vendor/github.com/docker/libtrust/rsa_key.go - third_party/github.com/docker/docker - Git at Google

 package libtrust

 import (
 	"crypto"
 	"crypto/rand"
 	"crypto/rsa"
 	"crypto/x509"
 	"encoding/json"
 	"encoding/pem"
 	"errors"
 	"fmt"
 	"io"
 	"math/big"
 )

 /*
  * RSA DSA PUBLIC KEY
  */

 // rsaPublicKey implements a JWK Public Key using RSA digital signature algorithms.
 type rsaPublicKey struct {
 	*rsa.PublicKey
 	extended map[string]interface{}
 }

 func fromRSAPublicKey(cryptoPublicKey *rsa.PublicKey) *rsaPublicKey {
 	return &rsaPublicKey{cryptoPublicKey, map[string]interface{}{}}
 }

 // KeyType returns the JWK key type for RSA keys, i.e., "RSA".
 func (k *rsaPublicKey) KeyType() string {
 	return "RSA"
 }

 // KeyID returns a distinct identifier which is unique to this Public Key.
 func (k *rsaPublicKey) KeyID() string {
 	return keyIDFromCryptoKey(k)
 }

 func (k *rsaPublicKey) String() string {
 	return fmt.Sprintf("RSA Public Key <%s>", k.KeyID())
 }

 // Verify verifyies the signature of the data in the io.Reader using this Public Key.
 // The alg parameter should be the name of the JWA digital signature algorithm
 // which was used to produce the signature and should be supported by this
 // public key. Returns a nil error if the signature is valid.
 func (k *rsaPublicKey) Verify(data io.Reader, alg string, signature []byte) error {
 	// Verify the signature of the given date, return non-nil error if valid.
 	sigAlg, err := rsaSignatureAlgorithmByName(alg)
 	if err != nil {
 		return fmt.Errorf("unable to verify Signature: %s", err)
 	}

 	hasher := sigAlg.HashID().New()
 	_, err = io.Copy(hasher, data)
 	if err != nil {
 		return fmt.Errorf("error reading data to sign: %s", err)
 	}
 	hash := hasher.Sum(nil)

 	err = rsa.VerifyPKCS1v15(k.PublicKey, sigAlg.HashID(), hash, signature)
 	if err != nil {
 		return fmt.Errorf("invalid %s signature: %s", sigAlg.HeaderParam(), err)
 	}

 	return nil
 }

 // CryptoPublicKey returns the internal object which can be used as a
 // crypto.PublicKey for use with other standard library operations. The type
 // is either *rsa.PublicKey or *ecdsa.PublicKey
 func (k *rsaPublicKey) CryptoPublicKey() crypto.PublicKey {
 	return k.PublicKey
 }

 func (k *rsaPublicKey) toMap() map[string]interface{} {
 	jwk := make(map[string]interface{})
 	for k, v := range k.extended {
 		jwk[k] = v
 	}
 	jwk["kty"] = k.KeyType()
 	jwk["kid"] = k.KeyID()
 	jwk["n"] = joseBase64UrlEncode(k.N.Bytes())
 	jwk["e"] = joseBase64UrlEncode(serializeRSAPublicExponentParam(k.E))

 	return jwk
 }

 // MarshalJSON serializes this Public Key using the JWK JSON serialization format for
 // RSA keys.
 func (k *rsaPublicKey) MarshalJSON() (data []byte, err error) {
 	return json.Marshal(k.toMap())
 }

