/*
 * nimbus-jose-jwt
 *
 * Copyright 2012-2016, Connect2id Ltd.
 *
 * Licensed under the Apache License, Version 2.0 (the "License"); you may not use
 * this file except in compliance with the License. You may obtain a copy of the
 * License at
 *
 *    http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software distributed
 * under the License is distributed on an "AS IS" BASIS, WITHOUT WARRANTIES OR
 * CONDITIONS OF ANY KIND, either express or implied. See the License for the
 * specific language governing permissions and limitations under the License.
 */

package com.nimbusds.jose.jwk;


import java.io.Serializable;
import java.math.BigInteger;
import java.net.URI;
import java.security.*;
import java.security.cert.CertificateEncodingException;
import java.security.cert.X509Certificate;
import java.security.interfaces.RSAMultiPrimePrivateCrtKey;
import java.security.interfaces.RSAPrivateCrtKey;
import java.security.interfaces.RSAPrivateKey;
import java.security.interfaces.RSAPublicKey;
import java.security.spec.*;
import java.text.ParseException;
import java.util.*;

import net.jcip.annotations.Immutable;
import net.minidev.json.JSONArray;
import net.minidev.json.JSONObject;

import com.nimbusds.jose.Algorithm;
import com.nimbusds.jose.JOSEException;
import com.nimbusds.jose.util.Base64;
import com.nimbusds.jose.util.*;


/**
 * Public and private {@link KeyType#RSA RSA} JSON Web Key (JWK). This class is
 * immutable.
 *
 * <p>Provides RSA JWK import from / export to the following standard Java 
 * interfaces and classes:
 *
 * <ul>
 *     <li>{@link java.security.interfaces.RSAPublicKey}
 *     <li>{@link java.security.interfaces.RSAPrivateKey}
 *         <ul>
 *             <li>{@link java.security.interfaces.RSAPrivateCrtKey}
 *             <li>{@link java.security.interfaces.RSAMultiPrimePrivateCrtKey}
 *         </ul>
 *     <li>{@link java.security.PrivateKey} for an RSA key in a PKCS#11 store
 *     <li>{@link java.security.KeyPair}
 * </ul>
 *
 * <p>Example JSON object representation of a public RSA JWK:
 *
 * <pre>
 * { 
 *   "kty" : "RSA",
 *   "n"   : "0vx7agoebGcQSuuPiLJXZptN9nndrQmbXEps2aiAFbWhM78LhWx
 *            4cbbfAAtVT86zwu1RK7aPFFxuhDR1L6tSoc_BJECPebWKRXjBZCiFV4n3oknjhMs
 *            tn64tZ_2W-5JsGY4Hc5n9yBXArwl93lqt7_RN5w6Cf0h4QyQ5v-65YGjQR0_FDW2
 *            QvzqY368QQMicAtaSqzs8KJZgnYb9c7d0zgdAZHzu6qMQvRL5hajrn1n91CbOpbI
 *            SD08qNLyrdkt-bFTWhAI4vMQFh6WeZu0fM4lFd2NcRwr3XPksINHaQ-G_xBniIqb
 *            w0Ls1jF44-csFCur-kEgU8awapJzKnqDKgw",
 *   "e"   : "AQAB",
 *   "alg" : "RS256",
 *   "kid" : "2011-04-29"
 * }
 * </pre>
 *
 * <p>Example JSON object representation of a public and private RSA JWK (with 
 * both the first and the second private key representations):
 *
 * <pre>
 * { 
 *   "kty" : "RSA",
 *   "n"   : "0vx7agoebGcQSuuPiLJXZptN9nndrQmbXEps2aiAFbWhM78LhWx
 *            4cbbfAAtVT86zwu1RK7aPFFxuhDR1L6tSoc_BJECPebWKRXjBZCiFV4n3oknjhMs
 *            tn64tZ_2W-5JsGY4Hc5n9yBXArwl93lqt7_RN5w6Cf0h4QyQ5v-65YGjQR0_FDW2
 *            QvzqY368QQMicAtaSqzs8KJZgnYb9c7d0zgdAZHzu6qMQvRL5hajrn1n91CbOpbI
 *            SD08qNLyrdkt-bFTWhAI4vMQFh6WeZu0fM4lFd2NcRwr3XPksINHaQ-G_xBniIqb
 *            w0Ls1jF44-csFCur-kEgU8awapJzKnqDKgw",
 *   "e"   : "AQAB",
 *   "d"   : "X4cTteJY_gn4FYPsXB8rdXix5vwsg1FLN5E3EaG6RJoVH-HLLKD9
 *            M7dx5oo7GURknchnrRweUkC7hT5fJLM0WbFAKNLWY2vv7B6NqXSzUvxT0_YSfqij
 *            wp3RTzlBaCxWp4doFk5N2o8Gy_nHNKroADIkJ46pRUohsXywbReAdYaMwFs9tv8d
 *            _cPVY3i07a3t8MN6TNwm0dSawm9v47UiCl3Sk5ZiG7xojPLu4sbg1U2jx4IBTNBz
 *            nbJSzFHK66jT8bgkuqsk0GjskDJk19Z4qwjwbsnn4j2WBii3RL-Us2lGVkY8fkFz
 *            me1z0HbIkfz0Y6mqnOYtqc0X4jfcKoAC8Q",
 *   "p"   : "83i-7IvMGXoMXCskv73TKr8637FiO7Z27zv8oj6pbWUQyLPQBQxtPV
 *            nwD20R-60eTDmD2ujnMt5PoqMrm8RfmNhVWDtjjMmCMjOpSXicFHj7XOuVIYQyqV
 *            WlWEh6dN36GVZYk93N8Bc9vY41xy8B9RzzOGVQzXvNEvn7O0nVbfs",
 *   "q"   : "3dfOR9cuYq-0S-mkFLzgItgMEfFzB2q3hWehMuG0oCuqnb3vobLyum
 *            qjVZQO1dIrdwgTnCdpYzBcOfW5r370AFXjiWft_NGEiovonizhKpo9VVS78TzFgx
 *            kIdrecRezsZ-1kYd_s1qDbxtkDEgfAITAG9LUnADun4vIcb6yelxk",
 *   "dp"  : "G4sPXkc6Ya9y8oJW9_ILj4xuppu0lzi_H7VTkS8xj5SdX3coE0oim
 *            YwxIi2emTAue0UOa5dpgFGyBJ4c8tQ2VF402XRugKDTP8akYhFo5tAA77Qe_Nmtu
 *            YZc3C3m3I24G2GvR5sSDxUyAN2zq8Lfn9EUms6rY3Ob8YeiKkTiBj0",
 *   "dq"  : "s9lAH9fggBsoFR8Oac2R_E2gw282rT2kGOAhvIllETE1efrA6huUU
 *            vMfBcMpn8lqeW6vzznYY5SSQF7pMdC_agI3nG8Ibp1BUb0JUiraRNqUfLhcQb_d9
 *            GF4Dh7e74WbRsobRonujTYN1xCaP6TO61jvWrX-L18txXw494Q_cgk",
 *   "qi"  : "GyM_p6JrXySiz1toFgKbWV-JdI3jQ4ypu9rbMWx3rQJBfmt0FoYzg
 *            UIZEVFEcOqwemRN81zoDAaa-Bk0KWNGDjJHZDdDmFhW3AN7lI-puxk_mHZGJ11rx
 *            yR8O55XLSe3SPmRfKwZI6yU24ZxvQKFYItdldUKGzO6Ia6zTKhAVRU",
 *   "alg" : "RS256",
 *   "kid" : "2011-04-29"
 * }
 * </pre>
 *
 * <p>Use the builder to create a new RSA JWK:
 *
 * <pre>
 * RSAKey key = new RSAKey.Builder(n, e)
 *      .keyUse(KeyUse.SIGNATURE)
 *      .keyID("123")
 *      .build();
 * </pre>
 *
 * <p>See RFC 3447.
 *
 * <p>See http://en.wikipedia.org/wiki/RSA_%28algorithm%29
 *
 * @author Vladimir Dzhuvinov
 * @author Justin Richer
 * @author Cedric Staub
 * @version 2019-04-15
 */
@Immutable
public final class RSAKey extends JWK implements AsymmetricJWK {


        private static final long serialVersionUID = 1L;


