Example: $X$ and $Y$ have the same asymmetric key pair. $X$ communicates with $A$ and gives it's PubKey to $A$.

So, $A$ has the PubKey of $X$ which also happens to be the PubKey/PrivKey of $Y$. Now intentionally or by chance, can $A$ intercept and read the signed messages of $Y$?

If this can happen, does this not make the PKI vulnerable?

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    $\begingroup$ It is not really clear what you are asking. Can you edit to add more detail? $\endgroup$ – mikeazo Sep 18 '17 at 13:07
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    $\begingroup$ "Can" is ambiguous: is that from the standpoint of having the PKI work (and then is that from a mathematical or security/legal standpoint?). Or are you asking if this can happen by chance/or accident? $\endgroup$ – fgrieu Sep 18 '17 at 13:16
  • $\begingroup$ Sure, but what is the chance? $\endgroup$ – Dr_Bunsen Sep 18 '17 at 13:55
  • $\begingroup$ Are the keypair in question used for encryption or signing? $\endgroup$ – DannyNiu Sep 19 '17 at 6:15
  • $\begingroup$ what will be the case for each of these transactions i.e. signing and encryption? $\endgroup$ – Yash Sep 19 '17 at 6:33

If this can happen, does this not make the PKI vulnerable?

if by chance they tend to generate THE SAME key (p,q) and private parameter (d), indeed they could read/sign each other's messages

However - it is the same probability as with guessing any key - choosing a random value from the whole set, there's a small probability the chosen value will be the key. The keyspace (say 128 bit for symmetric / 2048 bit for assymetric keys) should be big enough that the probability is negligible.

Now A has the PuKey of X which also happens to be the PuKey/PrKey of Y

A shall have no access to the private key of Y. As it happens, that's not problem of PKI, that's problem of Y not protecting its key

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