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Encryption using AES 256 and SHA-2

I have a situation where I need to first encrypt a message using a public key and vector, that is already provided. Also as per requirement I need to use SHA-2 as well. For now, I am assuming that I need to hash the encrypted message and then send to the server. I have two questions related to this
1. Is it wise to hash the encrypted message? Also, will sending the encrypted message and hashed value to the server be a good idea?
2. I have done a lot search on internet, but whenever I try to get some example of using AES 256 and SHA-2 together, I actually land up where the difference between the two is explained. Can any help me with some sample code?
Thanks in Advance!!!

Let's break down the stuff first.
Public Key Cryptography
Allows a given pair (Kpriv, Kpub) to be used on a cipher to encrypt and decrypt data.
Any data encrypted with Kpriv can only be decrypted with Kpub and any data encrypted with Kpub can only be decrypted with Kpriv.
A nice and well known example of a public key cipher is RSA.
Asymmetric cryptography requires extremely large keys in order to be secure, such that it's extremely slow to execute! You should never encrypt large amount of data with Asymetric keys cryptography. You can use it in the beginning of a connecition to exchange a symetric key Ks, though.
Symetric Key Cryptography
Allows a Ks to be used on a cipher to encrypt and decrypt data.
An example of a symetric cipher is AES. AES is in fact so versatile you can change lots of parameters, such as, (as you mention) the Blocksize which can be of 128, 192 or 256 bits. AES256 is the AES cipher configured with a blocksize of 256 bits.
The block size is what's used against the provided Ks to perform the actual encryption. Note that your data can be larger than the block size (the algorithm will still work, It'l simply reuse the same Ks). Simply reusing the key every block is known as ECB mode and can reveal patterns if your data is repetitive. An alternative is to use modes like CBC or CTR which rely on also using previous block data and XORing with the next block data, to eliminate such patterns. What mode should you use depends on your data.
Note that, according to your cipher mode, you eventually will need padding. I'm assuming you are already quite familiar with this terms when you asked that question.
Guarantees By Cryptography
Cryptography does guarantee that the encrypted data is confidential but that's just it. It does not give any other guarantees such as whether the data is authentic or whether it has been tampered with or not.
While tampering data will most likely result in unintelligible text even after decryption, in cryptography, there's no such thing as invalid plaintext. As such, you need some mechanism to know if your data is valid or not.
A secure hash algorithm such as SHA can help you know whether your decrypted data is valid or not.
However, for these purposes, you usually shouldn't directly use a Digest algorithm. Try to instead use a MAC. That MAC can use SHA256 algorithm but MAC's and Hashes are not exactly the same.
How To Do It In Practice
If all you want is confidentiality and tampering detection, you would use the cipher and digest (or hash) algorithm as such:
E ks ( SHA(data) || data )
Where E is a symmetric cipher, ks is the shared symmetric key, SHA(data) is the digest of data using a secure hash algorithm, || means concatenation and data is a byte array.
A more safer approach would be:
E ks ( MAC mk(data) || data )
Where mk is the MAC's secret key.
Now just search how to "java symetric cipher" and "java hash byte array" and use the two as I'm describing above.

JAVA - Encrypt a text with windows username in AES 128

I would like to use Windows username like a key to crypt and decrypt a text.
But when i try, the key is too short :
Invalid AES key length: 7 bytes
I must encode the key before use ?
Thanks in advance

Technically - the key needs to be 128 bit long (or 192 or 256 bits). The easiest way is using padding (append some constant values to extend to the required length).
This approach is HIGHLY INSECURE, you could consider this approach as encoding, not encryption.
Proper encryption would need
having the key secret. Indeed this is very difficult to achieve if the code/configuration resides on the user's workstation, but using the username sounds as a VERY bad idea from the security perspective
Salt the key. Create an encryption key from a password using random salt and a hash function (e.g. PBKDF2).

Convert a string into a key

Is there a way to convert a string into a key so I can have the same encryption key for encoding/decoding? I do not mean using SecretKey btw.
Key symKey = KeyGenerator.getInstance(algorithm).generateKey();
I am using the DES algorithm btw

You probably mean a key derivation from a password. A popular approach is to use PBKDF2 in Java as discussed for example here. Don't forget to set the iteration count high to complicate brute-force attacks and set a random salt to prevent dictionary attacks.
A random salt is mandatory by now during user authentication, but it may also be used during encryption. It is just another public value that is added to the ciphertext like the IV for CBC mode.
Other alternatives would be bcrypt and scrypt which require additional libraries in Java.

