Circular External Difference Families (CEDF)

• CEDF is a collection of disjoint, equal-size subsets of a finite group whose cyclic external differences cover every nonzero element with a fixed multiplicity.
• CEDFs directly correspond to optimal weak circular AMD codes, enabling unconditionally secure, non-malleable threshold schemes through precise combinatorial design.
• Research on CEDFs spans explicit cyclotomic and graph-labelling constructions, existence and nonexistence results in cyclic and noncyclic groups, and several open combinatorial problems.

A circular external difference family (CEDF) is a collection of pairwise disjoint equal-size subsets of a finite group whose prescribed cyclic external differences cover every nonzero group element with a fixed multiplicity. In the form introduced by Veitch and Stinson, CEDFs arose as the exact combinatorial structures underlying optimal weak circular algebraic manipulation detection (AMD) codes and hence unconditionally secure non-malleable threshold schemes (Veitch et al., 2023). Subsequent work by Paterson and Stinson, Huczynska, Jefferson and McCartney, and others developed existence and non-existence theory, graceful-labelling and cyclotomic constructions, digraph-defined generalizations, and extensions to noncyclic abelian and nonabelian groups (Paterson et al., 2023, Huczynska et al., 29 Apr 2025).

1. Formal definition and parameter constraints

Let $G$ be a finite additive abelian group of order $n$. Fix integers $m \ge 2$, $1 \le c \le m-1$, and positive integers $\ell,\lambda$. Suppose

$A_0,A_1,\dots,A_{m-1}\subseteq G$

are pairwise disjoint and satisfy $|A_j|=\ell$ for all $j$. For two disjoint subsets $X,Y\subseteq G$, define the multiset of external differences

$D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$

Then $n$0 is an $n$1–$n$2-circular external difference family if

$n$3

as multisets; equivalently, every nonzero $n$4 appears exactly $n$5 times among all differences $n$6 with $n$7 and $n$8 (Veitch et al., 2023).

A necessary size condition is

$n$9

In the common case $m \ge 2$0, this forces $m \ge 2$1 (Veitch et al., 2023). When $m \ge 2$2, the family is called cyclic (Burgess et al., 2 Sep 2025). The terminology “CEDF” is often used for the case $m \ge 2$3.

A broader formulation, developed in group-ring language, allows a set of shifts $m \ge 2$4. In multiplicative notation, an $m \ge 2$5–$m \ge 2$6–CEDF satisfies

$m \ge 2$7

in $m \ge 2$8 (Wu et al., 2023). This places ordinary CEDFs, $m \ge 2$9-CEDFs, and related external difference families in a common algebraic framework.

2. Equivalence with circular AMD codes and non-malleable threshold schemes

The central structural theorem identifies CEDFs with optimal circular AMD codes. In the weak circular AMD game, the adversary picks a nonzero $1 \le c \le m-1$0; a source index $1 \le c \le m-1$1 is chosen uniformly; a codeword $1 \le c \le m-1$2 is drawn uniformly; and the adversary wins if $1 \le c \le m-1$3. Such a code is $1 \le c \le m-1$4-optimal if the maximum success probability is

$1 \le c \le m-1$5

Veitch and Stinson proved that $1 \le c \le m-1$6 is an $1 \le c \le m-1$7-optimal weak $1 \le c \le m-1$8-circular $1 \le c \le m-1$9-AMD code if and only if it is an $\ell,\lambda$0–$\ell,\lambda$1-CEDF with

$\ell,\lambda$2

In particular, in the tight case $\ell,\lambda$3, CEDFs are exactly the optimal weak circular AMD codes (Veitch et al., 2023).

This equivalence feeds directly into secret sharing. On the secret space $\ell,\lambda$4, define

$\ell,\lambda$5

Feeding an $\ell,\lambda$6-optimal $\ell,\lambda$7-circular AMD code into Shamir’s scheme yields a $\ell,\lambda$8 threshold scheme that is $\ell,\lambda$9-non-malleable with respect to $A_0,A_1,\dots,A_{m-1}\subseteq G$0 (Veitch et al., 2023). In the threshold construction, the dealer chooses a secret $A_0,A_1,\dots,A_{m-1}\subseteq G$1 uniformly, picks $A_0,A_1,\dots,A_{m-1}\subseteq G$2 uniformly, distributes shares of $A_0,A_1,\dots,A_{m-1}\subseteq G$3 via Shamir over $A_0,A_1,\dots,A_{m-1}\subseteq G$4, and reconstructs by recovering $A_0,A_1,\dots,A_{m-1}\subseteq G$5 from $A_0,A_1,\dots,A_{m-1}\subseteq G$6 shares and outputting the unique $A_0,A_1,\dots,A_{m-1}\subseteq G$7 such that $A_0,A_1,\dots,A_{m-1}\subseteq G$8 (Veitch et al., 2023).