 // PEMBlock serializes this Public Key to DER-encoded PKIX format.
 func (k *rsaPublicKey) PEMBlock() (*pem.Block, error) {
 	derBytes, err := x509.MarshalPKIXPublicKey(k.PublicKey)
 	if err != nil {
 		return nil, fmt.Errorf("unable to serialize RSA PublicKey to DER-encoded PKIX format: %s", err)
 	}
 	k.extended["kid"] = k.KeyID() // For display purposes.
 	return createPemBlock("PUBLIC KEY", derBytes, k.extended)
 }

 func (k *rsaPublicKey) AddExtendedField(field string, value interface{}) {
 	k.extended[field] = value
 }

 func (k *rsaPublicKey) GetExtendedField(field string) interface{} {
 	v, ok := k.extended[field]
 	if !ok {
 		return nil
 	}
 	return v
 }

 func rsaPublicKeyFromMap(jwk map[string]interface{}) (*rsaPublicKey, error) {
 	// JWK key type (kty) has already been determined to be "RSA".
 	// Need to extract 'n', 'e', and 'kid' and check for
 	// consistency.

 	// Get the modulus parameter N.
 	nB64Url, err := stringFromMap(jwk, "n")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Public Key modulus: %s", err)
 	}

 	n, err := parseRSAModulusParam(nB64Url)
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Public Key modulus: %s", err)
 	}

 	// Get the public exponent E.
 	eB64Url, err := stringFromMap(jwk, "e")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Public Key exponent: %s", err)
 	}

 	e, err := parseRSAPublicExponentParam(eB64Url)
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Public Key exponent: %s", err)
 	}

 	key := &rsaPublicKey{
 		PublicKey: &rsa.PublicKey{N: n, E: e},
 	}

 	// Key ID is optional, but if it exists, it should match the key.
 	_, ok := jwk["kid"]
 	if ok {
 		kid, err := stringFromMap(jwk, "kid")
 		if err != nil {
 			return nil, fmt.Errorf("JWK RSA Public Key ID: %s", err)
 		}
 		if kid != key.KeyID() {
 			return nil, fmt.Errorf("JWK RSA Public Key ID does not match: %s", kid)
 		}
 	}

 	if _, ok := jwk["d"]; ok {
 		return nil, fmt.Errorf("JWK RSA Public Key cannot contain private exponent")
 	}

 	key.extended = jwk

 	return key, nil
 }

 /*
  * RSA DSA PRIVATE KEY
  */

 // rsaPrivateKey implements a JWK Private Key using RSA digital signature algorithms.
 type rsaPrivateKey struct {
 	rsaPublicKey
 	*rsa.PrivateKey
 }

 func fromRSAPrivateKey(cryptoPrivateKey *rsa.PrivateKey) *rsaPrivateKey {
 	return &rsaPrivateKey{
 		*fromRSAPublicKey(&cryptoPrivateKey.PublicKey),
 		cryptoPrivateKey,
 	}
 }

 // PublicKey returns the Public Key data associated with this Private Key.
 func (k *rsaPrivateKey) PublicKey() PublicKey {
 	return &k.rsaPublicKey
 }

 func (k *rsaPrivateKey) String() string {
 	return fmt.Sprintf("RSA Private Key <%s>", k.KeyID())
 }

 // Sign signs the data read from the io.Reader using a signature algorithm supported
 // by the RSA private key. If the specified hashing algorithm is supported by
 // this key, that hash function is used to generate the signature otherwise the
 // the default hashing algorithm for this key is used. Returns the signature
 // and the name of the JWK signature algorithm used, e.g., "RS256", "RS384",
 // "RS512".
 func (k *rsaPrivateKey) Sign(data io.Reader, hashID crypto.Hash) (signature []byte, alg string, err error) {
 	// Generate a signature of the data using the internal alg.
 	sigAlg := rsaPKCS1v15SignatureAlgorithmForHashID(hashID)
 	hasher := sigAlg.HashID().New()

