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If $v1\neq v2$ then Alice and Bob can not agree to same key. So $v1$ and $v2$ must be same. In this case your problem is Diffie Hellman key exchange problem that are well known in cryptography. For more detail you can search "Diffie Hellman key exchange" in google. Edit: If your computation is not in $\mathbb Z_n$ then this system is not secure. You can't ...
In the context of the papers you reference, Generalized Mersenne prime numbers and Pseudo-Mersenne prime numbers are indeed two different things. A Pseudo-Mersenne prime number has the form $2^{\alpha}-\gamma$ for a small integer $\gamma \gt 0$. The term Generalized Mersenne prime number is defined by example in the referenced paper, and the examples given ...