        /**
         * Other Primes Info, represents the private {@code oth} parameter of a
         * RSA JWK. This class is immutable.
         */
        @Immutable
        public static class OtherPrimesInfo implements Serializable {


                private static final long serialVersionUID = 1L;


                 /**
                  * The prime factor.
                  */
                private final Base64URL r;

                
                /**
                 * The factor Chinese Remainder Theorem (CRT) exponent.
                 */
                private final Base64URL d;
        

                /**
                 * The factor Chinese Remainder Theorem (CRT) coefficient.
                 */
                private final Base64URL t;


                /**
                 * Creates a new JWK Other Primes Info with the specified 
                 * parameters.
                 *
                 * @param r The prime factor. Must not be {@code null}.
                 * @param d The factor Chinese Remainder Theorem (CRT) 
                 *          exponent. Must not be {@code null}.
                 * @param t The factor Chinese Remainder Theorem (CRT) 
                 *          coefficient. Must not be {@code null}.
                 */
                public OtherPrimesInfo(final Base64URL r, final Base64URL d, final Base64URL t) {

                        if (r == null) {

                                throw new IllegalArgumentException("The prime factor must not be null");
                        }

                        this.r = r;

                        if (d == null) {

                                throw new IllegalArgumentException("The factor CRT exponent must not be null");
                        }

                        this.d = d;

                        if (t == null) {

                                throw new IllegalArgumentException("The factor CRT coefficient must not be null");
                        }
                        
                        this.t = t;
                }


                /**
                 * Creates a new JWK Other Primes Info from the specified
                 * {@code java.security.spec.RSAOtherPrimeInfo} instance.
                 *
                 * @param oth The RSA Other Primes Info instance. Must not be 
                 *            {@code null}.
                 */
                public OtherPrimesInfo(final RSAOtherPrimeInfo oth) {

                        r = Base64URL.encode(oth.getPrime());
                        d = Base64URL.encode(oth.getExponent());
                        t = Base64URL.encode(oth.getCrtCoefficient());
                }
       
        
                /**
                 * Gets the prime factor ({@code r}).
                 *
                 * @return The prime factor.
                 */
                public Base64URL getPrimeFactor() {

                        return r;
                }


                /**
                 * Gets factor Chinese Remainder Theorem (CRT) exponent
                 * ({@code d}).
                 *
                 * @return The factor Chinese Remainder Theorem (CRT) exponent.
                 */
                public Base64URL getFactorCRTExponent() {

                        return d;
                }


                /**
                 * The factor Chinese Remainder Theorem (CRT) coefficient
                 * ({@code t}).
                 *
                 * @return The factor Chinese Remainder Theorem (CRT) 
                 *         coefficient.
                 */
                public Base64URL getFactorCRTCoefficient() {

                        return t;
                }


                /**
                 * Converts the specified array of 
                 * {@code java.security.spec.RSAOtherPrimeInfo} instances to a
                 * list of JWK Other Prime Infos.
                 *
                 * @param othArray Array of RSA Other Primes Info instances. 
                 *                 May be be {@code null}.
                 *
                 * @return The corresponding list of JWK Other Prime Infos, or
                 *         empty list of the array was {@code null}.
                 */
                public static List<OtherPrimesInfo> toList(final RSAOtherPrimeInfo[] othArray) {

                        List<OtherPrimesInfo> list = new ArrayList<>();

                        if (othArray == null) {

                                // Return empty list
                                return list;
                        }

                        for (RSAOtherPrimeInfo oth: othArray) {

                                list.add(new OtherPrimesInfo(oth));
                        }

                        return list;
                }
        }


        /**
         * Builder for constructing RSA JWKs.
         *
         * <p>Example usage:
         *
         * <pre>
         * RSAKey key = new RSAKey.Builder(n, e).
         *              privateExponent(d).
         *              algorithm(JWSAlgorithm.RS512).
         *              keyID("456").
         *              build();
         * </pre>
         */
        public static class Builder {


                // Public RSA params

                /**
                 * The modulus value for the RSA key.
                 */
                private final Base64URL n;


                /**
                 * The public exponent of the RSA key.
                 */
                private final Base64URL e;


                // Private RSA params, 1st representation       

                /**
                 * The private exponent of the RSA key.
                 */
                private Base64URL d;

                
                // Private RSA params, 2nd representation

                /**
                 * The first prime factor of the private RSA key.
                 */
                private Base64URL p;

                
                /**
                 * The second prime factor of the private RSA key.
                 */
                private Base64URL q;

                
                /**
                 * The first factor Chinese Remainder Theorem exponent of the 
                 * private RSA key.
                 */
                private Base64URL dp;

                
                /**
                 * The second factor Chinese Remainder Theorem exponent of the
                 * private RSA key.
                 */
                private Base64URL dq;

                
                /**
                 * The first Chinese Remainder Theorem coefficient of the private RSA
                 * key.
                 */
                private Base64URL qi;

                
                /**
                 * The other primes information of the private RSA key, should
                 * they exist. When only two primes have been used (the normal
                 * case), this parameter MUST be omitted. When three or more 
                 * primes have been used, the number of array elements MUST be 
                 * the number of primes used minus two.
                 */
                private List<OtherPrimesInfo> oth;
                
                
                // Private RSA key, as PKCS#11 handle
                
                /**
                 * The private RSA key, as PKCS#11 handle.
                 */
                private PrivateKey priv;


                /**
                 * The key use, optional.
                 */
                private KeyUse use;


                /**
                 * The key operations, optional.
                 */
                private Set<KeyOperation> ops;


                /**
                 * The intended JOSE algorithm for the key, optional.
                 */
                private Algorithm alg;


                /**
                 * The key ID, optional.
                 */
                private String kid;


                /**
                 * X.509 certificate URL, optional.
                 */
                private URI x5u;


                /**
                 * X.509 certificate SHA-1 thumbprint, optional.
                 */
                @Deprecated
                private Base64URL x5t;


                /**
                 * X.509 certificate SHA-256 thumbprint, optional.
                 */
                private Base64URL x5t256;


                /**
                 * The X.509 certificate chain, optional.
                 */
                private List<Base64> x5c;
                
                
                /**
                 * Reference to the underlying key store, {@code null} if none.
                 */
                private KeyStore ks;


                /**
                 * Creates a new RSA JWK builder.
                 *
                 * @param n The the modulus value for the public RSA key. It is 
                 *          represented as the Base64URL encoding of value's 
                 *          big endian representation. Must not be 
                 *          {@code null}.
                 * @param e The exponent value for the public RSA key. It is 
                 *          represented as the Base64URL encoding of value's 
                 *          big endian representation. Must not be 
                 *          {@code null}.
                 */
                public Builder(final Base64URL n, final Base64URL e) {

                        // Ensure the public params are defined

                        if (n == null) {
                                throw new IllegalArgumentException("The modulus value must not be null");
                        }

                        this.n = n;


                        if (e == null) {
                                throw new IllegalArgumentException("The public exponent value must not be null");
                        }

                        this.e = e;
                }


                /**
                 * Creates a new RSA JWK builder.
                 * 
                 * @param pub The public RSA key to represent. Must not be 
                 *            {@code null}.
                 */
                public Builder(final RSAPublicKey pub) {

                        n = Base64URL.encode(pub.getModulus());
                        e = Base64URL.encode(pub.getPublicExponent());
                }
                
                
                /**
                 * Creates a new RSA JWK builder.
                 *
                 * @param rsaJWK The RSA JWK to start with. Must not be
                 *               {@code null}.
                 */
                public Builder(final RSAKey rsaJWK) {
                        
