Strong external difference families in abelian and non-abelian groups

Strong external difference families (SEDFs) have applications to cryptography and are rich combinatorial structures in their own right. We extend the definition of SEDF from abelian groups to all finite groups, and introduce the concept of equivalence. We prove new recursive constructions for SEDFs and generalized SEDFs (GSEDFs) in cyclic groups, and present the first family of non-abelian SEDFs. We prove there exist at least two non-equivalent (k2 + 1,2,k,1)-SEDFs for every k > 2, and begin the task of enumerating SEDFs, via a computational approach which yields complete results for all groups up to order 24.


Introduction
There has been considerable recent interest in strong external difference families (SEDFs), which have applications to cryptography and are rich combinatorial structures in their own right (see for example [1], [4], [6], [7], [9], [10]).Most of this activity has centred around identifying parameters for which the existence or non-existence of SEDFs can be established, and obtaining constructions, often driven by the application of "classic" techniques such as cyclotomy.Up till now, all SEDFs have been in abelian groups.
We are motivated by the following questions: what is the situation for SEDFs in general finite groups, not simply abelian groups?For given parameters, which groups contain SEDFs with these parameters?What are all possible structures for such SEDFs?How many "different" SEDFs exist with the same parameters?
In this paper, we characterize the order of those groups which contain admissible parameters for SEDFs.We analyse and develop non-existence and existence results for abelian and nonabelian groups, several of which extend known results, and we present the first examples of non-abelian SEDFs (including a construction for an infinite non-abelian family).We introduce the concept of equivalence for EDFs and SEDFs, and begin the task of enumerating SEDFs.We present complete results for all groups up to order 24, underpinned by a computational approach using constraint satisfaction programming.

Strong External Difference Families
External difference families were first introduced in [8] in relation to AMD codes, while strong EDFs were subsequently introduced in [9] to correspond to strong AMD codes.They were defined in abelian groups.
In this paper, we extend the concept of EDF and SEDF to hold in the natural way in any group of order n, whether abelian or non-abelian.The definitions precisely correspond to the originals, with the removal of the word "abelian".Since the differences are defined in terms of ordered pairs, there is no ambiguity in this definition.However, various results proved in the context of abelian EDFs will not hold in the non-abelian context, since their proofs often (implicitly or explicitly) rely on commutativity of the group operation.In previous literature, following the usual convention for abelian groups, additive notation was used for EDFs and SEDFs.However in this paper, we will focus mainly on the non-abelian and general cases, so we will adopt multiplicative notation (unless otherwise stated).Definition 2.1 (External difference family).Let G be an additive group of order n.An (n, m, k, λ)-external difference family (or (n, m, k, λ)-EDF) is a set of m ≥ 2 disjoint ksubsets of G, say A 1 , . . ., A m , such that the multiset M = {xy −1 : x ∈ A i , y ∈ A j , i = j} comprises λ occurrences of each non-identity element of G. Definition 2.2 (Strong external difference family).Let G be an additive group of order n.
For every group G, there is at least one SEDF.This has k = 1 and is often referred to as the trivial SEDF.We give the construction below; this is well-known in the abelian case, but can also be shown to hold in the non-abelian case.
To our knowledge, the existing literature contains only one example of a non-trivial nonabelian EDF (in [5]), and no examples of non-trivial non-abelian SEDFs.

Non-existence results from the parameter equation
The following theorem expresses necessary relationships between the parameters of an EDF or SEDF in any group (abelian or non-abelian).Although originally established (in [8] and [9]) in the context of abelian groups, the conditions remain valid in a non-abelian setting.(ii) n ≤ mk; (2) Necessary conditions for the existence of an (n, m, k, λ)-SEDF in a group G of order n are (i) m ≥ 2; Parameter sets that satisfy these conditions will be called admissible.
Proof.In each case, part (i) is in the definition, while part (ii) is due to the fact that the m k-sets are disjoint.Part (iii) follows in each case from double-counting the ordered pairs which correspond to the external differences; in particular this does not assume commutativity of the group operation.
