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Is RijndaelManaged key length the same as AES[key length]?

RijndaelManaged is an algorithm while AES is the standard. When referring to AES256, does that mean the key length I'm using with RijndaelManaged must be 256 characters?
If I have a key like this:
key = "mytestkey";
which is only 9 characters, does that mean I'm using AES9?

Aes/Rijndael key sizes are in bits, not characters; AES256 uses a 256-bit key. You must give it a key that is exactly 256 bits. The AES standard and Rijndael only accept key sizes that are either 128, 192, or 256 bits.
You should not interchange AES and Rijndael. The AES standard is derived from Rijndael, but they are not exactly the same. If you wish to use AES, use AES for all operations; if you wish to use Rijndael, use Rijndael for all operations.
"mytestkey" cannot directly be a key to AES256 nor a similar Rijndael mode; it is not 256 bits long. In order to use it as a key, you'll have to transform it into a block of bytes that is 256 bits long.
If you're working with passwords, one typical means to do this is with key stretching, using hash algorithms such as PBKDF2 or Scrypt. PBKDF stands for "Password-based key derivation function", which is basically exactly what you're doing - deriving a key from a password.
Theoretically you could also just hash the password with SHA256 (which always has a 256-bit output) and use that 256-bit block as the key to AES; doing so is unwise from a security standpoint because it is relatively easy to precompute SHA hashes of passwords.
Please keep in mind that if you use a password that has very little entropy, then the security of your encryption suffers - the time it'll take for someone to guess the key could be short. "mytestkey" has at most ~42 bits of entropy - you're only using lower case letters, so 26 values per place, and there are 9 places (9 characters). Thus the number of bits theoretically needed to stores this is log_2( 26^9 ) = 42.3. In this circumstance, you'd be using AES256 with a key that has only ~42 bits of entropy.
I should note that the advice given here is an incomplete treatment of how to turn passwords into keys, from a security perspective. If you want to better understand how to properly generate keys from passwords, I suggest you start with some reading such as the Password Storage Cheat Sheet at owasp.org.

Private key encryption in .NET 4.0 of a very large number

In my scenario, I would like to encrypt a very big number (10^27) using a private key and later be able to decrypt it using a public key. The problem I have is that I want to keep the size of the encrypted text as small as possible.
I know that .NET has support for public key encryption (RSACryptoServiceProvider), but the encrypted text gets so huge.
Would it work to instead treat the private key as a public key?
Would Elliptic curve cryptography produce a smaller output?

First of all, if you want to achieve confidentiality you should always encrypt with the public key, not the private key. RSA encryption is not defined for encryption with the private key, and the results may vary (especially the kind of padding that is applied).
For direct RSA encryption, the size of the encrypted message is identical to the modulus. Now the modulus should be at least 2048 bits by now, and your message is only about (27/3)*10=90 bits. So RSA would have a large overhead, independent on the key used. Using ECIES is therefore likely to give significant benefits.

C# BouncyCastle RSA Encryption and Decryption

There are many topics on RSA Encryption and Decryption using BouncyCastle, however I'm encountering some unexpected behaviour.
I'm attempting to encrypt a 64 byte data blocking using a private key of size 64 bytes
I compute the RSA Encryption as followings:
public byte[] Encrypt(byte[] data, AsymmetricKeyParameter key)
{
var engine = new RsaEngine();
engine.Init(true, key);
var blockSize = engine.GetInputBlockSize();
return engine.ProcessBlock(data, 0, blockSize );
}
I compute the decryption using a public key as follows
public byte[] Decrypt(byte[] data, AsymmetricKeyParameter key)
{
var engine = new RsaEngine();
engine.Init(false, key);
var blockSize = engine.GetInputBlockSize();
return engine.ProcessBlock(data, 0, blockSize );
}
What I'm finding is that when I encrypt my 64 data using a 64 byte Private Key I get back a 64 byte encrypted dataBlock.
However when I decode the 64 byte array using a 64 byte public key I get back a data block of size 62 bytes. What is stranger is that the values contained in the 62 byte array equal the values of the 64 byte original array (pre encryption) however the decoded array is missing the first index of the original data and the final index.
I've tried using different keys and different sets of data and the same thing happens.
I must be doing something wrong, but I can't see it.
Cheers.