                        n = rsaJWK.n;
                        e = rsaJWK.e;
                        d = rsaJWK.d;
                        p = rsaJWK.p;
                        q = rsaJWK.q;
                        dp = rsaJWK.dp;
                        dq = rsaJWK.dq;
                        qi = rsaJWK.qi;
                        oth = rsaJWK.oth;
                        priv = rsaJWK.privateKey;
                        use = rsaJWK.getKeyUse();
                        ops = rsaJWK.getKeyOperations();
                        alg = rsaJWK.getAlgorithm();
                        kid = rsaJWK.getKeyID();
                        x5u = rsaJWK.getX509CertURL();
                        x5t = rsaJWK.getX509CertThumbprint();
                        x5t256 = rsaJWK.getX509CertSHA256Thumbprint();
                        x5c = rsaJWK.getX509CertChain();
                        ks = rsaJWK.getKeyStore();
                }


                /**
                 * Sets the private exponent ({@code d}) of the RSA key.
                 *
                 * @param d The private RSA key exponent. It is represented as 
                 *          the Base64URL encoding of the value's big endian 
                 *          representation. {@code null} if not specified (for 
                 *          a public key or a private key using the second 
                 *          representation only).
                 *
                 * @return This builder.
                 */
                public Builder privateExponent(final Base64URL d) {

                        this.d = d;
                        return this;
                }


                /**
                 * Sets the private RSA key, using the first representation.
                 * 
                 * @param priv The private RSA key, used to obtain the private
                 *             exponent ({@code d}). Must not be {@code null}.
                 *
                 * @return This builder.
                 */
                public Builder privateKey(final RSAPrivateKey priv) {

                        if (priv instanceof RSAPrivateCrtKey) {
                                return this.privateKey((RSAPrivateCrtKey) priv);
                        } else if (priv instanceof RSAMultiPrimePrivateCrtKey) {
                                return this.privateKey((RSAMultiPrimePrivateCrtKey) priv);
                        } else {
                                this.d = Base64URL.encode(priv.getPrivateExponent());
                                return this;
                        }
                }
                
                
                /**
                 * Sets the private RSA key, typically for a key located in a
                 * PKCS#11 store that doesn't expose the private key parameters
                 * (such as a smart card or HSM).
                 *
                 * @param priv The private RSA key reference. Its algorithm
                 *             must be "RSA". Must not be {@code null}.
                 *
                 * @return This builder.
                 */
                public Builder privateKey(final PrivateKey priv) {
                        if (priv instanceof RSAPrivateKey) {
                                return privateKey((RSAPrivateKey) priv);
                        }

                        if (! "RSA".equalsIgnoreCase(priv.getAlgorithm())) {
                                throw new IllegalArgumentException("The private key algorithm must be RSA");
                        }
                        
                        this.priv = priv;
                        return this;
                }


                /**
                 * Sets the first prime factor ({@code p}) of the private RSA
                 * key. 
                 *
                 * @param p The RSA first prime factor. It is represented as 
                 *          the Base64URL encoding of the value's big endian 
                 *          representation. {@code null} if not specified (for 
                 *          a public key or a private key using the first 
                 *          representation only).
                 *
                 * @return This builder.
                 */
                public Builder firstPrimeFactor(final Base64URL p) {

                        this.p = p;
                        return this;
                }


                /**
                 * Sets the second prime factor ({@code q}) of the private RSA 
                 * key.
                 *
                 * @param q The RSA second prime factor. It is represented as 
                 *          the Base64URL encoding of the value's big endian 
                 *          representation. {@code null} if not specified (for 
                 *          a public key or a private key using the first 
                 *          representation only).
                 *
                 * @return This builder.
                 */
                public Builder secondPrimeFactor(final Base64URL q) {

                        this.q = q;
                        return this;
                }


                /**
                 * Sets the first factor Chinese Remainder Theorem (CRT) 
                 * exponent ({@code dp}) of the private RSA key.
                 *
                 * @param dp The RSA first factor CRT exponent. It is 
                 *           represented as the Base64URL encoding of the 
                 *           value's big endian representation. {@code null} 
                 *           if not specified (for a public key or a private
                 *           key using the first representation only).
                 *
                 * @return This builder.
                 */
                public Builder firstFactorCRTExponent(final Base64URL dp) {

                        this.dp = dp;
                        return this;
                }


                /**
                 * Sets the second factor Chinese Remainder Theorem (CRT) 
                 * exponent ({@code dq}) of the private RSA key.
                 *
                 * @param dq The RSA second factor CRT exponent. It is 
                 *           represented as the Base64URL encoding of the 
                 *           value's big endian representation. {@code null} if 
                 *           not specified (for a public key or a private key 
                 *           using the first representation only).
                 *
                 * @return This builder.
                 */
                public Builder secondFactorCRTExponent(final Base64URL dq) {

                        this.dq = dq;
                        return this;
                }


                /**
                 * Sets the first Chinese Remainder Theorem (CRT) coefficient
                 * ({@code qi}) of the private RSA key.
                 *
                 * @param qi The RSA first CRT coefficient. It is represented 
                 *           as the Base64URL encoding of the value's big 
                 *           endian representation. {@code null} if not 
                 *           specified (for a public key or a private key using 
                 *           the first representation only).
                 *
                 * @return This builder.
                 */
                public Builder firstCRTCoefficient(final Base64URL qi) {

                        this.qi = qi;
                        return this;
                }


                /**
                 * Sets the other primes information ({@code oth}) for the 
                 * private RSA key, should they exist.
                 *
                 * @param oth The RSA other primes information, {@code null} or 
                 *            empty list if not specified.
                 *
                 * @return This builder.
                 */
                public Builder otherPrimes(final List<OtherPrimesInfo> oth) {

                        this.oth = oth;
                        return this;
                }


                /**
                 * Sets the private RSA key, using the second representation 
                 * (see RFC 3447, section 3.2).
                 * 
                 * @param priv The private RSA key, used to obtain the private
                 *             exponent ({@code d}), the first prime factor
                 *             ({@code p}), the second prime factor 
                 *             ({@code q}), the first factor CRT exponent 
                 *             ({@code dp}), the second factor CRT exponent
                 *             ({@code dq}) and the first CRT coefficient 
                 *             ({@code qi}). Must not be {@code null}.
                 *
                 * @return This builder.
                 */
                public Builder privateKey(final RSAPrivateCrtKey priv) {

                        d = Base64URL.encode(priv.getPrivateExponent());
                        p = Base64URL.encode(priv.getPrimeP());
                        q = Base64URL.encode(priv.getPrimeQ());
                        dp = Base64URL.encode(priv.getPrimeExponentP());
                        dq = Base64URL.encode(priv.getPrimeExponentQ());
                        qi = Base64URL.encode(priv.getCrtCoefficient());

                        return this;
                }


                /**
                 * Sets the private RSA key, using the second representation, 
                 * with optional other primes info (see RFC 3447, section 3.2).
                 * 
                 * @param priv The private RSA key, used to obtain the private
                 *             exponent ({@code d}), the first prime factor
                 *             ({@code p}), the second prime factor 
                 *             ({@code q}), the first factor CRT exponent 
                 *             ({@code dp}), the second factor CRT exponent
                 *             ({@code dq}), the first CRT coefficient 
                 *             ({@code qi}) and the other primes info
                 *             ({@code oth}). Must not be {@code null}.
                 *
                 * @return This builder.
                 */
                public Builder privateKey(final RSAMultiPrimePrivateCrtKey priv) {
                        
                        d = Base64URL.encode(priv.getPrivateExponent());
                        p = Base64URL.encode(priv.getPrimeP());
                        q = Base64URL.encode(priv.getPrimeQ());
                        dp = Base64URL.encode(priv.getPrimeExponentP());
                        dq = Base64URL.encode(priv.getPrimeExponentQ());
                        qi = Base64URL.encode(priv.getCrtCoefficient());
                        oth = OtherPrimesInfo.toList(priv.getOtherPrimeInfo());

                        return this;
                }


                /**
                 * Sets the use ({@code use}) of the JWK.
                 *
                 * @param use The key use, {@code null} if not specified or if
                 *            the key is intended for signing as well as
                 *            encryption.
                 *
                 * @return This builder.
                 */
                public Builder keyUse(final KeyUse use) {