Analysis of the conditions of Proposition 2.4 can be used to rule-out certain sets of parameters for SEDFs.Since the analysis is purely number theoretical on the parameter equation, these results apply equally to abelian and non-abelian groups.
The following result about λ and k was proved in [4], and holds for any group of order n: Lemma 2.5.For an (n, m, k, λ)-SEDF, either With a view to investigating which groups can contain SEDFs, and reducing the search space for computation, we are particularly interested in non-existence results which focus on the order n of the group.Previous literature has mainly focussed on specifying values of m or λ.
In [6], the following result is given for abelian groups; we may restate it in the general setting: Lemma 2.6.If gcd(k, n − 1) = 1, then a non-trivial (n, m, k, λ)-SEDF does not exist.
The following new result significantly restricts the orders of SEDF-containing groups.
Clearly the RHS must be an integer as n − 1 is, ie k 2 (m − 1) is divisible by λ.Moreover, n − 1 is square-free.If p α is a prime power divisor of k, then it must be a divisor of λ (otherwise λ would be divisible by p 2 ).So λ = ak for some positive integer a, i.e. k divides λ.But by Lemma 2.5, for any SEDF we must have either k = 1 and λ = 1 or k > 1 and λ < k.Since k | λ, we have k ≤ λ and so it is not possible to have k > 1.So k = 1 and λ = 1, ie n = m, and the SEDF is trivial.
We note two corollaries: Corollary 2.8.(i) A group of order n = p + 1, where p is a prime number, contains no non-trivial (n, m, k, λ)-SEDF.
We observe that part(i) of this corollary could also be proved using Lemma 2.6 to deal with the case when k = p.
An integer is said to be square-free if it is divisible by no square other than 1.We may prove a partial converse of Proposition 2.7, in the following sense: for every n − 1 which is not square-free, we can show there exists at least one set of valid non-trivial admissible parameters (n, m, k, λ) for an SEDF.Proposition 2.9.Let n(> 2) be such that n − 1 is not square-free.Then there exists a set of admissible parameters (n, m, k, λ) for a non-trivial SEDF.
Proof.Write n − 1 = a 2 • b, where a 2 is the largest square dividing n − 1 and b ≥ 1.Since n − 1 is not square-free, there exists some prime p such that p 2 | a 2 , so certainly a 2 > 1 and hence a > 1.We will take the tuple of parameters (n, m, k, λ) = (n, b + 1, a, 1).We must show that this satisfies all three conditions of Proposition 2.4 (2), and the non-trivial condition k > 1. Taking λ = 1 reduces the parameter equation from Proposition 2.4 (2)(iii) to n − 1 = k 2 (m − 1), ie a 2 b = k 2 (m − 1).We may set a = k and b = m − 1 to obtain results which satisfy this equation.From above k = a > 1.Since b ≥ 1, m ≥ 2. We must check that n ≥ km; in fact we prove that n − 1 = a 2 b ≥ km = a(b + 1).We have For a given n − 1 which is not square-free, there may of course be other sets of valid parameters for an (n, m, k, λ)-SEDF, beyond those given in the above proof.
Table 1 contains all admissible SEDF parameter sets for n up to 64.These were found computationally.It can be confirmed that the values of n which appear are precisely those for which n − 1 is not square-free, and that for each such n there is a set of the form (n, b + 1, a, 1) as described above.3 Beyond admissible parameters: combinatorial and algebraic approaches In the previous section, we have identified the orders of those groups which can contain SEDFs.However, it is not necessarily the case that an SEDF will exist for all admissible parametersmany of these potential parameters can be ruled-out in certain classes of group using combinatorial or algebraic arguments.
For SEDFs in abelian groups, although the landscape is far from being fully understood, the literature contains numerous results establishing the non-existence of SEDFs for classes of admissible parameters.For the non-abelian group situation, each such result raises various natural questions: does the result also hold for non-abelian groups, either by an adaptation or generalization of the proof in the abelian case, or via a different route?Alternatively, perhaps the result does not hold, in which case there is the possibility of finding SEDFs in non-abelian groups for parameters which cannot be realized in abelian groups.