AES-256 encryption: Generate 256bits(32bytes) key from my fixed string

I would like to use AES256 to encrypt a text, I'd like to use my email test#gmail.com as the key to encrypt it.
This is what I tried:
String key = "test#gmail.com";
SecretKeySpec keySpec = new SecretKeySpec(key.getBytes("UTF-8"), "AES");
byte iv[] = SOME_RANDOM_32_BYTES;
IvParameterSpec ivSpec = new IvParameterSpec(iv);
Cipher cipher = Cipher.getInstance("AES/CBC/PKCS5Padding");
cipher.init(Cipher.DECRYPT_MODE, keySpec, ivSpec);
byte[] encryptedResult = cipher.doFinal(text.getBytes("UTF-8"));
When I run above code, I got InvalidKeyException:
java.security.InvalidKeyException: Key length not 128/192/256 bits.
I checked on internet, the reason is my key is not 128/192/256 bits. My question is, how can I generate a 256bits(32bytes) key from my email string test#gmail.com ?

You can hash your key string (test#gmail.com) to a 256bit value using SHA256.
MessageDigest md = MessageDigest.getInstance("SHA-256");
md.update(yourEmail.getBytes());
byte[] encryptionKey = md.digest();

There can be multiple reasons for this. One of them is below.
Usually this error comes when you don't have the update policy in your JRE.
Java by default provide AES with 128, for 256 we have to put new policies provided by Java.

You should not bake your own crypto! (Unless you are very knowledgeable on the matter, that is.)
You should use an existing (and audited) encryption library.
Also, you should not use a guessable string such as your e-mail address as a password. Please, look for advice on how to choose a good password.
Now that I have said this, here are more details.
The proper way to implement password based encryption is to use a KDF (Key Derivation Function) to generate an encryption key from your password. Here are a few KDFs that you can use for this task: Argon2, Scrypt, Bcrypt and PBKDF2.
Key derivation functions include mechanisms to defend against know attacks such as rainbow tables and dictionary attacks, notably a "salt" and a work factor. Modern KDFs such as Argon2 also attempt to prevent attackers from gaining an advantage by using hardware more suitable to the task.
Generally speaking, here how this is used:
Select a work factor (the largest you can afford)
Generate the salt using a CSPRNG
Generate the encryption key and a MAC secret using your chosen KDF with the password, salt and work factor.
Generate an IV (initialization vector) using a CSPRNG
Encrypt the data to be protected using the generated encryption key.
Compute the MAC of the encrypted message using the generated secret.
Serialize the salt, the work factor, the computed MAC and the encrypted data. (Optionally, identifiers indicating what are the chosen KDF, encryption scheme and MAC should also be included if these are not fixed.)
Your encrypted message is the serialized data produced in step 7. Get any of the steps wrong (and that is easy) and your encryption code will probably break in horrible ways.
Perhaps now you get a sense of why you should use an existing library?
Note: the current best practice is to use AEAD (Authenticated Encryption with Associated Data) instead of encrypt-then-MAC as described above. Look this up if you are interested: I am not going to discuss this here.

Check encryption key correctness in JAVA

I'm using BouncyCastle to encrypt/decrypt some files using AES and PKCS5 padding in CBC mode :
Cipher c = Cipher.getInstance("AES/CBC/PKCS5Padding", "BC");
Now two questions:
How can I check that the provided key for decrypting data is correct or not ?
How Can I check encrypted input is untouched (e.g. not changed by user using an HEX editor)?
Thanks

You can use an AEAD mode, like CCM or GCM, in place of CBC. These modes authenticate an encrypted message, so if the wrong key is used, or the cipher text has been altered, you can detect it. You wouldn't be able to distinguish these cases though.
There is support in Java 7's cryptography API for GCM, but the SunJCE provider that ships with Oracle's Java implementation doesn't support it yet. You can get support through third-party providers like BouncyCastle.
You can achieve the same things if you use additional cryptographic services, like a digital signature or message authentication code.

Encryption is not just about the algorithm and the encryption key, it's also a lot about
the system organization.
In general, you can't determine that the key is correct. Any key can be used to decrypt the
data that's supposed to be decrypted, but it's up to some other mechanism to tell you if that
is the "correct" result.
In general, you can't determine if the data to be decrypted is untouched, except through some
external check. It's a property of most encryption systems that changing any of the encrypted
data would change the decrypted output drastically, probably into something you'd interpret
as garbage.

