This article introduces a novel cryptographic paradigm based on nonderived polyadic algebraic structures. Traditional cryptosystems rely on binary operations within groups, rings, or fields, whose well-understood properties can be exploited in cryptanalysis. To overcome these vulnerabilities, we propose a shift to polyadic rings, which generalize classical rings by allowing operations of higher arity: an $m$-ary addition and an $n$-ary multiplication. The foundation of our approach is the construction of polyadic integers -- congruence classes of ordinary integers endowed with such $m$-ary and $n$-ary operations. A key innovation is the parameter-to-arity mapping $Φ(a,b)=(m,n)$, which links the parameters $(a,b)$ defining a congruence class to the specific arities required for algebraic closure. This mapping is mathematically intricate: it is non-injective, non-surjective, and multivalued. This complex, non-unique relationship forms the core of the proposed cryptosystem's security. We present two concrete encryption procedures that leverage this structure by encoding plaintext within the parameters of polyadic rings and transmitting information via polyadically quantized analog signals. In one method, plaintext is linked to the additive arity $m_{i}$ and secured using the summation of such signals; in the other, it is linked to a ring parameter $a_{i}$ and secured using their multiplication. In both cases, the "quantized" nature of polyadic operations generates systems of equations that are straightforward for a legitimate recipient with the correct key but exceptionally difficult for an attacker without it. The resulting framework promises a substantial increase in cryptographic security. This work establishes the theoretical foundation for this new class of encryption schemes and highlights their potential for constructing robust, next-generation cryptographic protocols.

翻译：本文提出了一种基于非导出多元代数结构的新型密码学范式。传统密码系统依赖于群、环或域内的二元运算，这些结构的已知性质可能被密码分析所利用。为克服这些脆弱性，我们建议转向多元环——通过允许更高元数的运算（$m$元加法与$n$元乘法）来推广经典环。该方法的基础在于构造多元整数：赋予此类$m$元与$n$元运算的普通整数同余类。核心创新在于参数到元数的映射$\Phi(a,b)=(m,n)$，该映射将定义同余类的参数$(a,b)$与代数封闭所需的具体元数相关联。该映射在数学上具有复杂性：它非单射、非满射且为多值函数。这种复杂且非唯一的关系构成了所提密码系统安全性的核心。我们提出了两种具体的加密流程，通过将明文编码于多元环的参数中，并利用多元量化模拟信号传输信息来利用该结构。在第一种方法中，明文与加法元数$m_{i}$相关联，并通过此类信号的求和进行加密；在第二种方法中，明文与环参数$a_{i}$相关联，并通过信号的乘积进行加密。两种情况下，多元运算的“量化”特性所生成的方程组对持有正确密钥的合法接收者而言易于求解，但对未持有密钥的攻击者则异常困难。所构建的框架有望大幅提升密码安全性。本工作为这类新型加密方案奠定了理论基础，并凸显了其在构建稳健的下一代密码协议方面的潜力。