# Impossibility of Three Pass Protocol using Public Abelian Groups

## TL;DR

This paper proves that the three-pass protocol cannot securely transport secret keys over public Abelian groups, highlighting fundamental limitations for such cryptographic schemes in post-quantum contexts.

## Contribution

It establishes the impossibility of implementing the three-pass protocol over public Abelian groups, clarifying theoretical constraints in key transport protocols.

## Key findings

- Proves the impossibility of three-pass key transport over public Abelian groups.
- Highlights limitations for post-quantum cryptography using this protocol.
- Provides theoretical insights into the structure of key transport protocols.

## Abstract

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.