Example 14

EXAMPLE 14 Linear Cipher

To illustrate a linear cipher, we encrypt the message ADVANCE using the formula $21 x + 7$ and decrypt the message ANQANHQ that was encrypted using the formula $21 x + 7$ .

Message	A	D	V	A	N	C	E
Position	0	3	21	0	13	2	4
$Position \times 21 + 7$	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 $5 (x - 7) = 5 x - 35$ . To make calculations easier, it is better to replace $- 35$ by a positive integer by observing that when using modulo 26, we may add 26 to any integer without changing the value. Thus, $- 35 = - 35 + 26 = - 9 = - 9 + 26 = 17$ , and our formula becomes $5 x + 17$ . (Or use Google to obtain — $35 mod 26 = 17$ .)

Message	A	N	Q	A	N	H	Q
Position	0	13	16	0	13	7	16
$Position \times 5 + 17$	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