How to load the RSA public key from file in C#

Umar Jamil picture Umar Jamil · Jul 16, 2012 · Viewed 76.5k times · Source

I need to load the following RSA public key from a file for use with the RSACryptoServiceProvider class. How can I do this?

-----BEGIN PUBLIC KEY-----
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX/syEKqEkMtQL0+d
XXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXXX+izR
KbGMRtur2TYklnyVkjeeHfAggo8vWQmWesnOG55vQYHbOOFoJbk0EkwEr5R/PbKm
byXPPN8zwnS5/XXXXXXXXXXXX
-----END PUBLIC KEY-----

This code works with my pub key: http://www.jensign.com/opensslkey/

Here is the code I am using

        static string RSA(string input)
        {
            RSACryptoServiceProvider rsa = DecodeX509PublicKey(Convert.FromBase64String(GetKey()));

            return (Convert.ToBase64String(rsa.Encrypt(Encoding.ASCII.GetBytes(input), false)));
        }

        static string GetKey()
        {
            return File.ReadAllText("master.pub").Replace("-----BEGIN PUBLIC KEY-----", "").Replace("-----END PUBLIC KEY-----", "");
            //.Replace("\n", "");
        }

        private static bool CompareBytearrays(byte[] a, byte[] b)
        {
            if (a.Length != b.Length)
                return false;
            int i = 0;
            foreach (byte c in a)
            {
                if (c != b[i])
                    return false;
                i++;
            }
            return true;
        }

        public static RSACryptoServiceProvider DecodeX509PublicKey(byte[] x509key)
        {
            // encoded OID sequence for  PKCS #1 rsaEncryption szOID_RSA_RSA = "1.2.840.113549.1.1.1"
            byte[] SeqOID = { 0x30, 0x0D, 0x06, 0x09, 0x2A, 0x86, 0x48, 0x86, 0xF7, 0x0D, 0x01, 0x01, 0x01, 0x05, 0x00 };
            byte[] seq = new byte[15];
            // ---------  Set up stream to read the asn.1 encoded SubjectPublicKeyInfo blob  ------
            MemoryStream mem = new MemoryStream(x509key);
            BinaryReader binr = new BinaryReader(mem);    //wrap Memory Stream with BinaryReader for easy reading
            byte bt = 0;
            ushort twobytes = 0;

            try
            {

                twobytes = binr.ReadUInt16();
                if (twobytes == 0x8130) //data read as little endian order (actual data order for Sequence is 30 81)
                    binr.ReadByte();    //advance 1 byte
                else if (twobytes == 0x8230)
                    binr.ReadInt16();   //advance 2 bytes
                else
                    return null;

                seq = binr.ReadBytes(15);       //read the Sequence OID
                if (!CompareBytearrays(seq, SeqOID))    //make sure Sequence for OID is correct
                    return null;

                twobytes = binr.ReadUInt16();
                if (twobytes == 0x8103) //data read as little endian order (actual data order for Bit String is 03 81)
                    binr.ReadByte();    //advance 1 byte
                else if (twobytes == 0x8203)
                    binr.ReadInt16();   //advance 2 bytes
                else
                    return null;

                bt = binr.ReadByte();
                if (bt != 0x00)     //expect null byte next
                    return null;

                twobytes = binr.ReadUInt16();
                if (twobytes == 0x8130) //data read as little endian order (actual data order for Sequence is 30 81)
                    binr.ReadByte();    //advance 1 byte
                else if (twobytes == 0x8230)
                    binr.ReadInt16();   //advance 2 bytes
                else
                    return null;

                twobytes = binr.ReadUInt16();
                byte lowbyte = 0x00;
                byte highbyte = 0x00;

                if (twobytes == 0x8102) //data read as little endian order (actual data order for Integer is 02 81)
                    lowbyte = binr.ReadByte();  // read next bytes which is bytes in modulus
                else if (twobytes == 0x8202)
                {
                    highbyte = binr.ReadByte(); //advance 2 bytes
                    lowbyte = binr.ReadByte();
                }
                else
                    return null;
                byte[] modint = { lowbyte, highbyte, 0x00, 0x00 };   //reverse byte order since asn.1 key uses big endian order
                int modsize = BitConverter.ToInt32(modint, 0);

