Cryptography and Cryptanalysis: Affine Cipher Decoding

Last week's answer:  Taz fp D ovdsx kadh?

Last week we learned how to encode affine ciphers. This week we learn to decode.

In affine ciphers the decryption function for a letter denoted by x is
d(x) = a^-1(x - b) mod m

First of all, we need to understand a few rules:

• m is the size of the alphabet - in English it's 26
• b is in the range (0,m-1)

So, we know that in the equation:
d(which, in this case, stands for decrypt)(x) = a^-1(x - b)(mod 26)

Next, let's find 'a^-1'.
I'm not going to explain how to find the inverse to a number (it's a long complicated process - however, if anyone does have a question about this, I will be happy to send you the answer in an email) but for 'a' with {1,3,5,7,9,11,15,17,19,21,23,25} the inverse 'a^-1 is {1,9,21,15,3,19,7,23,11,5,17,25}. 




Here's an example:

Lets use (5,8) that means a=5 and b=8. However, the inverse of 5 is 21.

ciphertext:	I	H	H	W	V	V	C	S	W	F	R	C	P
y:	8	7	7	22	21	21	2	18	22	5	17	2	15

Now, the next step is to compute 21(y − 8), and then take the remainder when that result is divided by 26. The following table shows the results of both computations.

ciphertext:	I	H	H	W	V	C	S	W	F	R	C	P
y:	8	7	7	22	21	2	18	22	5	17	2	15
21(y-8):	0	-21	-21	294	273	-126	210	294	-63	189	-126	147
(21(y-8)) mod 26:	0	5	5	8	13	4	2	8	15	7	4	17

The final step in decrypting the ciphertext is to use the table to convert numeric values back into letters. The plaintext in this decryption is AFFINECIPHER. Below is the table with the final step completed.

ciphertext:	I	H	H	W	V	C	S	W	F	R	C	P
y:	8	7	7	22	21	2	18	22	5	17	2	15
21(y-8):	0	-21	-21	294	273	-126	210	294	-63		-126	147
(21(y-8)) mod 26:	0	5	5	8	13	4	2	8	15	7	4	17
plaintext:	A	F	F	I	N	E	C	I	P	H	E	R

Now your turn:

Decode this with the decryption key (a,b) = (7,5) with the inverse of a being (15,5) and ignore the capitals, spaces and question mark:

Gfi rzpubheo zs ich mftx!

(I will post the answer next week!)

 
Example source: Wikipedia
Picture source: Woot.com