You got the essential concepts wrong.
512 bit RSA is very weak, use at least 1024 bits
A private key is not for encryption. It's for decryption and signing. The public key is for encryption and verification.
Padding is essential for RSA security. A typical padding scheme requires several dozen bytes.
Even with textbook RSA, RSA can only work on values smaller than the modulus. So a 512 bit modulus can't operate on arbitrary 64 byte / 512 bit values. But only on 511 bits.
You should take a step back, and describe what you actually want to achieve, so we can find a scheme that fits your needs. Only after that you should worry about implementing it.

How to create Encryption Key for Encryption Algorithms?

I want to use encryption algorithm available in .Net Security namespace, however I am trying to understand how to generate the key, for example AES algorithm needs 256 bits, that 16 bytes key, and some initialization vector, which is also few bytes.
Can I use any combination of values in my Key and IV? e.g. all zeros in Key and IV are valid or not? I know the detail of algorithm which does lots of xors, so zero wont serve any good, but are there any restrictions by these algorithms?
Or Do I have to generate the key using some program and save it permanently somewhere?
I want to store data in database after encryption, the secure profile data like username, password, phone number etc, and the key will be available to database user mentioned in connection string only, and to the administrator.

You really ought to do this the correct way :)
1) Use a securely generated random IV
2) Use a securely generated random key
3) Don't use ECB mode - EVER
AesManaged aes = new AesManaged();
aes.GenerateKey();
aes.GenerateIV();
The code above will correctly and securely generate a random IV and random key for you.

Sounds like you need to read into the Rfc2898DeriveBytes class.
Rfc2898DeriveBytes.GetBytes();
It has a method(above) that allows you to tailor the size of byte arrays that are fed into the .Key and .IV properties on a symmetric encryption algorithm, simply by feeding an int value. The MS official 70-536 book suggests doing this pro-grammatically by dividing the KeySize property / 8.
I.e TripleDes or AESManaged. Whatever you use, the algorithm itself will have some pre-reqs that will need meeting first. I.e satisfying the key size conditions. The RunTime will automatically fill the properties and fields and etc the best and most strongest values for you. But the IV and Key needs to come from you. This how you can do the following:
RijndaelManaged myAlg = new RiRijndaelManaged();
byte[] salt = Encoding.ASCII.GetBytes("Some salt value");
Rfc2898DeriveBytes key = new Rfc2898DeriveBytes("some password", salt);
myAlg.Key = key.GetBytes( myAlg.KeySize / 8);
myAlg.IV = key.GetBytes( myAlg.BlockSize / 8);
// myAld should now fully set-up.
Above you can see what I mean by doing it pro-grammatically, as it should pretty much
do it all for you, without you even really having to bat an eye-lid as to meeting it's pre-reqs.
The Microsoft 70-536 book states that the .Key properties expect the byte arrays you supply
to them in bytes and not bits. The RFC class works in bytes where as an algorithms KeySize property works in bits. 1 byte = 8 bits. Can you see where this is going ... ?
This should give you an idea as to why the above sample peice of code is done the way it is! I studied it and it makes pretty darn good sense to me!
The above answer should allow you to create your algorithm object with supplied password and a static salt value that can be hard code at both ends. Only thing you need to do is worry about how you going to make sure that the byte arrays stored at .Key and .IV are safely transported to a recipient so that can successfully decrypt the message you encrypted. By safely reconstructing the same algorithm object.
OBTW:
AESManaged has a keysize req': 128Bits = 16 Bytes !!!
(8*8 = 64, 64Bit / 8bits per Byte = 8 Bytes) Therefore
64*2 = 128Bit, 8*2, ==> 16bytes key size !
256Bit = 32Bytes !!!!
According to the 70-536 official training kit book, Aes is limited to having keysize of 128bits in size. 256bits,192 and 128 key size for example can be used with the Rijndael class.
You could on the other hand completely forget all that crap and simply use .GenerateKey and GenerateIV methods instead to save you all the hassle of sorting out a pre-shared and agreed password and static salt values. Your only concern is figuring out a way of storing and retrieving the key and IV byte arrays. Binary Formatter? .

