TLS is using 3 way handshake. Isn't that possible to use such 3 way handshake protocol to share a secret without using a public key scheme, for instance as follows: Let A, B be private secrets of Alice and Bob respectively, where A,B are elements in a commutative groups. Alice wants to share the secret X with Bob, and sends Bob AX. Bob multiplies it by his secret B to create BAX, and sends it back to Alice, that Multiplied it By 1/A, creating BX, that is sent back to Bob, that multiplies it by 1/B, getting X, the secret Alice wanted to share. Is there a principle problem with this scheme?
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The attacker sees AX and BAX, and just divides them to compute B. Then they see BX and divide again to compute X.
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$\begingroup$ Lets say A, B are nxn matrix, while the dimension of X is n. AX is a vector of dimension n, and so BAX. So I think that he cannot divide BAX by AX, nor to solve for the components of A, B, as there are not enough equations $\endgroup$ Oct 27, 2020 at 19:40
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1$\begingroup$ @Evgen Vaknin: what about substituting "divides (them)" by "solves that linear system"? $\endgroup$– fgrieu ♦Oct 29, 2020 at 17:32