                        this.use = use;
                        return this;
                }


                /**
                 * Sets the operations ({@code key_ops}) of the JWK (for a
                 * non-public key).
                 *
                 * @param ops The key operations, {@code null} if not
                 *            specified.
                 *
                 * @return This builder.
                 */
                public Builder keyOperations(final Set<KeyOperation> ops) {

                        this.ops = ops;
                        return this;
                }


                /**
                 * Sets the intended JOSE algorithm ({@code alg}) for the JWK.
                 *
                 * @param alg The intended JOSE algorithm, {@code null} if not
                 *            specified.
                 *
                 * @return This builder.
                 */
                public Builder algorithm(final Algorithm alg) {

                        this.alg = alg;
                        return this;
                }

                /**
                 * Sets the ID ({@code kid}) of the JWK. The key ID can be used
                 * to match a specific key. This can be used, for instance, to
                 * choose a key within a {@link JWKSet} during key rollover.
                 * The key ID may also correspond to a JWS/JWE {@code kid}
                 * header parameter value.
                 *
                 * @param kid The key ID, {@code null} if not specified.
                 *
                 * @return This builder.
                 */
                public Builder keyID(final String kid) {

                        this.kid = kid;
                        return this;
                }


                /**
                 * Sets the ID ({@code kid}) of the JWK to its SHA-256 JWK
                 * thumbprint (RFC 7638). The key ID can be used to match a
                 * specific key. This can be used, for instance, to choose a
                 * key within a {@link JWKSet} during key rollover. The key ID
                 * may also correspond to a JWS/JWE {@code kid} header
                 * parameter value.
                 *
                 * @return This builder.
                 *
                 * @throws JOSEException If the SHA-256 hash algorithm is not
                 *                       supported.
                 */
                public Builder keyIDFromThumbprint()
                        throws JOSEException {

                        return keyIDFromThumbprint("SHA-256");
                }


                /**
                 * Sets the ID ({@code kid}) of the JWK to its JWK thumbprint
                 * (RFC 7638). The key ID can be used to match a specific key.
                 * This can be used, for instance, to choose a key within a
                 * {@link JWKSet} during key rollover. The key ID may also
                 * correspond to a JWS/JWE {@code kid} header parameter value.
                 *
                 * @param hashAlg The hash algorithm for the JWK thumbprint
                 *                computation. Must not be {@code null}.
                 *
                 * @return This builder.
                 *
                 * @throws JOSEException If the hash algorithm is not
                 *                       supported.
                 */
                public Builder keyIDFromThumbprint(final String hashAlg)
                        throws JOSEException {

                        // Put mandatory params in sorted order
                        LinkedHashMap<String,String> requiredParams = new LinkedHashMap<>();
                        requiredParams.put("e", e.toString());
                        requiredParams.put("kty", KeyType.RSA.getValue());
                        requiredParams.put("n", n.toString());
                        this.kid = ThumbprintUtils.compute(hashAlg, requiredParams).toString();
                        return this;
                }


                /**
                 * Sets the X.509 certificate URL ({@code x5u}) of the JWK.
                 *
                 * @param x5u The X.509 certificate URL, {@code null} if not
                 *            specified.
                 *
                 * @return This builder.
                 */
                public Builder x509CertURL(final URI x5u) {

                        this.x5u = x5u;
                        return this;
                }


                /**
                 * Sets the X.509 certificate SHA-1 thumbprint ({@code x5t}) of
                 * the JWK.
                 *
                 * @param x5t The X.509 certificate SHA-1 thumbprint,
                 *            {@code null} if not specified.
                 *
                 * @return This builder.
                 */
                @Deprecated
                public Builder x509CertThumbprint(final Base64URL x5t) {

                        this.x5t = x5t;
                        return this;
                }
                
                
                /**
                 * Sets the X.509 certificate SHA-256 thumbprint
                 * ({@code x5t#S256}) of the JWK.
                 *
                 * @param x5t256 The X.509 certificate SHA-256 thumbprint,
                 *               {@code null} if not specified.
                 *
                 * @return This builder.
                 */
                public Builder x509CertSHA256Thumbprint(final Base64URL x5t256) {
                        
                        this.x5t256 = x5t256;
                        return this;
                }
                

                /**
                 * Sets the X.509 certificate chain ({@code x5c}) of the JWK.
                 *
                 * @param x5c The X.509 certificate chain as a unmodifiable
                 *            list, {@code null} if not specified.
                 *
                 * @return This builder.
                 */
                public Builder x509CertChain(final List<Base64> x5c) {

                        this.x5c = x5c;
                        return this;
                }
                
                
                /**
                 * Sets the underlying key store.
                 *
                 * @param keyStore Reference to the underlying key store,
                 *                 {@code null} if none.
                 *
                 * @return This builder.
                 */
                public Builder keyStore(final KeyStore keyStore) {
                        
                        this.ks = keyStore;
                        return this;
                }
                

                /**
                 * Builds a new RSA JWK.
                 *
                 * @return The RSA JWK.
                 *
                 * @throws IllegalStateException If the JWK parameters were
                 *                               inconsistently specified.
                 */
                public RSAKey build() {

                        try {
                                // The full constructor
                                return new RSAKey(n, e, d, p, q, dp, dq, qi, oth,
                                                  priv,
                                                  use, ops, alg, kid, x5u, x5t, x5t256, x5c,
                                                  ks);

                        } catch (IllegalArgumentException e) {

                                throw new IllegalStateException(e.getMessage(), e);
                        }
                }
        }


        // Public RSA params

        /**
         * The modulus value of the RSA key.
         */
        private final Base64URL n;


        /**
         * The public exponent of the RSA key.
         */
        private final Base64URL e;


        // Private RSA params, 1st representation       

        /**
         * The private exponent of the RSA key.
         */
        private final Base64URL d;

        
        // Private RSA params, 2nd representation

        /**
         * The first prime factor of the private RSA key.
         */
        private final Base64URL p;

        
        /**
         * The second prime factor of the private RSA key.
         */
        private final Base64URL q;

        
        /**
         * The first factor Chinese Remainder Theorem exponent of the private 
         * RSA key.
         */
        private final Base64URL dp;

        
        /**
         * The second factor Chinese Remainder Theorem exponent of the private
         * RSA key.
         */
        private final Base64URL dq;

        
        /**
         * The first Chinese Remainder Theorem coefficient of the private RSA
         * key.
         */
        private final Base64URL qi;

        
        /**
         * The other primes information of the private RSA key, should they
         * exist. When only two primes have been used (the normal case), this 
         * parameter MUST be omitted. When three or more primes have been used,
         * the number of array elements MUST be the number of primes used minus
         * two.
         */
        private final List<OtherPrimesInfo> oth;
        
        
        // Private RSA PKCS#11 key handle
        
        /**
         * Private PKCS#11 key handle.
         */
        private final PrivateKey privateKey;


        /**
         * Creates a new public RSA JSON Web Key (JWK) with the specified 
         * parameters.
         *
         * @param n      The the modulus value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param e      The exponent value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final Base64URL n, final Base64URL e,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {

                // Call the full constructor, all private key parameters are null
                this(n, e, null, null, null, null, null, null, null, null, use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c,
                     ks);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by its first
         * representation (see RFC 3447, section 3.2).
         * 
         * @param n      The the modulus value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param e      The exponent value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param d      The private exponent. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. Must not be {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final Base64URL n, final Base64URL e, final Base64URL d,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
            
