  (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.
