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New results on non-disjoint and classical strong external difference families

Sophie Huczynska, Sophie Hume

TL;DR

A range of new external difference structures are introduced in both the classical and non-disjoint SEDFs, and it is shown how these may be applied to various communications applications.

Abstract

Classical strong external difference families (SEDFs) are much-studied combinatorial structures motivated by information security applications; it is conjectured that only one classical abelian SEDF exists with more than two sets. Recently, non-disjoint SEDFs were introduced; it was shown that families of these exist with arbitrarily many sets. We present constructions for both classical and non-disjoint SEDFs, which encompass all known non-cyclotomic examples for either type (plus many new examples) using a sequence-based framework. Moreover, we introduce a range of new external difference structures (allowing set-sizes to vary, and sets to be replaced by multisets) in both the classical and non-disjoint case, and show how these may be applied to various communications applications.

New results on non-disjoint and classical strong external difference families

TL;DR

A range of new external difference structures are introduced in both the classical and non-disjoint SEDFs, and it is shown how these may be applied to various communications applications.

Abstract

Classical strong external difference families (SEDFs) are much-studied combinatorial structures motivated by information security applications; it is conjectured that only one classical abelian SEDF exists with more than two sets. Recently, non-disjoint SEDFs were introduced; it was shown that families of these exist with arbitrarily many sets. We present constructions for both classical and non-disjoint SEDFs, which encompass all known non-cyclotomic examples for either type (plus many new examples) using a sequence-based framework. Moreover, we introduce a range of new external difference structures (allowing set-sizes to vary, and sets to be replaced by multisets) in both the classical and non-disjoint case, and show how these may be applied to various communications applications.
Paper Structure (10 sections, 37 theorems, 39 equations, 1 figure)

This paper contains 10 sections, 37 theorems, 39 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Background
  3. Definitions and context
  4. Basic structural results
  5. New results on non-disjoint SEDFs and GSEDFs using sequence methods
  6. New results on non-disjoint SEDFs and GSEDFs using group theory
  7. Non-disjoint PSEDF and GPSEDF Constructions in External Direct Products
  8. New results on classical SEDFs and GSEDFs
  9. Applications in communications and cryptography
  10. Conclusion

Key Result

Proposition 2.7

Figures (1)

  • Figure 1: Diagrams of the classical and non-disjoint external difference structures

Theorems & Definitions (91)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Proposition 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 81 more