Affine cipher: Difference between revisions

The affine cipher is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. The formula used means that each letter encrypts to one other letter, and back again, meaning the cipher is essentially a standard substitution cipher with a rule governing which letter goes to which. As such, it has the weaknesses of all substitution ciphers. Each letter is enciphered with the function ${\displaystyle (ax+b)\mod (26)}$, where ${\displaystyle b}$ is the magnitude of the shift.

In these two examples, one encrypting and one decrypting, the alphabet is going to be the letters A through Z, and will have the corresponding values found in the following table.

In this encrypting example,^[1] the plaintext to be encrypted is "AFFINE CIPHER" using the table mentioned above for the numeric values of each letter, taking ${\displaystyle a}$ to be 5, ${\displaystyle b}$ to be 8, and ${\displaystyle m}$ to be 26 since there are 26 characters in the alphabet being used. Only the value of ${\displaystyle a}$ has a restriction since it has to be coprime with 26. The possible values that ${\displaystyle a}$ could be are 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, and 25. The value for ${\displaystyle b}$ can be arbitrary as long as ${\displaystyle a}$ does not equal 1 since this is the shift of the cipher. Thus, the encryption function for this example will be ${\displaystyle y=E(x)=(5x+8){\pmod {26}}}$. The first step in encrypting the message is to write the numeric values of each letter.

Now, take each value of x, and solve the first part of the equation, ${\displaystyle (5x+8)}$. After finding the value of ${\displaystyle (5x+8)}$ for each character, take the remainder when dividing the result of ${\displaystyle (5x+8)}$ by 26. The following table shows the first four steps of the encrypting process.

plaintext:	A	F	F	I	N	E	C	I	P	H	E	R
x:	0	5	5	8	13	4	2	8	15	7	4	17
${\displaystyle 5x+8}$	8	33	33	48	73	28	18	48	83	43	28	93
${\displaystyle (5x+8){\pmod {26}}}$	8	7	7	22	21	2	18	22	5	17	2	15

The final step in encrypting the message is to look up each numeric value in the table for the corresponding letters. In this example, the encrypted text would be IHHWVCSWFRCP. The table below shows the completed table for encrypting a message in the Affine cipher.

plaintext:	A	F	F	I	N	E	C	I	P	H	E	R
x:	0	5	5	8	13	4	2	8	15	7	4	17
${\displaystyle 5x+8}$	8	33	33	48	73	28	18	48	83	43	28	93
${\displaystyle (5x+8){\pmod {26}}}$	8	7	7	22	21	2	18	22	5	17	2	15
ciphertext:	I	H	H	W	V	C	S	W	F	R	C	P

To make encrypting and decrypting quicker, the entire alphabet can be encrypted to create a one to one map between the letters of the cleartext and the ciphertext. In this example, the one to one map would be the following:

Using the Python programming language, the following code can be used to create an encrypted alphabet using the Roman letters A through Z.

#Prints a transposition table for an affine cipher.
#a must be coprime to m=26.
def affine(a, b):
    for i in range(26):
        print chr(i+65) + ": " + chr(((a*i+b)%26)+65)

#An example call
affine(5, 8)

Or in Java:

public void Affine(int a, int b){
	    for (int num = 0; num < 26; num++)
	      System.out.println(((char)('A'+num)) + ":" + ((char)('A'+(a*num + b)% 26)) );
	}
Affine(5,8)

Or in Pascal:

Procedure Affine(a,b : Integer);
  begin
    for num := 0 to 25 do
      WriteLn(Chr(num+65) , ': ' , Chr(((a*num + b) mod 26) + 65);
  end;

begin
  Affine(5,8)
end.

• Affine functions
• Atbash code
• Caesar cipher
• ROT13
• Topics in cryptography
• Perl interface to "Affine cipher"