The cryptographic interpretation is that tampering attempts to move an encoding of source $A_0,A_1,\dots,A_{m-1}\subseteq G$9 into an encoding of source $|A_j|=\ell$0. Any tampering on a single share shifts $|A_j|=\ell$1, and winning the $|A_j|=\ell$2-malleability game requires $|A_j|=\ell$3 to land in $|A_j|=\ell$4; by $|A_j|=\ell$5-optimality this succeeds with probability at most $|A_j|=\ell$6 (Veitch et al., 2023). Paterson and Stinson restated the same equivalence for the case $|A_j|=\ell$7: an $|A_j|=\ell$8-optimal circular weak $|A_j|=\ell$9-AMD code is equivalent to an $j$0-CEDF (Paterson et al., 2023).

3. Existence and non-existence landscape

The known existence theory is most complete in the tight cyclic case $j$1, where $j$2. Several parameter regimes are settled, while others remain open.

Parameter regime	Cyclic status	Source
even $j$3, any $j$4	$j$5-CEDF exists	(Paterson et al., 2023)
$j$6 both odd	no cyclic $j$7-CEDF	(Paterson et al., 2023)
odd $j$8, $j$9	cyclic $X,Y\subseteq G$0-CEDF exists	(Burgess et al., 2 Sep 2025)
$X,Y\subseteq G$1, even $X,Y\subseteq G$2	cyclic $X,Y\subseteq G$3-CEDF exists	(Burgess et al., 2 Sep 2025)
$X,Y\subseteq G$4, shift $X,Y\subseteq G$5	$X,Y\subseteq G$6–2-CEDF exists in $X,Y\subseteq G$7	(Angus et al., 5 Mar 2026)

For cyclic groups, Paterson and Stinson proved that if $X,Y\subseteq G$8 is even and $X,Y\subseteq G$9 then an $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$0-CEDF exists, while if $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$1 and $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$2 are both odd then no such cyclic CEDF exists (Paterson et al., 2023). The odd-odd non-existence result is described in later work as a parity obstruction: the cyclic symmetry plus odd-odd parameters create a parity-sum contradiction (Burgess et al., 2 Sep 2025).

The 2025 paper “On circular external difference families” sharpened the cyclic existence picture when $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$3 is odd and $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$4 is even. It constructed cyclic $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$5-CEDFs for every odd $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$6, and cyclic $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$7-CEDFs for every even $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$8 (Burgess et al., 2 Sep 2025). As stated there, this fills the last gaps for $D(X,Y)=\{\,x-y:x\in X,\ y\in Y\,\}.$9 in the cyclic case and resolves the entire $n$00 row for even $n$01.

A separate line of work provides a uniform explicit cyclic family for all even $n$02 in $n$03, avoiding the earlier case-splitting according to $n$04 or $n$05 (Huczynska et al., 29 Apr 2025). More recently, graph-labelling methods produced the first explicit construction for an infinite family of $n$06-CEDFs, achieving all parameter sets for $n$07–$n$08-CEDFs with $n$09 sets (Angus et al., 5 Mar 2026).

Open cyclic cases remain. For general odd $n$10 and even $n$11, cyclic constructions beyond $n$12 for all odd $n$13 and beyond $n$14 for all even $n$15 remain open (Burgess et al., 2 Sep 2025).

4. Construction paradigms

Two construction paradigms dominate the literature: cyclotomy in finite fields and graph-labelling methods.

Veitch and Stinson’s field-based construction begins with a finite field $n$16 such that $n$17, a primitive element $n$18, and the subgroup

$n$19

of order $n$20. For $n$21, define

$n$22

Then $n$23 are disjoint $n$24-subsets of $n$25, and they form a $n$26–$n$27-CEDF if and only if

$n$28

is a full set of coset representatives of $n$29 in $n$30; in that case $n$31 (Veitch et al., 2023). The specialization $n$32 gives a particularly concrete criterion: if $n$33 and $n$34 is primitive, then the cosets

$n$35

form a $n$36–$n$37-CEDF if and only if $n$38 is a quadratic non-residue (Veitch et al., 2023).

This cyclotomic approach extends to arbitrary circular step $n$39. In the notation of Wang and coauthors, if $n$40, $n$41, $n$42, and $n$43, then $n$44 is a $n$45–$n$46-CEDF if and only if

$n$47

is a complete set of coset representatives of $n$48; for $n$49, this is equivalent to requiring $n$50 to be a nonsquare (Wu et al., 2023). The same paper also treats multi-shift $n$51-CEDFs and a lifting theorem from $n$52 to $n$53 when $n$54 (Wu et al., 2023).