                // Call the full constructor, the second private representation 
                // parameters are all null
                this(n, e, d, null, null, null, null, null, null, null, use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c, ks);

                if (d == null) {
                        throw new IllegalArgumentException("The private exponent must not be null");
                }
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by its
         * second representation (see RFC 3447, section 3.2).
         * 
         * @param n      The the modulus value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param e      The exponent value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param p      The first prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. Must not be {@code null}.
         * @param q      The second prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. Must not be {@code null}.
         * @param dp     The first factor Chinese Remainder Theorem exponent.
         *               It is represented as the Base64URL encoding of the
         *               value's big endian representation. Must not be
         *               {@code null}.
         * @param dq     The second factor Chinese Remainder Theorem exponent.
         *               It is represented as the Base64URL encoding of the
         *               value's big endian representation. Must not be
         *               {@code null}.
         * @param qi     The first Chinese Remainder Theorem coefficient. It is
         *               represented as the Base64URL encoding of the value's
         *               big endian representation. Must not be {@code null}.
         * @param oth    The other primes information, should they exist,
         *               {@code null} or an empty list if not specified.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final Base64URL n, final Base64URL e, 
                      final Base64URL p, final Base64URL q, 
                      final Base64URL dp, final Base64URL dq, final Base64URL qi, 
                      final List<OtherPrimesInfo> oth,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
            
                // Call the full constructor, the first private representation 
                // d param is null
                this(n, e, null, p, q, dp, dq, qi, oth, null, use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c,
                     ks);

                if (p == null) {
                        throw new IllegalArgumentException("The first prime factor must not be null");
                }

                if (q == null) {
                        throw new IllegalArgumentException("The second prime factor must not be null");
                }

                if (dp == null) {
                        throw new IllegalArgumentException("The first factor CRT exponent must not be null");
                }

                if (dq == null) {
                        throw new IllegalArgumentException("The second factor CRT exponent must not be null");
                }

                if (qi == null) {
                        throw new IllegalArgumentException("The first CRT coefficient must not be null");
                }
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by both its
         * first and second representations (see RFC 3447, section 3.2).
         *
         * <p>A valid first private RSA key representation must specify the
         * {@code d} parameter.
         *
         * <p>A valid second private RSA key representation must specify all 
         * required Chinese Remainder Theorem (CRT) parameters - {@code p},
         * {@code q}, {@code dp}, {@code dq} and {@code qi}, else an
         * {@link java.lang.IllegalArgumentException} will be thrown.
         * 
         * @param n      The the modulus value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param e      The exponent value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param d      The private exponent. It is represented as the Base64URL
         *               encoding of the value's big endian representation. May
         *               be {@code null}.
         * @param p      The first prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. May be {@code null}.
         * @param q      The second prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. May be {@code null}.
         * @param dp     The first factor Chinese Remainder Theorem exponent. It
         *               is represented as the Base64URL encoding of the value's
         *               big endian representation. May be {@code null}.
         * @param dq     The second factor Chinese Remainder Theorem exponent. It
         *               is represented as the Base64URL encoding of the value's
         *               big endian representation. May be {@code null}.
         * @param qi     The first Chinese Remainder Theorem coefficient. It is
         *               represented as the Base64URL encoding of the value's big
         *               endian representation. May be {@code null}.
         * @param oth    The other primes information, should they exist,
         *               {@code null} or an empty list if not specified.
         * @param use    The key use, {@code null} if not specified or if the key
         *               is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null} if
         *               not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         */
        @Deprecated
        public RSAKey(final Base64URL n, final Base64URL e,
                      final Base64URL d, 
                      final Base64URL p, final Base64URL q, 
                      final Base64URL dp, final Base64URL dq, final Base64URL qi, 
                      final List<OtherPrimesInfo> oth,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c) {
            
                this(n, e, d, p, q, dp, dq, qi, oth, null, use, ops, alg, kid, x5u, x5t, x5t256, x5c, null);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the
         * specified parameters. The private RSA key can be specified by its
         * first representation, its second representation (see RFC 3447,
         * section 3.2), or by a PKCS#11 handle as {@link PrivateKey}.
         *
         * <p>A valid first private RSA key representation must specify the
         * {@code d} parameter.
         *
         * <p>A valid second private RSA key representation must specify all
         * required Chinese Remainder Theorem (CRT) parameters - {@code p},
         * {@code q}, {@code dp}, {@code dq} and {@code qi}, else an
         * {@link java.lang.IllegalArgumentException} will be thrown.
         *
         * @param n      The the modulus value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param e      The exponent value for the public RSA key. It is
         *               represented as the Base64URL encoding of value's big
         *               endian representation. Must not be {@code null}.
         * @param d      The private exponent. It is represented as the Base64URL
         *               encoding of the value's big endian representation. May
         *               be {@code null}.
         * @param p      The first prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. May be {@code null}.
         * @param q      The second prime factor. It is represented as the
         *               Base64URL encoding of the value's big endian
         *               representation. May be {@code null}.
         * @param dp     The first factor Chinese Remainder Theorem exponent. It
         *               is represented as the Base64URL encoding of the value's
         *               big endian representation. May be {@code null}.
         * @param dq     The second factor Chinese Remainder Theorem exponent. It
         *               is represented as the Base64URL encoding of the value's
         *               big endian representation. May be {@code null}.
         * @param qi     The first Chinese Remainder Theorem coefficient. It is
         *               represented as the Base64URL encoding of the value's big
         *               endian representation. May be {@code null}.
         * @param oth    The other primes information, should they exist,
         *               {@code null} or an empty list if not specified.
         * @param use    The key use, {@code null} if not specified or if the key
         *               is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null} if
         *               not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final Base64URL n, final Base64URL e,
                      final Base64URL d,
                      final Base64URL p, final Base64URL q,
                      final Base64URL dp, final Base64URL dq, final Base64URL qi,
                      final List<OtherPrimesInfo> oth,
                      final PrivateKey prv,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
            
                super(KeyType.RSA, use, ops, alg, kid, x5u, x5t, x5t256, x5c, ks);


                // Ensure the public params are defined

                if (n == null) {
                        throw new IllegalArgumentException("The modulus value must not be null");
                }
                this.n = n;


                if (e == null) {
                        throw new IllegalArgumentException("The public exponent value must not be null");
                }
                this.e = e;

                if (getParsedX509CertChain() != null) {
                        if (! matches(getParsedX509CertChain().get(0)))
                                throw new IllegalArgumentException("The public subject key info of the first X.509 certificate in the chain must match the JWK type and public parameters");
                }

                // Private params, 1st representation

                this.d = d;


                // Private params, 2nd representation, check for consistency

                if (p != null && q != null && dp != null && dq != null && qi != null) {

                        // CRT params fully specified
                        this.p = p;
                        this.q = q;
                        this.dp = dp;
                        this.dq = dq;
                        this.qi = qi;

                        // Other RSA primes info optional, default to empty list
                        if (oth != null) {
                                this.oth = Collections.unmodifiableList(oth);
                        } else {
                                this.oth = Collections.emptyList();
                        }

                } else if (p == null && q == null && dp == null && dq == null && qi == null && oth == null) {

                        // No CRT params
                        this.p = null;
                        this.q = null;
                        this.dp = null;
                        this.dq = null;
                        this.qi = null;

                        this.oth = Collections.emptyList();

