Higher power polyadic group rings
For mathematicians and cryptographers, this provides a novel algebraic structure that could enable new cryptographic schemes, though the work is purely theoretical and incremental in nature.
This paper introduces polyadic group rings, a higher-arity generalization of classical group rings, and develops their theory including operations, quantization conditions, and algebraic properties. It establishes a new foundation with potential applications in cryptography and coding theory.
This paper introduces and systematically develops the theory of polyadic group rings, a higher arity generalization of classical group rings $\mathcal{R}[\mathsf{G}]$. We construct the fundamental operations of these structures, defining the $\mathbf{m}_{r}$-ary addition and $\mathbf{n}_{r} $-ary multiplication for a polyadic group ring $\mathrm{R}^{[\mathbf{m} _{r},\mathbf{n}_{r}]}=\mathcal{R}^{[m_{r},n_{r}]}[\mathsf{G}^{[n_{g}]}]$ built from a nonderived $(m_{r},n_{r})$-ring and a nonderived $n_{g}$-ary group. A central result is the derivation of the "quantization" conditions that interrelate these arities, governed by the arity freedom principle, which also extends to operations with higher polyadic powers. We establish key algebraic properties, including conditions for total associativity and the existence of a zero element and identity. The concepts of the polyadic augmentation map and augmentation ideal are generalized, providing a bridge to the classical theory. The framework is illustrated with explicit examples, solidifying the theoretical constructions. This work establishes a new foundation in ring theory with potential applications in cryptography and coding theory, as evidenced by recent schemes utilizing polyadic structures.