                byte firstbyte = binr.ReadByte();
                binr.BaseStream.Seek(-1, SeekOrigin.Current);

                if (firstbyte == 0x00)
                {   //if first byte (highest order) of modulus is zero, don't include it
                    binr.ReadByte();    //skip this null byte
                    modsize -= 1;   //reduce modulus buffer size by 1
                }

                byte[] modulus = binr.ReadBytes(modsize);   //read the modulus bytes

                if (binr.ReadByte() != 0x02)            //expect an Integer for the exponent data
                    return null;
                int expbytes = (int)binr.ReadByte();        // should only need one byte for actual exponent data (for all useful values)
                byte[] exponent = binr.ReadBytes(expbytes);

                // ------- create RSACryptoServiceProvider instance and initialize with public key -----
                RSACryptoServiceProvider RSA = new RSACryptoServiceProvider();
                RSAParameters RSAKeyInfo = new RSAParameters();
                RSAKeyInfo.Modulus = modulus;
                RSAKeyInfo.Exponent = exponent;
                RSA.ImportParameters(RSAKeyInfo);
                return RSA;
            }
            catch (Exception)
            {
                return null;
            }

            finally { binr.Close(); }

        }

Just call the 'RSA' method with the text you want to encrypt and you're done.

Answer

Sergey picture Sergey · Aug 27, 2015

You can create an RSACryptoServiceProvider from a PEM file using the following class (GetRSAProviderFromPemFile method).

Warning: Don't just copy code from StackOverflow without verification! Especially not crypto code! This code has bugs (see comments). You may want to write and run tests before using this in production (if you really have no better option). I refuse to edit the code to fix it, as it would be just as unreliable as before without tests and an active maintainer.

Source: This code seems to be taken from opensslkey from this site. Copyright (c) 2000 JavaScience Consulting, Michel Gallant. The original package was released under a BSD-like license, so it is probably OK to use it (but you may want to double-check). There is also a NuGet package by the same author.

Here is the copy-pasted source code originally posted to this answer:

RSACryptoServiceProvider provider = PemKeyUtils.GetRSAProviderFromPemFile(  @"public_key.pem" );


public class PemKeyUtils
{
    const String pemprivheader = "-----BEGIN RSA PRIVATE KEY-----";
    const String pemprivfooter = "-----END RSA PRIVATE KEY-----";
    const String pempubheader = "-----BEGIN PUBLIC KEY-----";
    const String pempubfooter = "-----END PUBLIC KEY-----";
    const String pemp8header = "-----BEGIN PRIVATE KEY-----";
    const String pemp8footer = "-----END PRIVATE KEY-----";
    const String pemp8encheader = "-----BEGIN ENCRYPTED PRIVATE KEY-----";
    const String pemp8encfooter = "-----END ENCRYPTED PRIVATE KEY-----";

    static bool verbose = false;

    public static RSACryptoServiceProvider GetRSAProviderFromPemFile( String pemfile )
    {
        bool isPrivateKeyFile = true;
        string pemstr = File.ReadAllText( pemfile ).Trim();
        if (pemstr.StartsWith( pempubheader ) && pemstr.EndsWith( pempubfooter ))
            isPrivateKeyFile = false;

        byte[] pemkey;
        if (isPrivateKeyFile)
            pemkey = DecodeOpenSSLPrivateKey( pemstr );
        else
            pemkey = DecodeOpenSSLPublicKey( pemstr );

        if (pemkey == null)
            return null;

        if (isPrivateKeyFile)
            return DecodeRSAPrivateKey( pemkey );
        else
            return DecodeX509PublicKey( pemkey );

    }



    //--------   Get the binary RSA PUBLIC key   --------
    static byte[] DecodeOpenSSLPublicKey( String instr )
    {
        const String pempubheader = "-----BEGIN PUBLIC KEY-----";
        const String pempubfooter = "-----END PUBLIC KEY-----";
        String pemstr = instr.Trim();
        byte[] binkey;
        if (!pemstr.StartsWith( pempubheader ) || !pemstr.EndsWith( pempubfooter ))
            return null;
        StringBuilder sb = new StringBuilder( pemstr );
        sb.Replace( pempubheader, "" );  //remove headers/footers, if present
        sb.Replace( pempubfooter, "" );

