Date: 2021-10-21

# How does cryptography work?

#### 2021-10-21

This page attempts to give a very basic conceptual introduction to cryptographic methods. Before we start the usual disclaimer:

I am not a cryptographer. This document is only for educational purposes. Crypto is hard, you should never trust your home-grown implementation. Unless you’re a cryptographer you will probably overlook some crucial details. Developers should only use the high-level functions that have been implemented by an actual cryptographer.

Now that we got this is out of the way, let’s start hacking :)

### The XOR operator

The bitwise XOR operator outputs `true` only when both inputs differ (one is `true`, the other is `false`). It is sometimes called an invertor because the output of a bit in `x` gets inverted if and only if the corresponding bit in `y` is true:

    # XOR two (8bit) bytes 'x' and 'y'
    x <- as.raw(0x7a)
    y <- as.raw(0xe4)
    z <- base::xor(x, y)
    dput(z)

    as.raw(0x9e)

    # Show the bits in each byte
    cbind(x = rawToBits(x), y = rawToBits(y), z = rawToBits(z))

          x  y  z
    [1,] 00 00 00
    [2,] 01 00 01
    [3,] 00 01 01
    [4,] 01 00 01
    [5,] 01 00 01
    [6,] 01 01 00
    [7,] 01 01 00
    [8,] 00 01 01

In cryptography we `xor` a message `x` with secret random data `y`. Because each bit in `y` is randomly `true` with probability 0.5, the `xor` output is completely random and uncorrelated to `x`. This is called perfect secrecy. Only if we know `y` we can decipher the message `x`.

    # Encrypt message using random one-time-pad
    msg <- charToRaw("TTIP is evil")
    one_time_pad <- random(length(msg))
    ciphertext <- base::xor(msg, one_time_pad)

    # It's really encrypted
    rawToChar(ciphertext)

    [1] "\xe8R\xd5\xc1\xed\031\t\xe7qJ\n\xc5"

    # Decrypt with same pad
    rawToChar(base::xor(ciphertext, one_time_pad))

    [1] "TTIP is evil"

This method is perfectly secure and forms the basis for most cryptograhpic methods. However the challenge is generating and communicating unique pseudo-random `y` data every time we want to encrypt something. One-time-pads as in the example are not very practical for large messages. Also we should never re-use a one-time-pad `y` for encrypting multiple messages, as this compromises the secrecy.

### Stream ciphers

The solution to this problem are stream ciphers. A stream cipher generates a unique stream of pseudo-random data based on a secret `key` and a unique `nonce`. For a given set of parameters the stream cipher always generates the same stream of data. Sodium implements a few popular stream ciphers:

    password <- "My secret passphrase"
    key <- hash(charToRaw(password))
    nonce <- random(8)
    chacha20(size = 20, key, nonce)

     [1] 6b ed fd 5d 30 77 29 95 b5 d9 ce ae 71 a5 fa b9 f4 96 e7 b5

Each stream requires a `key` and a `nonce`. The key forms the shared secret and should only be known to trusted parties. The `nonce` is not secret and is stored or sent along with the ciphertext. The purpose of the `nonce` is to make a random stream unique to protect gainst re-use attacks. This way you can re-use a your key to encrypt multiple messages, as long as you never re-use the same nonce.

    salsa20(size = 20, key, nonce)

     [1] 7e 87 53 c7 c8 6d 20 d9 8b f2 13 2f ba 3c 4d 90 5a d0 05 65

Over the years cryptographers have come up with many more variants. Many stream ciphers are based on a block cipher such as AES: a keyed permutation of fixed length amount of data. The block ciphers get chained in a particular mode of operation which repeatedly applies the cipher’s single-block operation to securely transform amounts of data larger than a block.

We are not going to discuss implementation details, but you could probably come up with something yourself. For example you could use a hash function such `sha256` as the block cipher and append counter which is incremented for each block (this is called CTR mode).