There are also several existence results for SEDFs in abelian groups, which exhibit specific constructions in specific groups or families of group.We may ask how many different nonequivalent SEDFs exist for a given set of parameters, in both abelian and non-abelian groups.
Of these, (1), ( 2), ( 3) and ( 6) are obtained using character theory in abelian groups, while (4) and (5) employ direct combinatorial arguments which rely on the commutativity of the operation.In [1] and [6], there are further useful results which prove the non-existence of SEDFs with certain technical conditions on n.There is also a result (Theorem 3.7 of [1]) which proves the nonexistence of a (p 2 , m, k, λ)-SEDF for m > 2 in the case when G is cyclic.
In the abelian setting, a characterization for the case when λ = 1 is given by Paterson and Stinson in [9].The following summarizes the main known constructive existence results in abelian groups.Proposition 3.3 (Proposition 1.1, [6]).An (n, m, k, λ)-SEDF exists in the group G in the following cases: ) and n is congruent to 1 mod 4, provided there exists an 4]).In particular, when n is a prime power, take the sets of squares and non-squares in the multiplicative group of the finite field of order n.
Proposition 3.4.Let A = {A 1 , . . ., A m } be a collection of k-sets (k > 1) which partition the non-identity elements of an abelian group G. Then A is a non-trivial (n, m, k, λ)-SEDF if and only if 243,11,22,20) and each A j is a non-trivial regular (243, 22, 1, 2) partial difference set in G for 1 ≤ j ≤ 11.

New results in general groups
Recall that, by Propositions 2.7 and 2.9, the groups with admissible SEDF parameter sets are precisely those of order n where n − 1 is not square-free.In other words, we need only consider n such that n = a 2 b + 1, where a > 1 is the largest square dividing n − 1.
We begin by considering the λ = 1 case in the setting of general groups.Recall that in abelian groups, by Proposition 3.2 , there exists an (n, m, k, 1)-SEDF in an abelian group if and only if n = k 2 + 1 and m = 2, or n = m and k = 1 (the trivial SEDF).In the notation of the proof of Proposition 2.9, this asserts that it is never possible to achieve b > 1 for a non-trivial (a 2 b + 1, b + 1, a, 1)-SEDF, but that a non-trivial (a 2 b + 1, b + 1, a, 1)-SEDF with b = 1 can always be found.
The forward direction is a consequence of the following result (Theorem 3.3) from [9], whose proof relies on the commutativity of the operation.Proposition 3.5.There does not exist an (n, m, k, 1)-SEDF with m ≥ 3 and k > 1.
For the reverse direction, constructions are provided in appropriate abelian groups: • Let G = Z n and take the trivial construction comprising all singleton sets.Unlike the abelian case, there does not exist a non-abelian group of order k 2 + 1 for every value of k -for example there exists no non-abelian group of order k 2 + 1 for k ∈ {2, 4, 6, 8, 10} -and so we could not expect a direct analogue of this result.However, we exhibit an infinite family of non-abelian (k 2 + 1, 2, k, 1)-SEDFs for k odd.To our knowledge, this is the first example of non-abelian SEDFs.
We will consider the dihedral group D n of order n as being generated by the elements s (reflection) and r (rotation by 4π n ), so that and the generators satisfy s 2 = 1, r  We now present the general construction.Theorem 3.7.Let k > 1 be odd.Let G = (D k 2 +1 , * ), the dihedral group of order n = k 2 + 1.There exists a (k 2 + 1, 2, k, 1)-SEDF in G.
Proof.We prove that the stated construction is a SEDF via Table 3, which displays the multiset of external differences A 1 A 2 −1 .In the table, the entry in the row labelled by x and the column labelled by y is xy −1 .Each non-identity group element occurs precisely once in this table.The multiset of external differences A 2 A 1 −1 consists of the inverses of the multiset in the table, and hence also comprises each non-identity element precisely once, which confirms that A = {A 1 , A 2 } is a (k 2 + 1, 2, k, 1)-SEDF.The ordering within the sets A 1 and A 2 in the table has been chosen to emphasise the structure of the situation.
