## Nihilist Substitution Cipher

The Nihilist Substitution is a poly-alphabetic cipher which means it uses multiple substitution alphabets and similar to the Vigenère Cipher.

The key consists of a 5×5 polybius square which has all the letters in the alphabet however I/J are treated the same and a second key.

 1 2 3 4 5 1 A B C D E 2 F G H I/J K 3 L M N O P 4 Q R S T U 5 V W X Y Z

The second key can be of any length, keep in mind that the longer the key the more secure it theoretically is, however the key should be memorable so a person could remember and use it. Examples:

MAGIC, KEY, DEFEND, POLYALPHABETIC etc.

# Encryption

Consider the polybius square created using the keyword CIPHER

 1 2 3 4 5 1 C I/J P H E 2 R A B D F 3 G K L M N 4 O Q S T U 5 V W X Y Z

and a second key of PAGE which defines the period as 4.

The second key is replaced with its position the polybius square (row then column), the numbers effectively become the key. Then each plaintext letter is  written in rows of the period length and it too also replaced with its position the polybius square (row then column). The cipher text is then the sum of the key and the cipher text numbers. If the number is greater than 99 (3 digit number) subtract 100. 105 becomes 05, 100 becomes 00.

P  A  G  E   P  A  G  E   P  A  G  E
13 22 31 15  13 22 31 15  13 22 31 15
-----------  -----------  -----------
W  H  E  N   S  T  R  I   E  A  T  T
52 14 15 35  43 44 21 12  15 22 44 44
65 36 46 50  56 66 52 27  28 44 75 59
-----------  -----------  -----------
T  H  E  C   K  E  S  T   A  C  K
44 14 15 11  32 15 43 44  22 11 32
57 36 46 26  45 37 74 59  35 33 63
-----------  -----------
L  O  C  K   W  E  L  V
33 41 11 32  52 15 33 51
46 63 42 47  65 37 64 66

WHENTHECLOCKSTRIKESTWELVEATTACK using these keys encrypts to 65364650573646264663424756665227453774596537646628447559353363.

# Decryption

To decrypt simply split the number text into blocks of 2 and write each block in rows of the period length then subtract the key numbers. If the result is less than 0 add 100.

There are however some serious flaws that significantly decrease the security of this cipher, these flaws can be used to break the Nihilist Substitution Cipher.

# Encryption

To encrypt in Hill a key first needs to be chosen, this will be a square matrix which has an inverse in modular 26. For the matrix to have an inverse the determinant must be co-prime to 26.

Here are two example matrices…
$M_{2}=\begin{bmatrix}2 & 3 \\5 & 3 \end{bmatrix}, M_{3}=\begin{bmatrix}25 & 2 & 11 \\19 & 5 & 12 \\21 & 22 & 6 \end{bmatrix}$
$det(M_{2})=-9$ (-9 mod 26) is 17 (-9+26×1), 17 is co-prime to 26
$det(M_{3})=-2131$ (-2131 mod 26) is 1 (-2131+26×82), 1 is co-prime to 26
Hence both these matrices are valid keys for the Hill cipher
*Co-prime means that the greatest common factor between the two numbers is 1.

If a 2 by 2 matrix is chosen for the key, the plaintext must be padded usually with an ‘X’ so that it is a multiple of 2. So for an N by N matrix the plaintext must be padded so that it that it is a multiple of N. E.G
For N=2, “CAREFUL” would become “CAREFULX”
For N=2, “SPORTS” would stay as “SPORTS”
For N=3, “CAREFUL” would become “CAREFULXX”
Once the text is a valid length, you take letters in blocks of N and convert them to a column vector. Letter ‘A’ has the value 0, ‘B’ is 1, ‘C’ is 2 … ‘Z’ is 25 etc.

Example:
$CA=\begin{bmatrix}2 \\0 \end{bmatrix}$, $RE=\begin{bmatrix}17 \\4 \end{bmatrix}$, $FU=\begin{bmatrix}5 \\20 \end{bmatrix}$, $LX=\begin{bmatrix}11 \\23 \end{bmatrix}$
$CAR=\begin{bmatrix}2 \\0 \\17 \end{bmatrix}$, $EFU=\begin{bmatrix}4 \\5 \\20 \end{bmatrix}$, $LXX=\begin{bmatrix}11 \\23 \\23 \end{bmatrix}$

You do this for each block of N characters. Then multiply the key matrix by each column vector. You will get a new column vector which can be converted back to letters.

$\begin{bmatrix}2&3 \\5&3 \end{bmatrix}\begin{bmatrix}2 \\0 \end{bmatrix}=\begin{bmatrix}2\times2+3\times0 \\5\times2+3\times0 \end{bmatrix}=\begin{bmatrix}4 \\10 \end{bmatrix}\bmod26$
“CAREFUL” encodes to “EKUTSHNU” using the key matrix M2

$\begin{bmatrix}25 & 2 & 11 \\19 & 5 & 12 \\21 & 22 & 6 \end{bmatrix}\begin{bmatrix}2 \\0 \\17 \end{bmatrix}=\begin{bmatrix}25\times2+2\times0+11\times17 \\19\times2+5\times0+12\times17 \\21\times2+22\times0+6\times17 \end{bmatrix}=\begin{bmatrix}3 \\8 \\14 \end{bmatrix}\bmod26$
“CAREFULX” encodes to “DIOSDCCCR” using the key matrix M3