even an entire hard drive (or solid state drive) – in a computer so that, if it is set aside (or stolen) an individual other than the owner cannot access the contents. (Apple and Windows both have encryption applications that use XTS‐AES‐128 encryption with a 256‐bit key. Apple's MacOS uses it in FileVault 2, [App18], and Windows 10 version 1511 uses it in their Bitlocker feature, [Win19].) We can fit this into our previous general situation with the owner of the computer playing the role of both A and B.
We have been silent on how A and B get their enciphering and deciphering keys. This is discussed in a later chapter, but will depend on the kind of encryption being used. The two fundamentally different possibilities for cryptography are as follows:
1 Symmetric Cryptography
2 Asymmetric Cryptography, i.e. Public Key Cryptography.
Figure 3.2 Symmetric encryption.
Recall that as before A, B are the communicating entities and A wants to send a secret message
1 The enciphering key used by the transmitter A is equal to the deciphering key used by the receiver B and this key is known only to A and B.
2 The enciphering algorithm, converting the plain text to cipher text, is such that the cipher text can be calculated immediately given and .
3 The deciphering algorithm, converting back to , can be calculated immediately given and .
The security depends on the fact that the secret key
Public key cryptography works differently (Figure 3.3). The procedure is as follows:
1 The enciphering key used by A (or anybody else) to send a message to B is publicly known, and is called the public key of B. However, the deciphering key used by B to decrypt the cipher text is known only to B and is the private key of B. So the two keys are quite different.
2 The enciphering procedure, converting the plain text to cipher text , can be immediately calculated given M and .
3 The deciphering procedure, converting back to M, can be calculated immediately by B using . However, it is not possible for somebody else who is not in possession of to convert back to M in a reasonable amount of time.The security of public key cryptography rests on the assumption that it is not feasible to convert back to M without knowledge of the private key .Thus, in public key cryptography each user B in a network has a public key and a private key , which are supplied by a public key authority (PKA).
Symmetric cryptography (encryption) is also called secret key cryptography (encryption). The security depends, as stated above, on the assumption that only the communicating parties A and B know the (common) key. Note that A, B could also denote groups of entities on a network and that B can also send a secret message to A using their common secret key.
Figure 3.3 Asymmetric or public encryption.
Historically, cryptography meant symmetric cryptography. Nowadays, important symmetric algorithms in use are AES and the One‐Time Pad which is sometimes derived from a shift register (see Chapter 16).
3.2 Public Key Cryptography and RSA on a Calculator
We now turn to some examples of asymmetric or public key cryptography. First, let us explain RSA, the main public key algorithm. As before, A wants to send a secret message
We want to find the (unique) remainder
where
(3.6)
Pushing the divide button on the calculator, we get
(3.7)