We use the following relations: for x, y ∈ G, the quantity xy −1 equals: • r i−j if x = r i and y = r j ; • sr i−j if x = sr i and y = r j ; • r j−i if x = sr i and y = sr j ; • sr j−i if x = r i and y = sr j .
Note also that r ( k 2 +1 2 ) = e; so for example, ) .To obtain the elements r i (0 2 ) in their natural order, the reader should traverse the first column of the bottom-right quadrant of the table from bottom to top (starting at r and ending at r ( k−1 2 ) ), then traverse the last column of the top-left quadrant from top to bottom (starting at r ( k+1 2 ) and ending at r k ), then return to the bottom-right quadrant at element r k−1 , repeating a similar process until all such elements have been encountered within these two quadrants.Similarly, to obtain the elements sr i (0 2 ) in their natural order, the reader should traverse the top-right quadrant of the table from bottom to top starting at s and ending at sr ( k−1 2 ) , then traverse the bottom-left quadrant of the table from top to bottom starting at sr ( k+1 2 ) and ending at sr k−1 , then move to element sr k in the top-right quadrant, repeating a similar process until all such elements have been encountered within these two quadrants. 2 ) .
given in Theorem 3.7 We now consider the reverse direction: is it the case in general that the only non-trivial SEDFs with λ = 1 are those with m = 2? As noted, the proof of the abelian result relies on the commutativity of the operation.However, we may introduce a new, but very natural, object in the setting of general groups, which allows us to prove a generalization of Proposition 3.5, as well as other results.Definition 3.8.Let G be a group of order n.Let A be a set of m ≥ 2 disjoint k-subsets of G, say A 1 , . . ., A m .We say that A is an (n, m, k, λ)-coEDF if the multiset Definition 3.9.Let G be a group of order n.Let A be a set of m ≥ 2 disjoint k-subsets of G, say A 1 , . . ., A m .We say that A is an (n, m, k, λ)-coSEDF if, for every i, 1 ≤ i ≤ m, the multiset (ii) An (n, m, k, λ)-coSEDF must satisfy precisely the same parameter conditions as an (n, m, k, λ)-SEDF.
For a set X in a group G, we denote Proof.Since the inverse of a group element is unique, the sets of A are disjoint precisely if the sets of Since this multiset comprises λ occurrences of each non-identity element, and the definitions of EDF and coEDF are symmetric in i and j, we have that A −1 is an (n, m, k, λ)-coEDF.A similar argument establishes the reverse implication.(ii) Suppose that A is an (n, m, k, λ)-SEDF.For a given i ∈ {1, . . ., m}, the multiset M i = {xy −1 : x ∈ A i , y ∈ A j , i = j} may be rewritten as Since the multiset M i comprises λ occurrences of each non-identity element, so too does the multiset {v −1 u : u ∈ A −1 i , v ∈ A −1 j }, and hence A −1 is an (n, m, k, λ)-coSEDF.A similar argument establishes the reverse implication.
In an abelian group, it is immediately clear that any translate of an EDF will yield an EDF with the same parameters, and similarly any translate of a SEDF will yield a SEDF.In a non-abelian group, this is less obvious, but in fact it turns out also to hold in this setting.Proof.If G is an abelian group (written additively) then for x, y ∈ G, the differences (g + x) − (g − y) and (x + g) − (y + g) may both be rearranged to x − y by commutativity.Hence the multiset of differences obtained from (g + A i ) − (g + A j ) and (A i + g) − (A j + g) precisely correspond to the multiset of differences from A i − A j .Now suppose G is non-abelian; we consider the differences (gx)(gy) −1 and (xg)(yg) −1 .First consider right translates: by the reversal rule for inverses, for x, y ∈ G we have (xg)(yg) −1 = (xg)(g −1 y −1 ) = x(gg −1 )y −1 = xy −1 .So for any A i g(A j g) −1 (i = j), the multiset of differences is precisely the same as the multiset corresponding to A i (A j ) −1 .
