How is a ciphertext encrypted using random numbers decrypted using a symmetric key algorithm

Question

We use true random number generators as a source of keys for encryption. What I can not wrap my head around is: How is this cipher text decrypted in case of symmetric key algorithm as we need the same random number to decrypt the cipher text.

What I could guess is:

• Is this random number used as a session key and transmitted using any asymmetric key algorithm?

• Are PRNGs and true random number generators used as a seed?

Answer 1

Usually the symmetric key is exchanged with Public-key cryptography (which is asymmetric).

Quote from Wikipedia:

Because of the computational complexity of asymmetric encryption, it is usually used only for small blocks of data, typically the transfer of a symmetric encryption key (e.g. a session key). This symmetric key is then used to encrypt the rest of the potentially long message sequence. The symmetric encryption/decryption is based on simpler algorithms and is much faster.

Concerning random-number-generators in cryptography:

Cryptographically secure pseudorandom number generators are typically used when the design of a cryptographic scheme needs a random number. These include i.e:

• One-time pad
• Elliptic Curve Digital Signature Algorithm
• Cryptographic nonce

Answer 2

As you have guessed any Probabilistic Encryption scheme (i.e. one which produces a randomized ciphertext output) requires that you transmit the seed for that randomness with the message. This value is called an Initialization Vector (IV) or a nonce and the combination of the Key and the IV/nonce is required to decrypt the message.

An picture of how this works for a block cipher in cipher block chaining (CBC) is depicted in the diagram:

The $\bigoplus$ symbol represents the XOR operation. From this diagram it should be fairly obvious that choosing a different random IV (and XORing it with the input) would result in very different ciphertext values.

Let's assume I am sending you a message and we have already shared a key. I would generate a random IV value and encrypt the message with that key/IV pair. I would transmit the IV and the ciphertext to you and then you would use the pre-shared key and the message-specific IV to decrypt the ciphertext.

The requirements on IV/nonce input vary depending on the encryption mode and cipher. In general IVs should be secure random values and nonces need only be unique (so a counter works as long as you don't repeat the same nonce for the same key).

Note that while the example above illustrates a block cipher, a probabilistic stream cipher will use similarly a IV/nonce along with a key. The IV would permute the initial state of the stream such that each encryption is unique.