  Key transport protocols are designed to transfer a secret key from an
initiating principal to other entities in a network. The three-pass protocol is
a key transport protocol developed by Adi Shamir in 1980 where Alice wants to
transport a secret message to Bob over an insecure channel, and they do not
have any pre-shared secret information. In this paper, we prove the
impossibility of secret key transportation from a principal to another entity
in a network by using the three pass protocol over public Abelian groups. If it
were possible to employ public Abelian groups to implement the three-pass
protocol, we could use it in post-quantum cryptography for transporting keys
providing information theoretic security without relying on any computationally
difficult problem.