 	_, err = io.Copy(hasher, data)
 	if err != nil {
 		return nil, "", fmt.Errorf("error reading data to sign: %s", err)
 	}
 	hash := hasher.Sum(nil)

 	signature, err = rsa.SignPKCS1v15(rand.Reader, k.PrivateKey, sigAlg.HashID(), hash)
 	if err != nil {
 		return nil, "", fmt.Errorf("error producing signature: %s", err)
 	}

 	alg = sigAlg.HeaderParam()

 	return
 }

 // CryptoPrivateKey returns the internal object which can be used as a
 // crypto.PublicKey for use with other standard library operations. The type
 // is either *rsa.PublicKey or *ecdsa.PublicKey
 func (k *rsaPrivateKey) CryptoPrivateKey() crypto.PrivateKey {
 	return k.PrivateKey
 }

 func (k *rsaPrivateKey) toMap() map[string]interface{} {
 	k.Precompute() // Make sure the precomputed values are stored.
 	jwk := k.rsaPublicKey.toMap()

 	jwk["d"] = joseBase64UrlEncode(k.D.Bytes())
 	jwk["p"] = joseBase64UrlEncode(k.Primes[0].Bytes())
 	jwk["q"] = joseBase64UrlEncode(k.Primes[1].Bytes())
 	jwk["dp"] = joseBase64UrlEncode(k.Precomputed.Dp.Bytes())
 	jwk["dq"] = joseBase64UrlEncode(k.Precomputed.Dq.Bytes())
 	jwk["qi"] = joseBase64UrlEncode(k.Precomputed.Qinv.Bytes())

 	otherPrimes := k.Primes[2:]

 	if len(otherPrimes) > 0 {
 		otherPrimesInfo := make([]interface{}, len(otherPrimes))
 		for i, r := range otherPrimes {
 			otherPrimeInfo := make(map[string]string, 3)
 			otherPrimeInfo["r"] = joseBase64UrlEncode(r.Bytes())
 			crtVal := k.Precomputed.CRTValues[i]
 			otherPrimeInfo["d"] = joseBase64UrlEncode(crtVal.Exp.Bytes())
 			otherPrimeInfo["t"] = joseBase64UrlEncode(crtVal.Coeff.Bytes())
 			otherPrimesInfo[i] = otherPrimeInfo
 		}
 		jwk["oth"] = otherPrimesInfo
 	}

 	return jwk
 }

 // MarshalJSON serializes this Private Key using the JWK JSON serialization format for
 // RSA keys.
 func (k *rsaPrivateKey) MarshalJSON() (data []byte, err error) {
 	return json.Marshal(k.toMap())
 }

 // PEMBlock serializes this Private Key to DER-encoded PKIX format.
 func (k *rsaPrivateKey) PEMBlock() (*pem.Block, error) {
 	derBytes := x509.MarshalPKCS1PrivateKey(k.PrivateKey)
 	k.extended["keyID"] = k.KeyID() // For display purposes.
 	return createPemBlock("RSA PRIVATE KEY", derBytes, k.extended)
 }

 func rsaPrivateKeyFromMap(jwk map[string]interface{}) (*rsaPrivateKey, error) {
 	// The JWA spec for RSA Private Keys (draft rfc section 5.3.2) states that
 	// only the private key exponent 'd' is REQUIRED, the others are just for
 	// signature/decryption optimizations and SHOULD be included when the JWK
 	// is produced. We MAY choose to accept a JWK which only includes 'd', but
 	// we're going to go ahead and not choose to accept it without the extra
 	// fields. Only the 'oth' field will be optional (for multi-prime keys).
 	privateExponent, err := parseRSAPrivateKeyParamFromMap(jwk, "d")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key exponent: %s", err)
 	}
 	firstPrimeFactor, err := parseRSAPrivateKeyParamFromMap(jwk, "p")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
 	}
 	secondPrimeFactor, err := parseRSAPrivateKeyParamFromMap(jwk, "q")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
 	}
 	firstFactorCRT, err := parseRSAPrivateKeyParamFromMap(jwk, "dp")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
 	}
 	secondFactorCRT, err := parseRSAPrivateKeyParamFromMap(jwk, "dq")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
 	}
 	crtCoeff, err := parseRSAPrivateKeyParamFromMap(jwk, "qi")
 	if err != nil {
 		return nil, fmt.Errorf("JWK RSA Private Key CRT coefficient: %s", err)
 	}

 	var oth interface{}
 	if _, ok := jwk["oth"]; ok {
 		oth = jwk["oth"]
 		delete(jwk, "oth")
 	}