You should add a MAC which first verifies the integrity of the message, and only then you should decrypt it. A common choice of MAC is HMAC with whatever hash function you prefer, such as SHA-2.
Instead of doing this yourself, it's often a good idea to use an authenticated cipher. AES-GCM is a common choice. But you need to be really careful to never reuse an IV in that case.

The JCE ciphers are usually very basic. If you need a full featured protection including integrity and key testing, you need to combine them. And as usual it is better to not device that yourself. So better opt for a more high level format like PKCS7/12 or PGP.
Depending on the Padding used some ciphers will give you a PaddingException when you try to decrypt it with the wrong key. For stronger integrity check I would use a padding consiting of HMAC bytes.
A pretty complete method is included in the JCE, it is the AESWrap algorithm. It requires padded data but will ensure integrity. It is best combined with a length byte as described in RFC 3537. Note, that this is only intended for smaller amounts of secrets (like symmetric keys). The RFC3537 padding is restricted to 255 bytes.
To use this with a password derived key, you can use this:
char[] pass = ... // your password
byte[] codeBytes = ... // up to 255 bytes you want to protect
// generate wrapping key from password
SecretKeyFactory f = SecretKeyFactory.getInstance("PBKDF2WithHmacSHA1");
SecureRandom rand = SecureRandom.getInstance("SHA1PRNG");
byte[] salt = new byte[16]; rand.nextBytes(salt);
SecretKey kek = f.generateSecret(new PBEKeySpec(pass, salt, 1000, 128));
kek = new SecretKeySpec(password.getEncoded(), "AES"); // convert into AES
// RFC3537 padding (lengthbyte)
byte[] wrappedCodeBytes = new byte[codeBytes + 1 % 8];
System.arraycopy(codeBytes,0,wrappedCodeBytes,1,wrappedCodeBytes.length);
paddedCodeBytes[0]=(byte)codeBytes.length;
byte[] pad = new byte[paddedCodeBytes.length - codeBytes.length -1]; rand.nextBytes(pad);
System.arraycopy(pad,0,paddedCodeBytes,codeBytes.length+1,pad.length);
// AESWrap is WRAP_MODE:needs a SecretKey
SecretKey paddedCodeKey = new SecretKeySpec(paddedCodeBytes, "RAW");
// now wrap the password with AESWrap kek is 128 bit
Cipher c = Cipher.getInstance("AESWrap"); // default IV
c.init(Cipher.WRAP_MODE, kek);
byte[] result = c.warp(paddedCodeKey);
The unwrapping is left for the reader as an exercise :) The example code uses 128bit keysize, since more entropy cant be expected from the PBKDF2 anyway.
Note that this will detect wrong passwords with high probability, and some critics will see this as a weakness of AESWrap.

Take a look at this tutorial on BC encryption, specifically the InitCiphers methods, and in detail at the second code block which specifies the actual type of cipher.
How can I check that the provided key for decrypting data is correct or not?
According to JCE Javadocs, specifically the constructor of Class SecretKeySpec:
This constructor does not check if the given bytes indeed specify a secret key of the specified algorithm. For example, if the algorithm is DES, this constructor does not check if key is 8 bytes long, and also does not check for weak or semi-weak keys. In order for those checks to be performed, an algorithm-specific key specification class (in this case: DESKeySpec) should be used.
Note that Interface KeySpec lists all implementing classes, basically a list of validation options.
How Can I check encrypted input is untouched (e.g. not changed by user using an HEX editor)?
Indeed. That's a good one. 'Input' is pretty generic. Do you mean the actual content to decrypt? Well, if it's munged I believe it will not decrypt properly. Does that make sense?
IFF you are talking about the case of a key with parity bits being altered, as described in item (6) at the Bouncy Castle FAQ, you will have to do an actual parity check on the key. Only the first 56 bytes of the key are used for the encryption ops, and the last 8 bytes are reserved for parity checking. So, essentially, the last part of the 'key' can be changed and the first part is still useful. To detect whether either the parity or the key have been altered, you would run a parity check. I found this little ditty on doing a parity check. And, for more info on how parity is set in these keys, see comments in the JDK7 Crypto Provider source for Class DESKeyGenerator by Jan Luehe (near bottom) which discuss parity setting.
I recently had some interaction with BC, and I hope this info helps.