKeyAgreeInit Phase Other party should send their public value and its length Saving the Object State Performing Elliptic Curve Operations Performing Elliptic Curve OperationsGenerating Elliptic Curve Parameters Implementation version Input of 0 defaults to fieldElementBitsInput of 0 defaults to Format of info supplied to BSetAlgorithmInfo Performing Elliptic Curve Operations No Update step is necessary for parameter generation GeneralFlag = Retrieving Elliptic Curve Parameters Elliptic curve coefficient Indicates type of base fieldIt is the prime number Case that fieldType = Ftfp If status = output-base.data == Nullptr != 0 break FreeECParamInfo&ecParamInfo If status = output-order.data == Nullptr != 0 breakAecparams ecParamInfo Sample code, FreeECParamInfo is implemented as follows Generating an Elliptic Curve Key Pair Becparams paramInfo Initialize There is no Update step for key generation Aecparams curveParams Retrieving an Elliptic Curve KeyKIECPublic gives a pointer to an Aecpublickey structure Public component Performing Elliptic Curve Operations End FreeECPubKeyInfo Generating Acceleration TablesGenerating a Generic Acceleration Table Aecparams *cryptocECParamInfo Aecparams ecParamInfo Retrieve the elliptic curve parametersFormat the information For odd prime field There is no Update step for building acceleration tables Item accelTableItem Allocate memoryBuild the acceleration table Retrieve the public key information Generating a Public-Key Acceleration Table FreeECPubKeyInfo&publicKeyInfo Put the information retrieved in the proper format Item pubKeyAccelTableItem unsigned int maxTableLen GeneralFlag = If status = BBuildTableFinal Performing EC Diffie-Hellman Key AgreementBuild the public-key acceleration table Item pubKeyAccelTableItem Create Optional Set Acceleration Table Info If status = alicePublicValue == Nullptr != 0 break If status = aliceSecretValue == Nullptr != 0 break Performing Ecdsa in Compliance with Ansi Generating an EC Key Pair Generating EC Parameters Computing a Digital Signature Create Item aTableItem Build an algorithm chooser with the appropriate AMs If status = signature == Nullptr != 0 break Now, using BSignUpdate, pass in the data to be signedAecparams *ecParamInfo Unsigned int signatureLen Initialized random algorithm in BSignFinalDestroy all objects that are no longer needed Use the same AI and digestInfo as you did for signing Optional Set Public Key Acceleration Table Info Performing Ecdsa with X9.62-Compliant BER 0x7a Item stockECParamsBERECParamsBER154 = 0x30 0x81 0x52 0xd1 StockECParamsBER.data = ECParamsBER stockECParamsBER.len = 0x79 DataToSign = 0x530x68 0x73 Aecparams *ecParamsInfo Use the same AI as you did for signing Tfreesignature BDestoryAlgorithmObject &ecDSAVerify Using Ecaes 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 = TmallocmaxDecryptedDataLen BDecryptUpdateOutputLenUpdate MaxDecryptedDataLen = outputLenTotalPage Generating Shares Secret Sharing Example in this section corresponds to the file scrtshar.c Secret Sharing #define Secretsize 16 #define Totalshares Tfree secretSharecount If status = secretSharecount == Nullptr != 0 breakBreak If status != 0 break Unsigned int outputLenFinal If status = BEncryptFinal Use the same AI, AIBSSecretSharing Reconstructing the Secret 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 Printf ================================================\n Generating Random BytesStatic unsigned char f4Data = 0x01, 0x00 Unsigned int status 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 != Return End GeneralSurrenderFunction Surrendering Control Printing the Buffer Contents Appendix a Overview of the Demos Command-Line Demo User’s Guide Command-Line Demo User’s GuideCommand Line mode Input Redirection mode 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 Running Bdemoec Using Bdemoec Table A-1Demo Program Source Files File ReferenceFile Reference BSLite Table A-1Demo Program Source Files BSLite BSLite Page Glossary Application Programming Interface Advanced Encryption StandardSeries of steps used to complete a task American National Standards Institute O s s a r y See XOR Generic security service application program interfaceElectronic business Data Interchange See ECC An algorithm that generates the subkeys in a block cipher Process of having a third party hold onto encryption keysProcess that creates a larger key from the original key Act of creating a key Private key in the RSA public-key cryptosystem Extra bits concatenated with a key, password, or plaintextData to be encrypted Secure Multipurpose Internet Mail Extensions Number extracted from a pseudo- random sequence Trusted Computer System Evaluation Criteria Simple Mail Transfer ProtocolSecure Wide Area Network See secret key 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