What happen if a asymmetric crypto-system deals with only one key Assume there is a crypto algorithm that deals with matrices to encrypt and decrypt. Regardless of the specification of such algorithm, what if the algorithm assumes that two parties can securely agree on a matrix and use it along with its inverse to encrypt and decrypt? What I am trying to understand, is that, such thing plays excatly the same role of asymmetric/public key cryptosystem. However, in this algorithm (unknown), there is only one key to encrypt and decrypt messages and is known to two parties who have never met before (agreed on it through a public channel). I would like to know in point of security, and consideration of public key cryptoo

Thanks in advance
Well, each party would know that the other party uses the same key, because they would probably have the same public key. 

Now, if some person I have never met before would use the same key as me, I would go into paranoid panic mode. That person could decrypt all messages that were only meant to be decrypted by me (and of course vice versa).

However, if I only use my public-private key pair for communication with this one person, and nobody else, it would be not that bad, because the messages are no secret for either of us anyway. BUT: you never know if some stranger takes care of his passwords the same way as you do. An attacker could capture the pub keys, see that you both use the same key, and in case the other person is a little sloppy, and the attacker would get hands of the other persons keys, he could not only decrypt messages meant for your partner, but also messages meant only for your eyes.

When it comes to security, it is always good to be as paranoid as possible, if it does not take that much extra effort, and avoiding to use the same key as someone else (as unlikely as this is) is definitely something that is easily avoidable, with quite a big impact. (And while there might be even fancier attacks for scenarios like that, these are the most obvious points, which are enough to say 'don't do that')