        String pubstr = sb.ToString().Trim();   //get string after removing leading/trailing whitespace

        try
        {
            binkey = Convert.FromBase64String( pubstr );
        }
        catch (System.FormatException)
        {       //if can't b64 decode, data is not valid
            return null;
        }
        return binkey;
    }

    static RSACryptoServiceProvider DecodeX509PublicKey(byte[] x509Key)
    {
        // encoded OID sequence for  PKCS #1 rsaEncryption szOID_RSA_RSA = "1.2.840.113549.1.1.1"
        byte[] seqOid = { 0x30, 0x0D, 0x06, 0x09, 0x2A, 0x86, 0x48, 0x86, 0xF7, 0x0D, 0x01, 0x01, 0x01, 0x05, 0x00 };
        // ---------  Set up stream to read the asn.1 encoded SubjectPublicKeyInfo blob  ------
        using (var mem = new MemoryStream(x509Key))
        {
            using (var binr = new BinaryReader(mem))    //wrap Memory Stream with BinaryReader for easy reading
            {
                try
                {
                    var twobytes = binr.ReadUInt16();
                    switch (twobytes)
                    {
                        case 0x8130:
                            binr.ReadByte();    //advance 1 byte
                            break;
                        case 0x8230:
                            binr.ReadInt16();   //advance 2 bytes
                            break;
                        default:
                            return null;
                    }

                    var seq = binr.ReadBytes(15);
                    if (!CompareBytearrays(seq, seqOid))  //make sure Sequence for OID is correct
                        return null;

                    twobytes = binr.ReadUInt16();
                    if (twobytes == 0x8103) //data read as little endian order (actual data order for Bit String is 03 81)
                        binr.ReadByte();    //advance 1 byte
                    else if (twobytes == 0x8203)
                        binr.ReadInt16();   //advance 2 bytes
                    else
                        return null;

                    var bt = binr.ReadByte();
                    if (bt != 0x00)     //expect null byte next
                        return null;

                    twobytes = binr.ReadUInt16();
                    if (twobytes == 0x8130) //data read as little endian order (actual data order for Sequence is 30 81)
                        binr.ReadByte();    //advance 1 byte
                    else if (twobytes == 0x8230)
                        binr.ReadInt16();   //advance 2 bytes
                    else
                        return null;

                    twobytes = binr.ReadUInt16();
                    byte lowbyte = 0x00;
                    byte highbyte = 0x00;

                    if (twobytes == 0x8102) //data read as little endian order (actual data order for Integer is 02 81)
                        lowbyte = binr.ReadByte();  // read next bytes which is bytes in modulus
                    else if (twobytes == 0x8202)
                    {
                        highbyte = binr.ReadByte(); //advance 2 bytes
                        lowbyte = binr.ReadByte();
                    }
                    else
                        return null;
                    byte[] modint = { lowbyte, highbyte, 0x00, 0x00 };   //reverse byte order since asn.1 key uses big endian order
                    int modsize = BitConverter.ToInt32(modint, 0);

                    byte firstbyte = binr.ReadByte();
                    binr.BaseStream.Seek(-1, SeekOrigin.Current);

                    if (firstbyte == 0x00)
                    {   //if first byte (highest order) of modulus is zero, don't include it
                        binr.ReadByte();    //skip this null byte
                        modsize -= 1;   //reduce modulus buffer size by 1
                    }

                    byte[] modulus = binr.ReadBytes(modsize); //read the modulus bytes

                    if (binr.ReadByte() != 0x02)            //expect an Integer for the exponent data
                        return null;
                    int expbytes = binr.ReadByte();        // should only need one byte for actual exponent data (for all useful values)
                    byte[] exponent = binr.ReadBytes(expbytes);

                    // We don't really need to print anything but if we insist to...
                    //showBytes("\nExponent", exponent);
                    //showBytes("\nModulus", modulus);