 	// JWK key type (kty) has already been determined to be "RSA".
 	// Need to extract the public key information, then extract the private
 	// key values.
 	publicKey, err := rsaPublicKeyFromMap(jwk)
 	if err != nil {
 		return nil, err
 	}

 	privateKey := &rsa.PrivateKey{
 		PublicKey: *publicKey.PublicKey,
 		D:         privateExponent,
 		Primes:    []*big.Int{firstPrimeFactor, secondPrimeFactor},
 		Precomputed: rsa.PrecomputedValues{
 			Dp:   firstFactorCRT,
 			Dq:   secondFactorCRT,
 			Qinv: crtCoeff,
 		},
 	}

 	if oth != nil {
 		// Should be an array of more JSON objects.
 		otherPrimesInfo, ok := oth.([]interface{})
 		if !ok {
 			return nil, errors.New("JWK RSA Private Key: Invalid other primes info: must be an array")
 		}
 		numOtherPrimeFactors := len(otherPrimesInfo)
 		if numOtherPrimeFactors == 0 {
 			return nil, errors.New("JWK RSA Privake Key: Invalid other primes info: must be absent or non-empty")
 		}
 		otherPrimeFactors := make([]*big.Int, numOtherPrimeFactors)
 		productOfPrimes := new(big.Int).Mul(firstPrimeFactor, secondPrimeFactor)
 		crtValues := make([]rsa.CRTValue, numOtherPrimeFactors)

 		for i, val := range otherPrimesInfo {
 			otherPrimeinfo, ok := val.(map[string]interface{})
 			if !ok {
 				return nil, errors.New("JWK RSA Private Key: Invalid other prime info: must be a JSON object")
 			}

 			otherPrimeFactor, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "r")
 			if err != nil {
 				return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
 			}
 			otherFactorCRT, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "d")
 			if err != nil {
 				return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
 			}
 			otherCrtCoeff, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "t")
 			if err != nil {
 				return nil, fmt.Errorf("JWK RSA Private Key CRT coefficient: %s", err)
 			}

 			crtValue := crtValues[i]
 			crtValue.Exp = otherFactorCRT
 			crtValue.Coeff = otherCrtCoeff
 			crtValue.R = productOfPrimes
 			otherPrimeFactors[i] = otherPrimeFactor
 			productOfPrimes = new(big.Int).Mul(productOfPrimes, otherPrimeFactor)
 		}

 		privateKey.Primes = append(privateKey.Primes, otherPrimeFactors...)
 		privateKey.Precomputed.CRTValues = crtValues
 	}

 	key := &rsaPrivateKey{
 		rsaPublicKey: *publicKey,
 		PrivateKey:   privateKey,
 	}

 	return key, nil
 }

 /*
  *	Key Generation Functions.
  */

 func generateRSAPrivateKey(bits int) (k *rsaPrivateKey, err error) {
 	k = new(rsaPrivateKey)
 	k.PrivateKey, err = rsa.GenerateKey(rand.Reader, bits)
 	if err != nil {
 		return nil, err
 	}

 	k.rsaPublicKey.PublicKey = &k.PrivateKey.PublicKey
 	k.extended = make(map[string]interface{})

 	return
 }

 // GenerateRSA2048PrivateKey generates a key pair using 2048-bit RSA.
 func GenerateRSA2048PrivateKey() (PrivateKey, error) {
 	k, err := generateRSAPrivateKey(2048)
 	if err != nil {
 		return nil, fmt.Errorf("error generating RSA 2048-bit key: %s", err)
 	}

 	return k, nil
 }

 // GenerateRSA3072PrivateKey generates a key pair using 3072-bit RSA.
 func GenerateRSA3072PrivateKey() (PrivateKey, error) {
 	k, err := generateRSAPrivateKey(3072)
 	if err != nil {
 		return nil, fmt.Errorf("error generating RSA 3072-bit key: %s", err)
 	}