                } else if (p != null || q != null || dp != null || dq != null || qi != null) {

                        if (p == null) {
                                throw new IllegalArgumentException("Incomplete second private (CRT) representation: The first prime factor must not be null");
                        } else if (q == null) {
                                throw new IllegalArgumentException("Incomplete second private (CRT) representation: The second prime factor must not be null");
                        } else if (dp == null) {
                                throw new IllegalArgumentException("Incomplete second private (CRT) representation: The first factor CRT exponent must not be null");
                        } else if (dq == null) {
                                throw new IllegalArgumentException("Incomplete second private (CRT) representation: The second factor CRT exponent must not be null");
                        } else {
                                throw new IllegalArgumentException("Incomplete second private (CRT) representation: The first CRT coefficient must not be null");
                        }
                } else {
                        // No CRT params
                        this.p = null;
                        this.q = null;
                        this.dp = null;
                        this.dq = null;
                        this.qi = null;
                        this.oth = Collections.emptyList();
                }
                
                this.privateKey = prv; // PKCS#11 handle
        }


        /**
         * Creates a new public RSA JSON Web Key (JWK) with the specified
         * parameters.
         * 
         * @param pub    The public RSA key to represent. Must not be
         *               {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final RSAPublicKey pub,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {

                this(Base64URL.encode(pub.getModulus()),
                        Base64URL.encode(pub.getPublicExponent()),
                        use, ops, alg, kid,
                        x5u, x5t, x5t256, x5c,
                        ks);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by its first
         * representation (see RFC 3447, section 3.2).
         * 
         * @param pub    The public RSA key to represent. Must not be
         *               {@code null}.
         * @param priv   The private RSA key to represent. Must not be
         *               {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final RSAPublicKey pub, final RSAPrivateKey priv,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
                
                this(Base64URL.encode(pub.getModulus()), 
                     Base64URL.encode(pub.getPublicExponent()), 
                     Base64URL.encode(priv.getPrivateExponent()),
                     use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c,
                     ks);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by its second
         * representation (see RFC 3447, section 3.2).
         * 
         * @param pub    The public RSA key to represent. Must not be
         *               {@code null}.
         * @param priv   The private RSA key to represent. Must not be
         *               {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final RSAPublicKey pub, final RSAPrivateCrtKey priv,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
                
                this(Base64URL.encode(pub.getModulus()), 
                     Base64URL.encode(pub.getPublicExponent()), 
                     Base64URL.encode(priv.getPrivateExponent()),
                     Base64URL.encode(priv.getPrimeP()),
                     Base64URL.encode(priv.getPrimeQ()),
                     Base64URL.encode(priv.getPrimeExponentP()),
                     Base64URL.encode(priv.getPrimeExponentQ()),
                     Base64URL.encode(priv.getCrtCoefficient()),
                     null,
                     null,
                     use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c,
                     ks);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the 
         * specified parameters. The private RSA key is specified by its second
         * representation, with optional other primes info (see RFC 3447, 
         * section 3.2).
         * 
         * @param pub    The public RSA key to represent. Must not be
         *               {@code null}.
         * @param priv   The private RSA key to represent. Must not be
         *               {@code null}.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final RSAPublicKey pub, final RSAMultiPrimePrivateCrtKey priv,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
                
                this(Base64URL.encode(pub.getModulus()), 
                     Base64URL.encode(pub.getPublicExponent()), 
                     Base64URL.encode(priv.getPrivateExponent()),
                     Base64URL.encode(priv.getPrimeP()),
                     Base64URL.encode(priv.getPrimeQ()),
                     Base64URL.encode(priv.getPrimeExponentP()),
                     Base64URL.encode(priv.getPrimeExponentQ()),
                     Base64URL.encode(priv.getCrtCoefficient()),
                     OtherPrimesInfo.toList(priv.getOtherPrimeInfo()),
                     null,
                     use, ops, alg, kid,
                     x5u, x5t, x5t256, x5c,
                     ks);
        }


        /**
         * Creates a new public / private RSA JSON Web Key (JWK) with the
         * specified parameters. The private RSA key is specified by a PKCS#11
         * handle.
         *
         * @param pub    The public RSA key to represent. Must not be
         *               {@code null}.
         * @param priv   The private RSA key as PKCS#11 handle, {@code null} if
         *               not specified.
         * @param use    The key use, {@code null} if not specified or if the
         *               key is intended for signing as well as encryption.
         * @param ops    The key operations, {@code null} if not specified.
         * @param alg    The intended JOSE algorithm for the key, {@code null}
         *               if not specified.
         * @param kid    The key ID. {@code null} if not specified.
         * @param x5u    The X.509 certificate URL, {@code null} if not
         *               specified.
         * @param x5t    The X.509 certificate SHA-1 thumbprint, {@code null}
         *               if not specified.
         * @param x5t256 The X.509 certificate SHA-256 thumbprint, {@code null}
         *               if not specified.
         * @param x5c    The X.509 certificate chain, {@code null} if not
         *               specified.
         * @param ks     Reference to the underlying key store, {@code null} if
         *               not specified.
         */
        public RSAKey(final RSAPublicKey pub, final PrivateKey priv,
                      final KeyUse use, final Set<KeyOperation> ops, final Algorithm alg, final String kid,
                      final URI x5u, final Base64URL x5t, final Base64URL x5t256, final List<Base64> x5c,
                      final KeyStore ks) {
                
                this(Base64URL.encode(pub.getModulus()),
                        Base64URL.encode(pub.getPublicExponent()),
                        null,
                        null,
                        null,
                        null,
                        null,
                        null,
                        null,
                        priv,
                        use, ops, alg, kid,
                        x5u, x5t, x5t256, x5c,
                        ks);
        }


        /**
         * Gets the modulus value ({@code n}) of the RSA key.
         *
         * @return The RSA key modulus. It is represented as the Base64URL 
         *         encoding of the value's big endian representation.
         */
        public Base64URL getModulus() {

                return n;
        }


        /**
         * Gets the public exponent ({@code e}) of the RSA key.
         *
         * @return The public RSA key exponent. It is represented as the 
         *         Base64URL encoding of the value's big endian representation.
         */
        public Base64URL getPublicExponent() {

                return e;
        }


        /**
         * Gets the private exponent ({@code d}) of the RSA key.
         *
         * @return The private RSA key exponent. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the second representation only).
         */
        public Base64URL getPrivateExponent() {

                return d;
        }


        /**
         * Gets the first prime factor ({@code p}) of the private RSA key. 
         *
         * @return The RSA first prime factor. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the first representation only).
         */
        public Base64URL getFirstPrimeFactor() {

                return p;
        }


        /**
         * Gets the second prime factor ({@code q}) of the private RSA key.
         *
         * @return The RSA second prime factor. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the first representation only).
         */
        public Base64URL getSecondPrimeFactor() {

                return q;
        }


        /**
         * Gets the first factor Chinese Remainder Theorem (CRT) exponent
         * ({@code dp}) of the private RSA key.
         *
         * @return The RSA first factor CRT exponent. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the first representation only).
         */
        public Base64URL getFirstFactorCRTExponent() {

                return dp;
        }


        /**
         * Gets the second factor Chinese Remainder Theorem (CRT) exponent 
         * ({@code dq}) of the private RSA key.
         *
         * @return The RSA second factor CRT exponent. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the first representation only).
         */
        public Base64URL getSecondFactorCRTExponent() {

                return dq;
        }


        /**
         * Gets the first Chinese Remainder Theorem (CRT) coefficient
         * ({@code qi})} of the private RSA key.
         *
         * @return The RSA first CRT coefficient. It is represented as the 
         *         Base64URL encoding of the value's big endian representation. 
         *         {@code null} if not specified (for a public key or a private
         *         key using the first representation only).
         */
        public Base64URL getFirstCRTCoefficient() {

                return qi;
        }


        /**
         * Gets the other primes information ({@code oth}) for the private RSA
         * key, should they exist.
         *
         * @return The RSA other primes information, {@code null} or empty list
         *         if not specified.
         */
        public List<OtherPrimesInfo> getOtherPrimes() {

                return oth;
        }

        
        /**
         * Returns a standard {@code java.security.interfaces.RSAPublicKey} 
         * representation of this RSA JWK.
         * 
         * @return The public RSA key.
         * 
         * @throws JOSEException If RSA is not supported by the underlying Java
         *                       Cryptography (JCA) provider or if the JWK
         *                       parameters are invalid for a public RSA key.
         */
        public RSAPublicKey toRSAPublicKey() 
                throws JOSEException {

                BigInteger modulus = n.decodeToBigInteger();
                BigInteger exponent = e.decodeToBigInteger();
                                
                RSAPublicKeySpec spec = new RSAPublicKeySpec(modulus, exponent);

                try {
                        KeyFactory factory = KeyFactory.getInstance("RSA");

                        return (RSAPublicKey) factory.generatePublic(spec);