                    // ------- create RSACryptoServiceProvider instance and initialize with public key -----
                    RSACryptoServiceProvider rsa = new RSACryptoServiceProvider();
                    RSAParameters rsaKeyInfo = new RSAParameters
                    {
                        Modulus = modulus,
                        Exponent = exponent
                    };
                    rsa.ImportParameters(rsaKeyInfo);
                    return rsa;
                }
                catch (Exception)
                {
                    return null;
                }
            }
        }
    }

    //------- Parses binary ans.1 RSA private key; returns RSACryptoServiceProvider  ---
    static RSACryptoServiceProvider DecodeRSAPrivateKey( byte[] privkey )
    {
        byte[] MODULUS, E, D, P, Q, DP, DQ, IQ;

        // ---------  Set up stream to decode the asn.1 encoded RSA private key  ------
        MemoryStream mem = new MemoryStream( privkey );
        BinaryReader binr = new BinaryReader( mem );    //wrap Memory Stream with BinaryReader for easy reading
        byte bt = 0;
        ushort twobytes = 0;
        int elems = 0;
        try
        {
            twobytes = binr.ReadUInt16();
            if (twobytes == 0x8130) //data read as little endian order (actual data order for Sequence is 30 81)
                binr.ReadByte();    //advance 1 byte
            else if (twobytes == 0x8230)
                binr.ReadInt16();   //advance 2 bytes
            else
                return null;

            twobytes = binr.ReadUInt16();
            if (twobytes != 0x0102) //version number
                return null;
            bt = binr.ReadByte();
            if (bt != 0x00)
                return null;


            //------  all private key components are Integer sequences ----
            elems = GetIntegerSize( binr );
            MODULUS = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            E = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            D = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            P = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            Q = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            DP = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            DQ = binr.ReadBytes( elems );

            elems = GetIntegerSize( binr );
            IQ = binr.ReadBytes( elems );

            Console.WriteLine( "showing components .." );
            if (verbose)
            {
                showBytes( "\nModulus", MODULUS );
                showBytes( "\nExponent", E );
                showBytes( "\nD", D );
                showBytes( "\nP", P );
                showBytes( "\nQ", Q );
                showBytes( "\nDP", DP );
                showBytes( "\nDQ", DQ );
                showBytes( "\nIQ", IQ );
            }

            // ------- create RSACryptoServiceProvider instance and initialize with public key -----
            RSACryptoServiceProvider RSA = new RSACryptoServiceProvider();
            RSAParameters RSAparams = new RSAParameters();
            RSAparams.Modulus = MODULUS;
            RSAparams.Exponent = E;
            RSAparams.D = D;
            RSAparams.P = P;
            RSAparams.Q = Q;
            RSAparams.DP = DP;
            RSAparams.DQ = DQ;
            RSAparams.InverseQ = IQ;
            RSA.ImportParameters( RSAparams );
            return RSA;
        }
        catch (Exception)
        {
            return null;
        }
        finally { binr.Close(); }
    }

    private static int GetIntegerSize( BinaryReader binr )
    {
        byte bt = 0;
        byte lowbyte = 0x00;
        byte highbyte = 0x00;
        int count = 0;
        bt = binr.ReadByte();
        if (bt != 0x02)     //expect integer
            return 0;
        bt = binr.ReadByte();

        if (bt == 0x81)
            count = binr.ReadByte();    // data size in next byte
        else
            if (bt == 0x82)
            {
                highbyte = binr.ReadByte(); // data size in next 2 bytes
                lowbyte = binr.ReadByte();
                byte[] modint = { lowbyte, highbyte, 0x00, 0x00 };
                count = BitConverter.ToInt32( modint, 0 );
            }
            else
            {
                count = bt;     // we already have the data size
            }



        while (binr.ReadByte() == 0x00)
        {   //remove high order zeros in data
            count -= 1;
        }
        binr.BaseStream.Seek( -1, SeekOrigin.Current );     //last ReadByte wasn't a removed zero, so back up a byte
        return count;
    }

    //-----  Get the binary RSA PRIVATE key, decrypting if necessary ----
    static byte[] DecodeOpenSSLPrivateKey( String instr )
    {
        const String pemprivheader = "-----BEGIN RSA PRIVATE KEY-----";
        const String pemprivfooter = "-----END RSA PRIVATE KEY-----";
        String pemstr = instr.Trim();
        byte[] binkey;
        if (!pemstr.StartsWith( pemprivheader ) || !pemstr.EndsWith( pemprivfooter ))
            return null;