 	return k, nil
 }

 // GenerateRSA4096PrivateKey generates a key pair using 4096-bit RSA.
 func GenerateRSA4096PrivateKey() (PrivateKey, error) {
 	k, err := generateRSAPrivateKey(4096)
 	if err != nil {
 		return nil, fmt.Errorf("error generating RSA 4096-bit key: %s", err)
 	}

 	return k, nil
 }
	package libtrust

	import (
	"crypto"
	"crypto/rand"
	"crypto/rsa"
	"crypto/x509"
	"encoding/json"
	"encoding/pem"
	"errors"
	"fmt"
	"io"
	"math/big"
	)

	/*
	* RSA DSA PUBLIC KEY
	*/

	// rsaPublicKey implements a JWK Public Key using RSA digital signature algorithms.
	type rsaPublicKey struct {
	*rsa.PublicKey
	extended map[string]interface{}
	}

	func fromRSAPublicKey(cryptoPublicKey rsa.PublicKey) rsaPublicKey {
	return &rsaPublicKey{cryptoPublicKey, map[string]interface{}{}}
	}

	// KeyType returns the JWK key type for RSA keys, i.e., "RSA".
	func (k *rsaPublicKey) KeyType() string {
	return "RSA"
	}

	// KeyID returns a distinct identifier which is unique to this Public Key.
	func (k *rsaPublicKey) KeyID() string {
	return keyIDFromCryptoKey(k)
	}

	func (k *rsaPublicKey) String() string {
	return fmt.Sprintf("RSA Public Key <%s>", k.KeyID())
	}

	// Verify verifyies the signature of the data in the io.Reader using this Public Key.
	// The alg parameter should be the name of the JWA digital signature algorithm
	// which was used to produce the signature and should be supported by this
	// public key. Returns a nil error if the signature is valid.
	func (k *rsaPublicKey) Verify(data io.Reader, alg string, signature []byte) error {
	// Verify the signature of the given date, return non-nil error if valid.
	sigAlg, err := rsaSignatureAlgorithmByName(alg)
	if err != nil {
	return fmt.Errorf("unable to verify Signature: %s", err)
	}

	hasher := sigAlg.HashID().New()
	_, err = io.Copy(hasher, data)
	if err != nil {
	return fmt.Errorf("error reading data to sign: %s", err)
	}
	hash := hasher.Sum(nil)

	err = rsa.VerifyPKCS1v15(k.PublicKey, sigAlg.HashID(), hash, signature)
	if err != nil {
	return fmt.Errorf("invalid %s signature: %s", sigAlg.HeaderParam(), err)
	}

	return nil
	}

	// CryptoPublicKey returns the internal object which can be used as a
	// crypto.PublicKey for use with other standard library operations. The type
	// is either rsa.PublicKey or ecdsa.PublicKey
	func (k *rsaPublicKey) CryptoPublicKey() crypto.PublicKey {
	return k.PublicKey
	}

	func (k *rsaPublicKey) toMap() map[string]interface{} {
	jwk := make(map[string]interface{})
	for k, v := range k.extended {
	jwk[k] = v
	}
	jwk["kty"] = k.KeyType()
	jwk["kid"] = k.KeyID()
	jwk["n"] = joseBase64UrlEncode(k.N.Bytes())
	jwk["e"] = joseBase64UrlEncode(serializeRSAPublicExponentParam(k.E))

	return jwk
	}

	// MarshalJSON serializes this Public Key using the JWK JSON serialization format for
	// RSA keys.
	func (k *rsaPublicKey) MarshalJSON() (data []byte, err error) {
	return json.Marshal(k.toMap())
	}

