Circular external difference families, graceful labellings and cyclotomy

TL;DR

This paper introduces circular external difference families (CEDFs) and their strong variants, providing new constructions based on graceful labellings and cyclotomy, with implications for cryptographic schemes like AMD codes.

Contribution

It presents novel constructions of CEDFs using graceful labellings of lexicographic products and explores cyclotomic number theory for approximate solutions, expanding the understanding of difference families.

Findings

CEDFs can be constructed from graceful labellings of lexicographic products.

Strong CEDFs with more than two subsets do not exist.

Cyclotomic numbers enable approximate constructions for cryptographic applications.

Abstract

(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.