EXAMPLE 14 Linear Cipher
To illustrate a linear cipher, we encrypt the message ADVANCE using the formula and decrypt the message ANQANHQ that was encrypted using the formula .
| Message | A | D | V | A | N | C | E |
| Position | 0 | 3 | 21 | 0 | 13 | 2 | 4 |
| 7 | 70 | 448 | 7 | 280 | 49 | 91 | |
| New position | 7 | 18 | 6 | 7 | 20 | 23 | 13 |
| Encrypted message | H | S | G | H | N | X | E |
To decrypt ANQANHQ, we see from Table 17.4 that the decryption value for 21 is 5, so our decryption formula is . To make calculations easier, it is better to replace by a positive integer by observing that when using modulo 26, we may add 26 to any integer without changing the value. Thus, , and our formula becomes . (Or use Google to obtain —.)
| Message | A | N | Q | A | N | H | Q |
| Position | 0 | 13 | 16 | 0 | 13 | 7 | 16 |
| 17 | 82 | 97 | 17 | 82 | 52 | 97 | |
| New position | 17 | 4 | 19 | 17 | 4 | 0 | 19 |
| Decrypted message | R | E | T | R | E | A | T |