Public-key cryptosystems eliminate the requirement for pre-shared secret keys by enabling encryption with a publicly disclosed key and decryption with a corresponding private key. In this article we generalize the public-key cryptosystems to ternary algebraic structures, with particular attention to ElGamal as a representative family. We introduce the necessary algebraic background for nonderived ternary structures, including special elements, ternary group rings, and a matrix ternarization procedure that maps binary rings and group rings to antidiagonal symbolic matrices closed under ternary multiplication. Building on these foundations, we formulate a ternary analogue of the ElGamal three-step protocol (key generation, ephemeral encryption, and decryption via querelements) and derive explicit ternary power and querelement formulas that enable correct decryption. Concrete instantiations and numerical examples over a ternary fraction field, a matrix-ternarized finite group ring, and a finite \((6,3)\)-ring (field) validate the construction and illustrate admissible word-length quantization and cycle behaviour of ternary powers. The ternary framework highlights two practical advantages: richer algebraic structure (querelements replace binary inverses) that increases algebraic complexity for attackers, and higher information density (matrix ternarization transfers paired/plaintext vectors). Formal hardness assumptions, optimized parameter choices, and comprehensive security and performance analyses remain necessary future work.

翻译：公钥密码体制通过使用公开密钥进行加密、对应私钥进行解密，消除了预先共享密钥的需求。本文将公钥密码体制推广至三元代数结构，特别以ElGamal作为代表性家族进行研究。为非导出型三元结构引入必要的代数基础，包括特殊元素、三元群环以及一种矩阵三元化过程——该过程将二元环与群环映射为在三元乘法下封闭的反对角符号矩阵。在此基础上，构建ElGamal三步协议（密钥生成、临时加密及基于拟元素的解密）的三元类比，并推导出显式的三元幂和拟元素公式以实现正确解密。通过三元分式域、矩阵三元化有限群环及有限(6,3)-环（域）上的具体实例和数值验证，验证该构造的可行性，并阐释可接受的字长量化与三元幂的循环特性。该三元框架凸显了两项实际优势：更丰富的代数结构（拟元素替代二元逆元）增加了攻击者的代数复杂度，以及更高的信息密度（矩阵三元化传输配对/明文向量）。严格的安全性假设、优化的参数选择以及全面的安全性与性能分析仍需未来进一步研究。