Existence of sequences satisfying bilinear type recurrence relations

We study sequences $\left\{A_n\right\}_{n=-\infty}^{+\infty}$ of elements of an arbitrary field $\mathbb{F}$ that satisfy decompositions of the form
$$ \begin{aligned}& A_{m+n} A_{m-n}=a_1(m) b_1(n)+a_2(m) b_2(n),\\ & A_{m+n+1} A_{m-n}=\tilde a_1(m) \tilde b_1(n)+\tilde a_2(m) \tilde b_2(n), \end{aligned} $$
where $a_1,a_2,b_1,b_2\colon \mathbb{Z}\to\mathbb{F}$. We prove some results concerning the existence and unique ness of such sequences. The results are used to construct analogs of the Diffie–Hellman and ElGamal cryptographic algorithms. The discrete logarithm problem is considered in the group $(S,+)$, where the set $S$ consists of quadruples $S(n)=(A_{n-1},A_n, A_{n+1}, A_{n+2})$, $n\in\mathbb{Z}$, and $S(n)+S(m)=S(n+m)$.