	// PEMBlock serializes this Public Key to DER-encoded PKIX format.
	func (k rsaPublicKey) PEMBlock() (pem.Block, error) {
	derBytes, err := x509.MarshalPKIXPublicKey(k.PublicKey)
	if err != nil {
	return nil, fmt.Errorf("unable to serialize RSA PublicKey to DER-encoded PKIX format: %s", err)
	}
	k.extended["kid"] = k.KeyID() // For display purposes.
	return createPemBlock("PUBLIC KEY", derBytes, k.extended)
	}

	func (k *rsaPublicKey) AddExtendedField(field string, value interface{}) {
	k.extended[field] = value
	}

	func (k *rsaPublicKey) GetExtendedField(field string) interface{} {
	v, ok := k.extended[field]
	if !ok {
	return nil
	}
	return v
	}

	func rsaPublicKeyFromMap(jwk map[string]interface{}) (*rsaPublicKey, error) {
	// JWK key type (kty) has already been determined to be "RSA".
	// Need to extract 'n', 'e', and 'kid' and check for
	// consistency.

	// Get the modulus parameter N.
	nB64Url, err := stringFromMap(jwk, "n")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Public Key modulus: %s", err)
	}

	n, err := parseRSAModulusParam(nB64Url)
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Public Key modulus: %s", err)
	}

	// Get the public exponent E.
	eB64Url, err := stringFromMap(jwk, "e")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Public Key exponent: %s", err)
	}

	e, err := parseRSAPublicExponentParam(eB64Url)
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Public Key exponent: %s", err)
	}

	key := &rsaPublicKey{
	PublicKey: &rsa.PublicKey{N: n, E: e},
	}

	// Key ID is optional, but if it exists, it should match the key.
	_, ok := jwk["kid"]
	if ok {
	kid, err := stringFromMap(jwk, "kid")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Public Key ID: %s", err)
	}
	if kid != key.KeyID() {
	return nil, fmt.Errorf("JWK RSA Public Key ID does not match: %s", kid)
	}
	}

	if _, ok := jwk["d"]; ok {
	return nil, fmt.Errorf("JWK RSA Public Key cannot contain private exponent")
	}

	key.extended = jwk

	return key, nil
	}

	/*
	* RSA DSA PRIVATE KEY
	*/

	// rsaPrivateKey implements a JWK Private Key using RSA digital signature algorithms.
	type rsaPrivateKey struct {
	rsaPublicKey
	*rsa.PrivateKey
	}

	func fromRSAPrivateKey(cryptoPrivateKey rsa.PrivateKey) rsaPrivateKey {
	return &rsaPrivateKey{
	*fromRSAPublicKey(&cryptoPrivateKey.PublicKey),
	cryptoPrivateKey,
	}
	}

	// PublicKey returns the Public Key data associated with this Private Key.
	func (k *rsaPrivateKey) PublicKey() PublicKey {
	return &k.rsaPublicKey
	}

	func (k *rsaPrivateKey) String() string {
	return fmt.Sprintf("RSA Private Key <%s>", k.KeyID())
	}

	// Sign signs the data read from the io.Reader using a signature algorithm supported
	// by the RSA private key. If the specified hashing algorithm is supported by
	// this key, that hash function is used to generate the signature otherwise the
	// the default hashing algorithm for this key is used. Returns the signature
	// and the name of the JWK signature algorithm used, e.g., "RS256", "RS384",
	// "RS512".
	func (k *rsaPrivateKey) Sign(data io.Reader, hashID crypto.Hash) (signature []byte, alg string, err error) {
	// Generate a signature of the data using the internal alg.
	sigAlg := rsaPKCS1v15SignatureAlgorithmForHashID(hashID)
	hasher := sigAlg.HashID().New()