        StringBuilder sb = new StringBuilder( pemstr );
        sb.Replace( pemprivheader, "" );  //remove headers/footers, if present
        sb.Replace( pemprivfooter, "" );

        String pvkstr = sb.ToString().Trim();   //get string after removing leading/trailing whitespace

        try
        {        // if there are no PEM encryption info lines, this is an UNencrypted PEM private key
            binkey = Convert.FromBase64String( pvkstr );
            return binkey;
        }
        catch (System.FormatException)
        {       //if can't b64 decode, it must be an encrypted private key
            //Console.WriteLine("Not an unencrypted OpenSSL PEM private key");  
        }

        StringReader str = new StringReader( pvkstr );

        //-------- read PEM encryption info. lines and extract salt -----
        if (!str.ReadLine().StartsWith( "Proc-Type: 4,ENCRYPTED" ))
            return null;
        String saltline = str.ReadLine();
        if (!saltline.StartsWith( "DEK-Info: DES-EDE3-CBC," ))
            return null;
        String saltstr = saltline.Substring( saltline.IndexOf( "," ) + 1 ).Trim();
        byte[] salt = new byte[saltstr.Length / 2];
        for (int i = 0; i < salt.Length; i++)
            salt[i] = Convert.ToByte( saltstr.Substring( i * 2, 2 ), 16 );
        if (!(str.ReadLine() == ""))
            return null;

        //------ remaining b64 data is encrypted RSA key ----
        String encryptedstr = str.ReadToEnd();

        try
        {   //should have b64 encrypted RSA key now
            binkey = Convert.FromBase64String( encryptedstr );
        }
        catch (System.FormatException)
        {  // bad b64 data.
            return null;
        }

        //------ Get the 3DES 24 byte key using PDK used by OpenSSL ----

        SecureString despswd = GetSecPswd( "Enter password to derive 3DES key==>" );
        //Console.Write("\nEnter password to derive 3DES key: ");
        //String pswd = Console.ReadLine();
        byte[] deskey = GetOpenSSL3deskey( salt, despswd, 1, 2 );    // count=1 (for OpenSSL implementation); 2 iterations to get at least 24 bytes
        if (deskey == null)
            return null;
        //showBytes("3DES key", deskey) ;

        //------ Decrypt the encrypted 3des-encrypted RSA private key ------
        byte[] rsakey = DecryptKey( binkey, deskey, salt ); //OpenSSL uses salt value in PEM header also as 3DES IV
        if (rsakey != null)
            return rsakey;  //we have a decrypted RSA private key
        else
        {
            Console.WriteLine( "Failed to decrypt RSA private key; probably wrong password." );
            return null;
        }
    }


    // ----- Decrypt the 3DES encrypted RSA private key ----------

    static byte[] DecryptKey( byte[] cipherData, byte[] desKey, byte[] IV )
    {
        MemoryStream memst = new MemoryStream();
        TripleDES alg = TripleDES.Create();
        alg.Key = desKey;
        alg.IV = IV;
        try
        {
            CryptoStream cs = new CryptoStream( memst, alg.CreateDecryptor(), CryptoStreamMode.Write );
            cs.Write( cipherData, 0, cipherData.Length );
            cs.Close();
        }
        catch (Exception exc)
        {
            Console.WriteLine( exc.Message );
            return null;
        }
        byte[] decryptedData = memst.ToArray();
        return decryptedData;
    }

    //-----   OpenSSL PBKD uses only one hash cycle (count); miter is number of iterations required to build sufficient bytes ---
    static byte[] GetOpenSSL3deskey( byte[] salt, SecureString secpswd, int count, int miter )
    {
        IntPtr unmanagedPswd = IntPtr.Zero;
        int HASHLENGTH = 16;    //MD5 bytes
        byte[] keymaterial = new byte[HASHLENGTH * miter];     //to store contatenated Mi hashed results


        byte[] psbytes = new byte[secpswd.Length];
        unmanagedPswd = Marshal.SecureStringToGlobalAllocAnsi( secpswd );
        Marshal.Copy( unmanagedPswd, psbytes, 0, psbytes.Length );
        Marshal.ZeroFreeGlobalAllocAnsi( unmanagedPswd );

