We study sequences \(\left\{A_n\right\}_{n=-\infty}^{+\infty}\) of elements of an arbitrary field \(\mathbb{F}\) that satisfy decompositions of the form

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 \(S(n)+S(m)=S(n+m)\).

我们研究满足分解的任意域\(\mathbb{F}\ ) 的元素的序列 \(\left\{A_n\right\}_{n=-\infty}^{+\infty}\)形式

其中\(a_1,a_2,b_1,b_2\colon \mathbb{Z}\to\mathbb{F}\)。我们证明了有关此类序列的存在性和唯一性的一些结果。结果用于构建 Diffie-Hellman 和 ElGamal 加密算法的类似物。离散对数问题在群\((S,+)\)中考虑，其中集合\(S\)由四元组组成\(S(n)=(A_{n-1},A_n, A_{n +1}、A_{n+2})\)、\(n\in\mathbb{Z}\)和\(S(n)+S(m)=S(n+m)\)。