	_, err = io.Copy(hasher, data)
	if err != nil {
	return nil, "", fmt.Errorf("error reading data to sign: %s", err)
	}
	hash := hasher.Sum(nil)

	signature, err = rsa.SignPKCS1v15(rand.Reader, k.PrivateKey, sigAlg.HashID(), hash)
	if err != nil {
	return nil, "", fmt.Errorf("error producing signature: %s", err)
	}

	alg = sigAlg.HeaderParam()

	return
	}

	// CryptoPrivateKey returns the internal object which can be used as a
	// crypto.PublicKey for use with other standard library operations. The type
	// is either rsa.PublicKey or ecdsa.PublicKey
	func (k *rsaPrivateKey) CryptoPrivateKey() crypto.PrivateKey {
	return k.PrivateKey
	}

	func (k *rsaPrivateKey) toMap() map[string]interface{} {
	k.Precompute() // Make sure the precomputed values are stored.
	jwk := k.rsaPublicKey.toMap()

	jwk["d"] = joseBase64UrlEncode(k.D.Bytes())
	jwk["p"] = joseBase64UrlEncode(k.Primes[0].Bytes())
	jwk["q"] = joseBase64UrlEncode(k.Primes[1].Bytes())
	jwk["dp"] = joseBase64UrlEncode(k.Precomputed.Dp.Bytes())
	jwk["dq"] = joseBase64UrlEncode(k.Precomputed.Dq.Bytes())
	jwk["qi"] = joseBase64UrlEncode(k.Precomputed.Qinv.Bytes())

	otherPrimes := k.Primes[2:]

	if len(otherPrimes) > 0 {
	otherPrimesInfo := make([]interface{}, len(otherPrimes))
	for i, r := range otherPrimes {
	otherPrimeInfo := make(map[string]string, 3)
	otherPrimeInfo["r"] = joseBase64UrlEncode(r.Bytes())
	crtVal := k.Precomputed.CRTValues[i]
	otherPrimeInfo["d"] = joseBase64UrlEncode(crtVal.Exp.Bytes())
	otherPrimeInfo["t"] = joseBase64UrlEncode(crtVal.Coeff.Bytes())
	otherPrimesInfo[i] = otherPrimeInfo
	}
	jwk["oth"] = otherPrimesInfo
	}

	return jwk
	}

	// MarshalJSON serializes this Private Key using the JWK JSON serialization format for
	// RSA keys.
	func (k *rsaPrivateKey) MarshalJSON() (data []byte, err error) {
	return json.Marshal(k.toMap())
	}

	// PEMBlock serializes this Private Key to DER-encoded PKIX format.
	func (k rsaPrivateKey) PEMBlock() (pem.Block, error) {
	derBytes := x509.MarshalPKCS1PrivateKey(k.PrivateKey)
	k.extended["keyID"] = k.KeyID() // For display purposes.
	return createPemBlock("RSA PRIVATE KEY", derBytes, k.extended)
	}

	func rsaPrivateKeyFromMap(jwk map[string]interface{}) (*rsaPrivateKey, error) {
	// The JWA spec for RSA Private Keys (draft rfc section 5.3.2) states that
	// only the private key exponent 'd' is REQUIRED, the others are just for
	// signature/decryption optimizations and SHOULD be included when the JWK
	// is produced. We MAY choose to accept a JWK which only includes 'd', but
	// we're going to go ahead and not choose to accept it without the extra
	// fields. Only the 'oth' field will be optional (for multi-prime keys).
	privateExponent, err := parseRSAPrivateKeyParamFromMap(jwk, "d")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key exponent: %s", err)
	}
	firstPrimeFactor, err := parseRSAPrivateKeyParamFromMap(jwk, "p")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
	}
	secondPrimeFactor, err := parseRSAPrivateKeyParamFromMap(jwk, "q")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
	}
	firstFactorCRT, err := parseRSAPrivateKeyParamFromMap(jwk, "dp")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
	}
	secondFactorCRT, err := parseRSAPrivateKeyParamFromMap(jwk, "dq")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
	}
	crtCoeff, err := parseRSAPrivateKeyParamFromMap(jwk, "qi")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key CRT coefficient: %s", err)
	}

	var oth interface{}
	if _, ok := jwk["oth"]; ok {
	oth = jwk["oth"]
	delete(jwk, "oth")
	}

