(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 $n$. We do not have any nontrivial examples of SCEDFs. 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）循环外差分族（简称CEDF与SCEDF）可用于构造不可延拓门限方案。作为近年来被广泛研究的（强）外差分族的变体，本文基于字典序乘积$C_n \boldsymbol{\cdot} K_{\ell}^c$的优美标号（$\alpha$-估值）给出了CEDF的多种构造方法，其中$C_n$表示长度为$n$的循环图。尽管目前尚未获得任何非平凡SCEDF实例，但利用有限域中的分圆数理论，我们构造了高度逼近方案（具体而言，即特定类型的循环代数篡改检测码）。