(Strong) circular external difference families (which we denote as CEDFs and SCEDFs) can be used to construct nonmalleable threshold schemes. They are a variation of (strong) external difference families, which have been extensively studied in recent years. We provide a variety of constructions for CEDFs based on graceful labellings ($\alpha$-valuations) of lexicographic products $C_n \boldsymbol{\cdot} K_{\ell}^c$, where $C_n$ denotes a cycle of length $n$. SCEDFs having more than two subsets do not exist. However, we can construct close approximations (more specifically, certain types of circular algebraic manipulation detection (AMD) codes) using the theory of cyclotomic numbers in finite fields.

翻译：（强）循环外部差族（记为CEDF和SCEDF）可用于构造不可窜改门限方案。作为近年来被广泛研究的（强）外部差族的变体，本文基于优美标号（$\alpha$-赋值）理论，为字典序乘积图$C_n \boldsymbol{\cdot} K_{\ell}^c$（其中$C_n$表示长度为$n$的圈）提供了CEDF的多种构造方法。当子集数量超过两个时，SCEDF并不存在。然而，利用有限域中的分圆数理论，我们可以构造其近似形式（更具体地说，是特定类型的循环代数操纵检测（AMD）码）。