Defining Symmetric vs Asymmetric cryptosystems I'm writing a project on the advantages and disadvantages of symmetric vs asymmetric cryptosystems. At the moment I am trying to define both systems using mathematical notation. Under the heading "Basic Cryptosystems" I have this:


  We can define the cryptogram functionally as
  \begin{equation}
\mathit{C=f(M,K)}
\end{equation}
  It will also be helpful to think of the cryptogram as a family of transformations with one parameter like this 
  \begin{equation}
\mathit{C=T_{k}M}
\end{equation}
  The transformation $T_{k}$ applied to the message $M$ produces cryptogram $C$. The operation of the transformation $T$ will depend on which cryptosystem is being used. The index $k$ corresponds to the particular key being used.
  Assume there are a finite number of keys, each with associated probability $P(T_{k})$. There will then be some discrete statistical process which chooses a transformation from the set $T_{1}, T_{2}, ..., T_{n}$. These transformations have associated probabilities $P(T_{1}), P(T_{2}), ..., P(T_{n})$ respectively. 
  We should also assume a finite number of possible messages $M_{1}, M_{2}, ..., M_{m}$ with associated a priori probabilities $P(M_{1}), P(M_{2}), ..., P(M_{m})$.
  Once $C$ is received, it can be deciphered with $k$ in order to reveal $M$. This is achieved with a unique inverse transformation $T^{-1}_{k}$ such that $T_{k}T^{-1}_{k}=I$, the identity transformation. So 
  \begin{equation}
M=T^{-1}_{k}C
\end{equation}
  The inverse transformation must be unique for every $C$ which can be deciphered with key $k$ to reveal $M$. 


My question is: does this definition apply to both symmetric and asymmetric systems? If not, how can I change it to account for asymmetric systems? 

thanks
Your definition only covers symmetric encryption since the same "index" $k$ is used in $T_k$ and $T^{-1}_k$ (i.e., encryption and decryption use the same key)

If you want to give a generic definition that covers both types of encryption, you could say that the transformations are $T_k$ and $T^{-1}_{k'}$ and that in the case of symmetric encryption $k'=k$.