Monoalphabetic cipher key

Question

I'm new to cryptography and have just completed homework on Monoalphabetic cipher.

I managed to decrypt the cipher text successfully but I cannot identify the key from my plaintext to ciphertext mapping table which is shown below (top is ciphertext and below in caps is what I have substituted with). No substitution were required for a, h and z. Notice r is substituted with B and y is substituted with y which I am a little confused with.

Although the decrypted message seems correct, I would like to understand what is the key for this.

a   b   c   d   e   f   g   h   i   j
    D   E   F   G   H   I       K   L
k   l   m   n   o   p   q   r   s   t
M   N   O   P   Q   R   S   B   T   U
u   v   w   x   y   z               
V   W   X   Y   C  

Answer 1

The key to a mono-alphabetic substitution cipher is a substitution table. Thus you already have (most of) the key for that cipher, it is

a   b   c   d   e   f   g   h   i   j
?   D   E   F   G   H   I   ?   K   L

k   l   m   n   o   p   q   r   s   t
M   N   O   P   Q   R   S   B   T   U

u   v   w   x   y   z               
V   W   X   Y   C   ?

where the question marks are unknown parts.

Often instead of defining the table manually, some simple generation algorithm is specified, such as “Shift all characters by $n$ positions to the right”. But not every substitution table can be produced in a simple way.

However, in your case, a simple algorithm would be (assuming a $\rightarrow$ A, h $\rightarrow$ J, z $\rightarrow$ Z):

Write the ciphertext alphabet A … Z in the first row and below write the corresponding plaintext characters, that is "a, r, y" and then the remaining ones in alphabetical order:

A  B  C  D  E  F  G  H ...
a  r  y  b  c  d  e  f ...

So in some way, you can consider "ary" to be the key but knowledge of the algorithm is also required.