The second major paradigm uses graceful labellings. Paterson and Stinson constructed CEDFs from $n$55-valuations of the lexicographic product graph $n$56. If a graph has an $n$57-valuation, then its blow-up by $n$58 does as well, and the labels read modulo $n$59 around the directed cycle produce an $n$60-CEDF (Paterson et al., 2023). This viewpoint has since been generalized to digraph-defined EDFs, in which a small labelled digraph $n$61 determines which difference multisets are aggregated; a $n$62-CEDF is exactly the case where $n$63 is the directed cycle $n$64 (Huczynska et al., 29 Apr 2025).

The 2026 graph-labelling framework broadens $n$65-valuations to near $n$66-valuations and oriented near $n$67-valuations. Combined with graph blow-up, this yields digraph-defined EDFs and, in particular, the infinite family of $n$68-CEDFs with $n$69 and $n$70 (Angus et al., 5 Mar 2026).

5. Representative examples and extensions beyond cyclic abelian groups

A standard small example is the $n$71–$n$72-CEDF in $n$73:

$n$74

Its three consecutive difference multisets are

$n$75

$n$76

$n$77

whose union is exactly $n$78. Hence $n$79, and indeed $n$80 (Veitch et al., 2023). The example was originally built via the primitive root $n$81 (Veitch et al., 2023).

Paterson and Stinson also exhibited small cyclic examples such as the $n$82-CEDF in $n$83

$n$84

and the singleton example $n$85 in $n$86, given by the cyclic order

$n$87

(Paterson et al., 2023).

A major development is that CEDFs are not confined to cyclic groups. Huczynska, Jefferson and McCartney constructed the first infinite family of $n$88-CEDFs in a noncyclic abelian group,

$n$89

for every odd $n$90 (Huczynska et al., 29 Apr 2025). They also produced, by computational search, the first CEDF in a nonabelian group: a $n$91-CEDF in the dihedral group

$n$92

with blocks

$n$93

(Huczynska et al., 29 Apr 2025).

These examples clarify a common misconception. The statement “if $n$94 and $n$95 are both odd, no CEDF exists” is only a cyclic non-existence theorem. The noncyclic abelian family in $n$96 shows that the odd-odd nonexistence hurdle in the cyclic case can be overcome by moving to a slightly larger noncyclic group, and the dihedral example shows that nonabelian CEDFs do exist (Huczynska et al., 29 Apr 2025).

6. Strong variants, generalizations, and open problems

The strong analogue of a CEDF is a strong circular external difference family (SCEDF). If $n$97 is an $n$98-SCEDF, then for each $n$99 one requires

$m \ge 2$00

Thus each adjacent pair alone must realize all nonzero differences with multiplicity $m \ge 2$01, and necessarily

$m \ge 2$02

(Paterson et al., 2023).

Although SCEDFs are a natural strengthening, the existence theory is degenerate. Paterson and Stinson showed that a family is an $m \ge 2$03-SCEDF if and only if each adjacent pair $m \ge 2$04 is an $m \ge 2$05-SEDF, and since no SEDF on more than two sets can exist in an abelian group, it follows that no SCEDF with $m \ge 2$06 exists (Paterson et al., 2023). Wang and coauthors sharpened this into the statement that all SCEDFs are exactly the cyclic re-packagings of two-set strong EDFs; there are no non-trivial SCEDFs (Wu et al., 2023).

This non-existence does not eliminate the strong circular AMD viewpoint. Paterson and Stinson used cyclotomic numbers to construct circular strong AMD codes whose success probability is

$m \ge 2$07

for cyclotomic classes $m \ge 2$08 of order $m \ge 2$09 in $m \ge 2$10, where the $m \ge 2$11-optimal bound would be $m \ge 2$12 (Paterson et al., 2023). For $m \ge 2$13, these codes do not reach the $m \ge 2$14-optimal bound, but they provide near-optimal strong circular AMD codes (Paterson et al., 2023).

Several open problems remain active. In the cyclic setting, for general odd $m \ge 2$15 and even $m \ge 2$16, the existence problem is open outside the families $m \ge 2$17 and $m \ge 2$18 settled in 2025 (Burgess et al., 2 Sep 2025). The more general notion of a $m \ge 2$19-CEDF is equivalent to the $m \ge 2$20 case whenever $m \ge 2$21, but systematic constructions for general $m \ge 2$22 are not yet known (Burgess et al., 2 Sep 2025). In the graph-labelling framework, open directions include extending the approach to non-bipartite graphs, obtaining $m \ge 2$23 families directly, and finding valuations for further graph families such as grids and higher $m \ge 2$24-CEDFs (Angus et al., 5 Mar 2026).

Across these developments, one conclusion has remained stable: circular external difference families provide the exact combinatorial structures for optimal weak $m \ge 2$25-circular AMD codes, and those AMD codes yield unconditionally secure, $m \ge 2$26-non-malleable threshold schemes with respect to the additive relation $m \ge 2$27 (Veitch et al., 2023).