    # Illustrative example.
    sha256_ctr <- function(size, key, nonce){
      n <- ceiling(size/32)
      output <- raw()
      for(i in 1:n){
        counter <- packBits(intToBits(i))
        block <- sha256(c(key, nonce, counter))
        output <- c(output, block)
      }
      return(output[1:size])
    }

This allows us to generate an arbitrary length stream from a single secret key:

    password <- "My secret passphrase"
    key <- hash(charToRaw(password))
    nonce <- random(8)
    sha256_ctr(50, key, nonce)

     [1] f4 35 17 96 6c 54 d7 78 28 28 83 ef 68 7b ad ab 79 27 5e d8 72 14 12 33 22
    [26] 62 18 20 8c 5e 7d 6a a9 c2 54 bc 4a 3e 1d af ad e1 10 b9 82 1e b6 9e d6 21

In practice, you should never write your own ciphers. A lot of research goes into studying the properties of block ciphers under various modes of operation. In the remainder we just use the standard Sodium ciphers: `chacha20`, `salsa20` or `xsalsa20`.

### Symmetric encryption

Symmetric encryption means that the same secret key is used for both encryption and decryption. All that is needed to implement symmetric encryption is `xor` and a stream cipher. For example to encrypt an arbitrary length `message` using `password`:

    # Encrypt 'message' using 'password'
    myfile <- file.path(R.home(), "COPYING")
    message <- readBin(myfile, raw(), file.info(myfile)$size)
    passwd <- charToRaw("My secret passphrase")

A hash function converts the password to a key of suitable size for the stream cipher, which we use to generate a psuedo random stream of equal length to the message:

    # Basic secret key encryption
    key <- hash(passwd)
    nonce8 <- random(8)
    stream <- chacha20(length(message), key, nonce8)
    ciphertext <- base::xor(stream, message)

Now the `ciphertext` is an encrypted version of the message. Only those that know the `key` and the `nonce` can re-generate the same keystream in order to `xor` the ciphertext back into the original message.

    # Decrypt with the same key
    key <- hash(charToRaw("My secret passphrase"))
    stream <- chacha20(length(ciphertext), key, nonce8)
    out <- base::xor(ciphertext, stream)

    # Print part of the message
    cat(substring(rawToChar(out), 1, 120))

                GNU GENERAL PUBLIC LICENSE
                   Version 2, June 1991

     Copyright (C) 1989, 1991 Free Software Foundation, Inc.

The Sodium functions `data_encrypt` and `data_decrypt` provide a more elaborate implementation of the above. This is what you should use in practice for secret key encryption.

Symmetric encryption can be used for e.g. encrypting local data. However because the same secret is used for both encryption and decryption, it is impractical for communication with other parties. For exchanging secure messages we need public key encryption.

### Public-key encryption and Diffie-Hellman

Rather than using a single secret-key, assymetric (public key) encryption requires a keypair, consisting of a public key for encryption and a private-key for decryption. Data that is encrypted using a given public key can only be decrypted using the corresponding private key.

The public key is not confidential and can be shared on e.g. a website or keyserver. This allows anyone to send somebody a secure message by encrypting it with the receivers public key. The encrypted message will only be readable by the owner of the corresponding private key.

    # Create keypair
    key <- keygen()
    pub <- pubkey(key)

    # Encrypt message for receiver using his/her public key
    msg <- serialize(iris, NULL)
    ciphertext <- simple_encrypt(msg, pub)

    # Receiver decrypts with his/her private key
    out <- simple_decrypt(ciphertext, key)
    identical(msg, out)

    [1] TRUE

How does this work? Public key encryption makes use of Diffie-Hellman (D-H): a method which allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure channel. In the most simple case, both parties generate a temporary keypair and exchange their public key over the insecure channel. Then both parties use the D-H function to calculcate the (same) shared secret key by combining their own private key with the other person’s public key:

    # Bob generates keypair
    bob_key <- keygen()
    bob_pubkey <- pubkey(bob_key)

    # Alice generates keypair
    alice_key <- keygen()
    alice_pubkey <- pubkey(alice_key)

    # After Bob and Alice exchange pubkey they can both derive the secret
    alice_secret <- diffie_hellman(alice_key, bob_pubkey)
    bob_secret <- diffie_hellman(bob_key, alice_pubkey)
    identical(alice_secret, bob_secret)

    [1] TRUE

Once the shared secret has been established, both parties can discard their temporary public/private key and use the shared secret to start encrypting communications with symmetric encryption as discussed earlier. Because the shared secret cannot be calculated using only the public keys, the process is safe from eavesdroppers.

The classical Diffie-Hellman method is based on the discrete logarithm problem with large prime numbers. Sodium uses curve25519, a state-of-the-art D-H function by Daniel Bernsteinan designed for use with the elliptic curve Diffie–Hellman (ECDH) key agreement scheme.