        //UTF8Encoding utf8 = new UTF8Encoding();
        //byte[] psbytes = utf8.GetBytes(pswd);

        // --- contatenate salt and pswd bytes into fixed data array ---
        byte[] data00 = new byte[psbytes.Length + salt.Length];
        Array.Copy( psbytes, data00, psbytes.Length );      //copy the pswd bytes
        Array.Copy( salt, 0, data00, psbytes.Length, salt.Length ); //concatenate the salt bytes

        // ---- do multi-hashing and contatenate results  D1, D2 ...  into keymaterial bytes ----
        MD5 md5 = new MD5CryptoServiceProvider();
        byte[] result = null;
        byte[] hashtarget = new byte[HASHLENGTH + data00.Length];   //fixed length initial hashtarget

        for (int j = 0; j < miter; j++)
        {
            // ----  Now hash consecutively for count times ------
            if (j == 0)
                result = data00;    //initialize 
            else
            {
                Array.Copy( result, hashtarget, result.Length );
                Array.Copy( data00, 0, hashtarget, result.Length, data00.Length );
                result = hashtarget;
                //Console.WriteLine("Updated new initial hash target:") ;
                //showBytes(result) ;
            }

            for (int i = 0; i < count; i++)
                result = md5.ComputeHash( result );
            Array.Copy( result, 0, keymaterial, j * HASHLENGTH, result.Length );  //contatenate to keymaterial
        }
        //showBytes("Final key material", keymaterial);
        byte[] deskey = new byte[24];
        Array.Copy( keymaterial, deskey, deskey.Length );

        Array.Clear( psbytes, 0, psbytes.Length );
        Array.Clear( data00, 0, data00.Length );
        Array.Clear( result, 0, result.Length );
        Array.Clear( hashtarget, 0, hashtarget.Length );
        Array.Clear( keymaterial, 0, keymaterial.Length );

        return deskey;
    }

    static SecureString GetSecPswd( String prompt )
    {
        SecureString password = new SecureString();

        Console.ForegroundColor = ConsoleColor.Gray;
        Console.Write( prompt );
        Console.ForegroundColor = ConsoleColor.Magenta;

        while (true)
        {
            ConsoleKeyInfo cki = Console.ReadKey( true );
            if (cki.Key == ConsoleKey.Enter)
            {
                Console.ForegroundColor = ConsoleColor.Gray;
                Console.WriteLine();
                return password;
            }
            else if (cki.Key == ConsoleKey.Backspace)
            {
                // remove the last asterisk from the screen...
                if (password.Length > 0)
                {
                    Console.SetCursorPosition( Console.CursorLeft - 1, Console.CursorTop );
                    Console.Write( " " );
                    Console.SetCursorPosition( Console.CursorLeft - 1, Console.CursorTop );
                    password.RemoveAt( password.Length - 1 );
                }
            }
            else if (cki.Key == ConsoleKey.Escape)
            {
                Console.ForegroundColor = ConsoleColor.Gray;
                Console.WriteLine();
                return password;
            }
            else if (Char.IsLetterOrDigit( cki.KeyChar ) || Char.IsSymbol( cki.KeyChar ))
            {
                if (password.Length < 20)
                {
                    password.AppendChar( cki.KeyChar );
                    Console.Write( "*" );
                }
                else
                {
                    Console.Beep();
                }
            }
            else
            {
                Console.Beep();
            }
        }
    }

    static bool CompareBytearrays( byte[] a, byte[] b )
    {
        if (a.Length != b.Length)
            return false;
        int i = 0;
        foreach (byte c in a)
        {
            if (c != b[i])
                return false;
            i++;
        }
        return true;
    }

    static void showBytes( String info, byte[] data )
    {
        Console.WriteLine( "{0}  [{1} bytes]", info, data.Length );
        for (int i = 1; i <= data.Length; i++)
        {
            Console.Write( "{0:X2}  ", data[i - 1] );
            if (i % 16 == 0)
                Console.WriteLine();
        }
        Console.WriteLine( "\n\n" );
    }

}