	// JWK key type (kty) has already been determined to be "RSA".
	// Need to extract the public key information, then extract the private
	// key values.
	publicKey, err := rsaPublicKeyFromMap(jwk)
	if err != nil {
	return nil, err
	}

	privateKey := &rsa.PrivateKey{
	PublicKey: *publicKey.PublicKey,
	D: privateExponent,
	Primes: []*big.Int{firstPrimeFactor, secondPrimeFactor},
	Precomputed: rsa.PrecomputedValues{
	Dp: firstFactorCRT,
	Dq: secondFactorCRT,
	Qinv: crtCoeff,
	},
	}

	if oth != nil {
	// Should be an array of more JSON objects.
	otherPrimesInfo, ok := oth.([]interface{})
	if !ok {
	return nil, errors.New("JWK RSA Private Key: Invalid other primes info: must be an array")
	}
	numOtherPrimeFactors := len(otherPrimesInfo)
	if numOtherPrimeFactors == 0 {
	return nil, errors.New("JWK RSA Privake Key: Invalid other primes info: must be absent or non-empty")
	}
	otherPrimeFactors := make([]*big.Int, numOtherPrimeFactors)
	productOfPrimes := new(big.Int).Mul(firstPrimeFactor, secondPrimeFactor)
	crtValues := make([]rsa.CRTValue, numOtherPrimeFactors)

	for i, val := range otherPrimesInfo {
	otherPrimeinfo, ok := val.(map[string]interface{})
	if !ok {
	return nil, errors.New("JWK RSA Private Key: Invalid other prime info: must be a JSON object")
	}

	otherPrimeFactor, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "r")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key prime factor: %s", err)
	}
	otherFactorCRT, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "d")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key CRT exponent: %s", err)
	}
	otherCrtCoeff, err := parseRSAPrivateKeyParamFromMap(otherPrimeinfo, "t")
	if err != nil {
	return nil, fmt.Errorf("JWK RSA Private Key CRT coefficient: %s", err)
	}

	crtValue := crtValues[i]
	crtValue.Exp = otherFactorCRT
	crtValue.Coeff = otherCrtCoeff
	crtValue.R = productOfPrimes
	otherPrimeFactors[i] = otherPrimeFactor
	productOfPrimes = new(big.Int).Mul(productOfPrimes, otherPrimeFactor)
	}

	privateKey.Primes = append(privateKey.Primes, otherPrimeFactors...)
	privateKey.Precomputed.CRTValues = crtValues
	}

	key := &rsaPrivateKey{
	rsaPublicKey: *publicKey,
	PrivateKey: privateKey,
	}

	return key, nil
	}

	/*
	* Key Generation Functions.
	*/

	func generateRSAPrivateKey(bits int) (k *rsaPrivateKey, err error) {
	k = new(rsaPrivateKey)
	k.PrivateKey, err = rsa.GenerateKey(rand.Reader, bits)
	if err != nil {
	return nil, err
	}

	k.rsaPublicKey.PublicKey = &k.PrivateKey.PublicKey
	k.extended = make(map[string]interface{})

	return
	}

	// GenerateRSA2048PrivateKey generates a key pair using 2048-bit RSA.
	func GenerateRSA2048PrivateKey() (PrivateKey, error) {
	k, err := generateRSAPrivateKey(2048)
	if err != nil {
	return nil, fmt.Errorf("error generating RSA 2048-bit key: %s", err)
	}

	return k, nil
	}

	// GenerateRSA3072PrivateKey generates a key pair using 3072-bit RSA.
	func GenerateRSA3072PrivateKey() (PrivateKey, error) {
	k, err := generateRSAPrivateKey(3072)
	if err != nil {
	return nil, fmt.Errorf("error generating RSA 3072-bit key: %s", err)
	}

	return k, nil
	}

	// GenerateRSA4096PrivateKey generates a key pair using 4096-bit RSA.
	func GenerateRSA4096PrivateKey() (PrivateKey, error) {
	k, err := generateRSAPrivateKey(4096)
	if err != nil {
	return nil, fmt.Errorf("error generating RSA 4096-bit key: %s", err)
	}

	return k, nil
	}