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Is MITM attack theoretically possible after public/private key pair exchange is done successfully. Say Alice and Bob have genuine public keys of each other. Is MITM possible further during authentication phase between Alice and Bob?

Please find protocol below:

Alice and Bob have key pairs.Assume Alice and Bob have the correct public key of the other party.(i.e. key exchange is already done successfully.)

Now,

Ia and Ib are the identities of Alice and Bob.
pu_a and pu_b are the public key of Alice and Bob.
pr_a and pr_b are the private key of Alice and Bob
No_a and No_a’ are the nonce generated by Alice; No_b is the nonce generated by Bob
Sig_a and Sig_b are the signatures generated by Alice and Bob respectively by encrypting the message content with their private key:
Sig_a = Encpr_a(No_a’, No_b, Ib); Sig_b = Encpr_b(No_b, No_a, Ia)
Alice verifies Bob by decrypting Sig_b with pu_b, and checking if No_a is there; while Bob verifies Alice in the same way.

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Can you clarify which protocol you mean? In general: If Alice and bob have the public key of the other side, then authentication is possible. But it depends on the protocol if it is secure against MITM. –  tylo Mar 6 at 10:06
    
Related to this other question –  fgrieu Mar 6 at 10:35
    
@tylo I have updated the question with protocol –  user40349 Mar 6 at 19:17
    
The main issue is that you are trying have Alice and Bob generate Sig_a and $\hspace{1.6 in}$ Sig_b by "encrypting the message content with their private key" rather than by $\hspace{1.3 in}$ "signing the message content with their private key". $\:$ –  Ricky Demer Mar 6 at 22:09
    
@RickyDemer Can you please elaborate? What is the exact difference? –  user40349 Mar 6 at 22:44

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