# A multiplicative cipher is defined on Z26 by the rule M5 (m)= 5x26 mA multiplicative cipher is defined on Z26 by the rule M5 (m)= 5x26 m messages in english are coded numerically using the...

A multiplicative cipher is defined on Z26 by the rule

M*5* (m)= 5x*26 m*

A multiplicative cipher is defined on Z26 by the rule

M*5* (m)= 5x*26 *m

messages in english are coded numerically using the correspondence below.

A 1 B 2 C 3 D 4 E 5 F 6 G 7................... Y 25 Z 26

1) encipher the message WHEN using M*5* (that is , find the ciphertext for this message.)

2) Write down the deciphering rule for this cipher justifying your answer.

3) A message is enciphered using M5 to give the ciphertext {18,23,23,18} use answer to 2) to find the middle two lwtters of the message. Using answer to part 1) or otherwise find the other two letters and write down the message.

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### 1 Answer

the cipher is:

multiply by 5, then subtract a multiple of 26 if your number is "too big"

so:

1 -> 5

2 -> 10

3 -> 15

4 -> 20

5 -> 25

6 -> 30: but 30 is too big (not between 1 and 26), so subtract 26 to obtain:

6 -> 4

7 -> 35: again, too big, subtract 26 and

7 -> 9

8 -> 14

9 -> 19

10 -> 24

11* 5 = 55, 55-26 = 29 is still too big, 55-52=3

11 -> 3

Thus your encryption is:

A -> 1 -> 5 -> E

B -> 2 -> 10 -> J

C -> 3 -> 15 -> O

...

F -> 6 -> 4 -> D

...

K-> 11-> 3 -> C

The deciphering rule is M21

that is, multiply by 21, then subtract a multiple of 26 if your number is "too big"

1 -> 21

2 -> 42: but 42 is too big, so subtract 26

2 -> 16

3 -> 63: 63 is too big, subtract 52

3 -> 11

4 -> 84: 84 is too big, subtract 78

4 -> 6

The reason this works is that 5*21 = 1 + multiple of 26

so, for example, to decipher M3 you use M9, since

3*9 = 1 + multiple of 26

ciphertext 18 23 23 18

18*21 - 26*14 = 14 -> N

23*21 - 26*18 = 15 -> O

your decrypted text is: NOON

PS: if you are interested in learning more about this (why you want 1+ multiple of 26), or why you can't use M2 or M4 to encrypt your text, try reading about "modular arithmetic", say on wikipedia