Asymmetric cryptography and hyperelliptic sequences

We study sequences $\{A_n \}_{n =-\infty}^{+\infty}$ of elements of a field $\mathbb F$ that satisfy decompositions of the form
$$ A_{m+n} A_{m-n} = a_1 (m) b_1 (n) + a_2 (m) b_2 (n), $$
where $ a_1, a_2, b_1, b_2: \mathbb Z \to \mathbb F $. The results are used to build analogues of the Diffie – Hellman and El-Gamal algorithms. The discrete logarithm problem is posed in the group $(S, +)$, where the set $S$ consists of fours $S(n) = (A_{n-1},A_n, A_{n+1}, A_{n+2})$, $n\in \mathbb Z$, and $S(n)+S(m) = S(n+m)$.