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

$$ \begin{aligned} A_{m+n} A_{m-n}&=a_1(m) b_1(n)+a_2(m) b_2(n),\\ A_{m+n+1} A_{m-n}&=\widetilde a_1(m) \widetilde b_1(n)+\widetilde a_2(m) \widetilde b_2(n), \end{aligned} $$

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