If you are using encryption to exchange data then you will need a key exchange protocol, but you don't make one yourself instead use one off-the-shelf like TLS or SSL.
If you use encryption to store data then you generate the IV using CryptGenRandom (or its .net equivalent RandomNumberGenerator.GetBytes) and save it along the document (in clear, no need to protect the IV). You never write down the key, the key is provided by the user. Usualy you derive the key from a password phrase using CryptDeriveKey, or its .Net equivalent PasswordDeriveKey.CryptDeriveKey.
Update
To store a secret in the database that is available only to the user and an administrator you need to use 3 keys:
one to encrypt the data with (call it the DK key)
one user key to encrypt the DK key (call it UK)
one administrator key to encrypt the DK key (call it AK)
In theory you encrypt the data with DK and then encrypt the DK with UK and save it, and encrypt the DK with AK and save it. This way the user can use again the UK to decrypt the DK and then decrypt the data, and the administrator can use the AK to decrypt the DK and then decrypt the data. The big problem is the fact that the system is always automated, so the system needs access to the administrator's key which means is not truly a persnal key of the administrator, but instead is a system key (it cannot be used for purposes of non-repudiation for instance).
As a heads up, knowledge of what IV is or how to use AES from C# and how cryptography algorithm work will get you exactly 0 (zero) traction in solving this kind of problems. The issue is never what IV and key to use, the issue is always key provisioning. For actual crypto operations, just use the built-in support from the database, see Cryptography in SQL Server. I can easily argue that the only facility you need is TDE (Transparent Data Encryption) to protect against accidental loss of media.

Generate a random letters / hex code in a specific length.
This function (taken from here) return a random key in a specific length:
private static string CreateSalt(int size)
{
//Generate a cryptographic random number.
RNGCryptoServiceProvider rng = new RNGCryptoServiceProvider();
byte[] buff = new byte[size];
rng.GetBytes(buff);
// Return a Base64 string representation of the random number.
return Convert.ToBase64String(buff);
}

Use System.Security.Cryptography.RandomNumberGenerator to generate random bytes:
var rnd = new System.Security.Cryptography.RandomNumberGenerator.Create();
var key = new byte[50];
rnd.GetBytes(key);

It really depends on what you ned to do with the key.
If the key is to be generated by the computer (and can be any random value) I generally take a SHA256 of a couple GUIDs. This is about as random as you're going to get without a hardware random number generator.
You can use keys with all 0s but obviously it won't be very secure.

Do encryption algorithms require an internal hashing algorithm?

When I use C# to implement the AES symmetric encryption cipher, I noticed:
PasswordDeriveBytes derivedPassword = new PasswordDeriveBytes(password, saltBytesArray, hashAlgorithmName, numPasswordIterations);
Why do I need to use a hashing algorithm for AES encryption? Aren't they separate? Or is the hashing algorithm only used to create a secure key?
The AES algorithm doesn't use a hashing algorithm internally does it?

PasswordDeriveBytes isn't part of AES. It implements an algorithm to derive encryption keys from a password. The algorithm involves the usage of a hash algorithm.

PasswordDeriveBytes is used to derived a symmetric key as well as the IV you are going to use for encryption/decryption. The PasswordDeriveBytes would take an passkey (password in your case), append the salt (saltBytesArray in your case) and hash it (with the algorithm you provided, e.g. SHA or MD5) 'n' many times (numPasswordIterations in your case) and give you the resulatant byte array.
This method is only used to derive the key, this has got nothing to do with encryption decryption per say.