2.7 Inversion

What is an inverse? The inverse of a process is its ``opposite''. For example, the opposite of increment by 2 is decrement by 2. For a given operation, the inverse of a value is its opposite value. For example, for addition, the inverse of 2 is -2. For cryptosystems, encryption and decryption are inverse processes. For the operation of transforming text, encryption and decryption keys are inverse ``values''.

To form a decryption key, you swap the top and bottom lines of an encryption key. More formally, swapping the top and bottom lines is called inverting:

• inverting an encryption key produces a decryption key
• inverting a decryption key produces an encryption key

Why the term inverse? If you transform plaintext using an encryption key, i.e. encrypt, you produce ciphertext. If you then transform the ciphertext using the decryption key, i.e., decrypt with the inverse of the encryption key, you produce the original plaintext. That is, the encryption key and decryption key ``undo'' each other and are inverses of each other. Mathematically speaking, for plaintext p,

encode(decode(p)) = encode(decode(p)) = p.

For instance, 'a' $\rightarrow$'d' followed by 'd' $\rightarrow$'a' yields the mapping 'a' $\rightarrow$'d' $\rightarrow$'a'.

  

Figure 9: Encryption and Decryption are Inverse Processes