                } catch (NoSuchAlgorithmException | InvalidKeySpecException e) {

                        throw new JOSEException(e.getMessage(), e);
                }
        }
        

        /**
         * Returns a standard {@code java.security.interfaces.RSAPrivateKey} 
         * representation of this RSA JWK.
         * 
         * @return The private RSA key, {@code null} if not specified by this
         *         JWK.
         * 
         * @throws JOSEException If RSA is not supported by the underlying Java
         *                       Cryptography (JCA) provider or if the JWK
         *                       parameters are invalid for a private RSA key.
         */
        public RSAPrivateKey toRSAPrivateKey() 
                throws JOSEException {
                
                if (d == null) {
                        // no private key
                        return null;
                }
                
                BigInteger modulus = n.decodeToBigInteger();
                BigInteger privateExponent = d.decodeToBigInteger();
                
                RSAPrivateKeySpec spec;

                if (p == null) {
                        // Use 1st representation
                        spec = new RSAPrivateKeySpec(modulus, privateExponent);

                } else {
                        // Use 2nd (CRT) representation
                        BigInteger publicExponent = e.decodeToBigInteger();
                        BigInteger primeP = p.decodeToBigInteger();
                        BigInteger primeQ = q.decodeToBigInteger();
                        BigInteger primeExponentP = dp.decodeToBigInteger();
                        BigInteger primeExponentQ = dq.decodeToBigInteger();
                        BigInteger crtCoefficient = qi.decodeToBigInteger();

                        if (oth != null && ! oth.isEmpty()) {
                                // Construct other info spec
                                RSAOtherPrimeInfo[] otherInfo = new RSAOtherPrimeInfo[oth.size()];

                                for (int i=0; i < oth.size(); i++) {

                                        OtherPrimesInfo opi = oth.get(i);

                                        BigInteger otherPrime = opi.getPrimeFactor().decodeToBigInteger();
                                        BigInteger otherPrimeExponent = opi.getFactorCRTExponent().decodeToBigInteger();
                                        BigInteger otherCrtCoefficient = opi.getFactorCRTCoefficient().decodeToBigInteger();

                                        otherInfo[i] = new RSAOtherPrimeInfo(otherPrime,
                                                                             otherPrimeExponent,
                                                                             otherCrtCoefficient);
                                }

                                spec = new RSAMultiPrimePrivateCrtKeySpec(modulus,
                                                                          publicExponent,
                                                                          privateExponent,
                                                                          primeP,
                                                                          primeQ,
                                                                          primeExponentP,
                                                                          primeExponentQ,
                                                                          crtCoefficient,
                                                                          otherInfo);
                        } else {
                                // Construct spec with no other info
                                spec = new RSAPrivateCrtKeySpec(modulus,
                                                                publicExponent,
                                                                privateExponent,
                                                                primeP,
                                                                primeQ,
                                                                primeExponentP,
                                                                primeExponentQ,
                                                                crtCoefficient);        
                        } 
                }

                try {
                        KeyFactory factory = KeyFactory.getInstance("RSA");

                        return (RSAPrivateKey) factory.generatePrivate(spec);

                } catch (InvalidKeySpecException | NoSuchAlgorithmException e) {

                        throw new JOSEException(e.getMessage(), e);
                }
        }


        @Override
        public PublicKey toPublicKey()
                throws JOSEException {

                return toRSAPublicKey();
        }


        @Override
        public PrivateKey toPrivateKey()
                throws JOSEException {
                
                PrivateKey prv = toRSAPrivateKey();
                
                if (prv != null) {
                        // Return private RSA key with key material
                        return prv;
                }
                
                // Return private RSA key as PKCS#11 handle, or null
                return privateKey;
        }


        /**
         * Returns a standard {@code java.security.KeyPair} representation of 
         * this RSA JWK.
         * 
         * @return The RSA key pair. The private RSA key will be {@code null} 
         *         if not specified.
         * 
         * @throws JOSEException If RSA is not supported by the underlying Java
         *                       Cryptography (JCA) provider or if the JWK
         *                       parameters are invalid for a public and / or
         *                       private RSA key.
         */
        @Override
        public KeyPair toKeyPair() 
                throws JOSEException {
                
                return new KeyPair(toRSAPublicKey(), toPrivateKey());
        }
        
        
        @Override
        public boolean matches(final X509Certificate cert) {
                
                RSAPublicKey certRSAKey;
                try {
                        certRSAKey = (RSAPublicKey) getParsedX509CertChain().get(0).getPublicKey();
                } catch (ClassCastException ex) {
                        return false;
                }
                if (! e.decodeToBigInteger().equals(certRSAKey.getPublicExponent())) {
                        return false;
                }
                if (! n.decodeToBigInteger().equals(certRSAKey.getModulus())) {
                        return false;
                }
                return true;
        }
        
        
        @Override
        public LinkedHashMap<String,?> getRequiredParams() {

                // Put mandatory params in sorted order
                LinkedHashMap<String,String> requiredParams = new LinkedHashMap<>();
                requiredParams.put("e", e.toString());
                requiredParams.put("kty", getKeyType().getValue());
                requiredParams.put("n", n.toString());
                return requiredParams;
        }


        @Override
        public boolean isPrivate() {

                // Check if 1st or 2nd form params are specified, or PKCS#11 handle
                return d != null || p != null || privateKey != null;
        }


        @Override
        public int size() {

                try {
                        return ByteUtils.safeBitLength(n.decode());
                } catch (IntegerOverflowException e) {
                        throw new ArithmeticException(e.getMessage());
                }
        }


        /**
         * Returns a copy of this RSA JWK with any private values removed.
         *
         * @return The copied public RSA JWK.
         */
        @Override
        public RSAKey toPublicJWK() {

                return new RSAKey(
                        getModulus(), getPublicExponent(),
                        getKeyUse(), getKeyOperations(), getAlgorithm(), getKeyID(),
                        getX509CertURL(), getX509CertThumbprint(), getX509CertSHA256Thumbprint(), getX509CertChain(),
                        getKeyStore());
        }
        
        
        @Override
        public JSONObject toJSONObject() {

                JSONObject o = super.toJSONObject();

                // Append public RSA key specific attributes
                o.put("n", n.toString());
                o.put("e", e.toString());
                if (d != null) {
                        o.put("d", d.toString());
                }
                if (p != null) {
                        o.put("p", p.toString());
                }
                if (q != null) {
                        o.put("q", q.toString());
                }
                if (dp != null) {
                        o.put("dp", dp.toString());
                }
                if (dq != null) {
                        o.put("dq", dq.toString());
                }
                if (qi != null) {
                        o.put("qi", qi.toString());
                }
                if (oth != null && !oth.isEmpty()) {

                        JSONArray a = new JSONArray();

                        for (OtherPrimesInfo other : oth) {

                                JSONObject oo = new JSONObject();
                                oo.put("r", other.r.toString());
                                oo.put("d", other.d.toString());
                                oo.put("t", other.t.toString());

                                a.add(oo);
                        }

                        o.put("oth", a);
                }

                return o;
        }


        /**
         * Parses a public / private RSA JWK from the specified JSON object
         * string representation.
         *
         * @param s The JSON object string to parse. Must not be {@code null}.
         *
         * @return The public / private RSA JWK.
         *
         * @throws ParseException If the string couldn't be parsed to an RSA
         *                        JWK.
         */
        public static RSAKey parse(final String s)
                throws ParseException {

                return parse(JSONObjectUtils.parse(s));
        }


        /**
         * Parses a public / private RSA JWK from the specified JSON object 
         * representation.
         *
         * @param jsonObject The JSON object to parse. Must not be 
         *                   {@code null}.
         *
         * @return The public / private RSA Key.
         *
         * @throws ParseException If the JSON object couldn't be parsed to an
         *                        RSA JWK.
         */
        public static RSAKey parse(final JSONObject jsonObject)
                throws ParseException {

                // Parse the mandatory public key parameters first
                Base64URL n = new Base64URL(JSONObjectUtils.getString(jsonObject, "n"));
                Base64URL e = new Base64URL(JSONObjectUtils.getString(jsonObject, "e"));

