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 $A_{m+n} A_{m-n}=a_1(m) b_1(n)+a_2(m) b_2(n)$, $A_{m+n+1} A_{m-n}=\tilde a_1(m) \tilde b_1(n)+\tilde a_2(m) \tilde b_2(n)$, where $a_1,a_2,b_1,b_2\colon \mathbb{Z}\to\mathbb{F}$. We prove some results concerning the existence and uniqueness 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 $S(n)+S(m)=S(n+m)$.

nonlinear recurrence sequences, Somos sequences, public-key cryptography