                // Check key type
                KeyType kty = KeyType.parse(JSONObjectUtils.getString(jsonObject, "kty"));
                if (kty != KeyType.RSA) {
                        throw new ParseException("The key type \"kty\" must be RSA", 0);
                }
                
                // Parse the optional private key parameters

                // 1st private representation
                Base64URL d = null;
                if (jsonObject.containsKey("d")) {
                        d = new Base64URL(JSONObjectUtils.getString(jsonObject, "d"));
                }

                // 2nd private (CRT) representation
                Base64URL p = null;
                if (jsonObject.containsKey("p")) {
                        p = new Base64URL(JSONObjectUtils.getString(jsonObject, "p"));
                }
                Base64URL q = null;
                if (jsonObject.containsKey("q")) {
                        q = new Base64URL(JSONObjectUtils.getString(jsonObject, "q"));
                }
                Base64URL dp = null;
                if (jsonObject.containsKey("dp")) {
                        dp = new Base64URL(JSONObjectUtils.getString(jsonObject, "dp"));
                }
                Base64URL dq= null;
                if (jsonObject.containsKey("dq")) {
                        dq = new Base64URL(JSONObjectUtils.getString(jsonObject, "dq"));
                }
                Base64URL qi = null;
                if (jsonObject.containsKey("qi")) {
                        qi = new Base64URL(JSONObjectUtils.getString(jsonObject, "qi"));
                }
                
                List<OtherPrimesInfo> oth = null;
                if (jsonObject.containsKey("oth")) {

                        JSONArray arr = JSONObjectUtils.getJSONArray(jsonObject, "oth");
                        oth = new ArrayList<>(arr.size());
                        
                        for (Object o : arr) {

                                if (o instanceof JSONObject) {
                                        JSONObject otherJson = (JSONObject)o;

                                        Base64URL r = new Base64URL(JSONObjectUtils.getString(otherJson, "r"));
                                        Base64URL odq = new Base64URL(JSONObjectUtils.getString(otherJson, "dq"));
                                        Base64URL t = new Base64URL(JSONObjectUtils.getString(otherJson, "t"));

                                        OtherPrimesInfo prime = new OtherPrimesInfo(r, odq, t);
                                        oth.add(prime);
                                }
                        }
                }

                try {
                        return new RSAKey(n, e, d, p, q, dp, dq, qi, oth, null,
                                JWKMetadata.parseKeyUse(jsonObject),
                                JWKMetadata.parseKeyOperations(jsonObject),
                                JWKMetadata.parseAlgorithm(jsonObject),
                                JWKMetadata.parseKeyID(jsonObject),
                                JWKMetadata.parseX509CertURL(jsonObject),
                                JWKMetadata.parseX509CertThumbprint(jsonObject),
                                JWKMetadata.parseX509CertSHA256Thumbprint(jsonObject),
                                JWKMetadata.parseX509CertChain(jsonObject),
                                null);
                
                } catch (IllegalArgumentException ex) {
                        // Inconsistent 2nd spec, conflicting 'use' and 'key_ops', etc.
                        throw new ParseException(ex.getMessage(), 0);
                }
        }
        
        
        /**
         * Parses a public RSA JWK from the specified X.509 certificate.
         *
         * <p><strong>Important:</strong> The X.509 certificate is not
         * validated!
         *
         * <p>Sets the following JWK parameters:
         *
         * <ul>
         *     <li>The JWK use inferred by {@link KeyUse#from}.
         *     <li>The JWK ID from the X.509 serial number (in base 10).
         *     <li>The JWK X.509 certificate chain (this certificate only).
         *     <li>The JWK X.509 certificate SHA-256 thumbprint.
         * </ul>
         *
         * @param cert The X.509 certificate. Must not be {@code null}.
         *
         * @return The public RSA key.
         *
         * @throws JOSEException If parsing failed.
         */
        public static RSAKey parse(final X509Certificate cert)
                throws JOSEException {
                
                if (! (cert.getPublicKey() instanceof RSAPublicKey)) {
                        throw new JOSEException("The public key of the X.509 certificate is not RSA");
                }
                
                RSAPublicKey publicKey = (RSAPublicKey)cert.getPublicKey();
                
                try {
                        MessageDigest sha256 = MessageDigest.getInstance("SHA-256");
                        
                        return new RSAKey.Builder(publicKey)
                                .keyUse(KeyUse.from(cert))
                                .keyID(cert.getSerialNumber().toString(10))
                                .x509CertChain(Collections.singletonList(Base64.encode(cert.getEncoded())))
                                .x509CertSHA256Thumbprint(Base64URL.encode(sha256.digest(cert.getEncoded())))
                                .build();
                } catch (NoSuchAlgorithmException e) {
                        throw new JOSEException("Couldn't encode x5t parameter: " + e.getMessage(), e);
                } catch (CertificateEncodingException e) {
                        throw new JOSEException("Couldn't encode x5c parameter: " + e.getMessage(), e);
                }
        }
        
        
        /**
         * Loads a public / private RSA JWK from the specified JCA key store.
         *
         * <p><strong>Important:</strong> The X.509 certificate is not
         * validated!
         *
         * @param keyStore The key store. Must not be {@code null}.
         * @param alias    The alias. Must not be {@code null}.
         * @param pin      The pin to unlock the private key if any, empty or
         *                 {@code null} if not required.
         *
         * @return The public / private RSA key, {@code null} if no key with
         *         the specified alias was found.
         *
         * @throws KeyStoreException On a key store exception.
         * @throws JOSEException     If RSA key loading failed.
         */
        public static RSAKey load(final KeyStore keyStore,
                                  final String alias,
                                  final char[] pin)
                throws KeyStoreException, JOSEException {
                
                java.security.cert.Certificate cert = keyStore.getCertificate(alias);
                
                if (cert == null || ! (cert instanceof X509Certificate)) {
                        return null;
                }
                
                X509Certificate x509Cert = (X509Certificate)cert;
                
                if (! (x509Cert.getPublicKey() instanceof RSAPublicKey)) {
                        throw new JOSEException("Couldn't load RSA JWK: The key algorithm is not RSA");
                }
                
                RSAKey rsaJWK = RSAKey.parse(x509Cert);
                
                // Let kid=alias
                rsaJWK = new RSAKey.Builder(rsaJWK).keyID(alias).keyStore(keyStore).build();
                
                // Check for private counterpart
                Key key;
                try {
                        key = keyStore.getKey(alias, pin);
                } catch (UnrecoverableKeyException | NoSuchAlgorithmException e) {
                        throw new JOSEException("Couldn't retrieve private RSA key (bad pin?): " + e.getMessage(), e);
                }
                
                if (key instanceof RSAPrivateKey) {
                        // Simple file based key store
                        return new RSAKey.Builder(rsaJWK)
                                .privateKey((RSAPrivateKey)key)
                                .build();
                } else if (key instanceof PrivateKey && "RSA".equalsIgnoreCase(key.getAlgorithm())) {
                        // PKCS#11 store
                        return new RSAKey.Builder(rsaJWK)
                                .privateKey((PrivateKey)key)
                                .build();
                } else {
                        return rsaJWK;
                }
        }

        
        @Override
        public boolean equals(Object o) {
                if (this == o) return true;
                if (!(o instanceof RSAKey)) return false;
                if (!super.equals(o)) return false;
                RSAKey rsaKey = (RSAKey) o;
                return Objects.equals(n, rsaKey.n) &&
                                Objects.equals(e, rsaKey.e) &&
                                Objects.equals(d, rsaKey.d) &&
                                Objects.equals(p, rsaKey.p) &&
                                Objects.equals(q, rsaKey.q) &&
                                Objects.equals(dp, rsaKey.dp) &&
                                Objects.equals(dq, rsaKey.dq) &&
                                Objects.equals(qi, rsaKey.qi) &&
                                Objects.equals(oth, rsaKey.oth) &&
                                Objects.equals(privateKey, rsaKey.privateKey);
        }

        
        @Override
        public int hashCode() {
                return Objects.hash(super.hashCode(), n, e, d, p, q, dp, dq, qi, oth, privateKey);
        }
}