For left translates we have (gx)(gy) −1 = gxy −1 g −1 , the conjugate of xy −1 by g −1 .So the multiset of differences corresponding to (gA i )(gA j ) −1 is the set of conjugates of the differences A i (A j ) −1 by (fixed) g −1 ∈ G.For an EDF A, the union of multisets of such differences for all i = j comprises each non-identity group element λ times; since conjugation by g −1 is an automorphism of the group G (and in particular fixes the identity), the corresponding union for gA also comprises each non-identity group element λ times.Similarly, if A is an SEDF, the multiset of such differences for a fixed i and all j = i comprises each non-identity group element λ times; since conjugation by g −1 is an automorphism of the group G, the corresponding multiset for gA also comprises each non-identity group element λ times.
It can be seen, by similar reasoning to the above, that conjugation by a fixed group element will also send an EDF to an EDF, and a SEDF to a SEDF (this is non-trivial only in the non-abelian case).In fact the same is true for any automorphism of G. Proposition 4.4.Let G be a group, let A = {A 1 , . . ., A m } be an (n, m, k, λ)-EDF (respectively, SEDF) over G and let α be an automorphism of G, Then α(A) = {α(A 1 ), . . ., α(A m )} forms an (n, m, k, λ)-EDF (respectively, SEDF) over G.
Proof.For x ∈ α(A i ) and y ∈ α(A j ), j = i, consider the difference xy −1 .We have x = α(a i ) and y = α(a j ) for some a i ∈ A i , a j ∈ A j .Hence xy −1 = α(a i )(α(a j )) −1 = α(a i a j −1 ) since α is an automorphism.For an EDF A, the union of multisets of external differences A i (A j ) −1 for all i = j comprises each non-identity group element λ times.Since α is an automorphism of the group G (and in particular fixes the identity), for α(A) we have that the union of multisets of external differences α(A i )(α(A j )) −1 = α(A i A −1 j ) (i = j) also comprises each non-identity group element λ times.Similarly, if A is an SEDF, the multiset of such differences for a fixed i and all j = i comprises each non-identity group element λ times; similarly, the corresponding multiset for α(A) also comprises each non-identity group element λ times.

SEDFs in groups of order up to 24
In this section, we discuss the situation for SEDFs in all groups of order up to 24.Computational results were obtained via a constraint satisfaction programming approach; the algorithm was implemented in GAP ( [3]).
Table 4 shows all admissible parameters sets for all groups (abelian and non-abelian) of order at most 24.
In the abelian case, results from the literature rule-out the existence of SEDFs for some of these combinations, resulting in a smaller number of cases to be tested by computational search.In the non-abelian case, it is still possible for some of the "forbidden sets" in the abelian case to be realised by non-abelian SEDFs.However, there are many integers for which no non-abelian groups exists.We first consider the setting of abelian groups.The following parameter sets are ruled-out by results from the literature as follows: • Proposition 3.1(1): (9, 3, 2, 1), (10,3,3,2) This leaves the following parameter sets: (5, 2, 2, 1), (9, 2, 4, 2), (10, 2, 3, 1), (13, 2, 6, 3), (17, 2, 4, 1), (17,2,8,4).For these, we ask: how many non-equivalent SEDFs exist for each parameter set?Which groups do these occur in?How many are already known in the literature?
Parameters Abelian group No of non-equiv.SEDFs Example Case (5, 2, 2, 1) Table 5: Non-equivalent SEDFs in abelian groups of order up to 24 Remark 5.1.We note the absence of an SEDF in the cyclic group of order 9.In [1], it is proved that there does not exist a (p 2 , m, k, λ)-SEDF in an abelian group G with p prime, k > 1, m > 2 and G cyclic; however we know of no such result in the literature for m = 2.
For non-abelian groups of order n ≤ 24, we cannot rule-out any of the parameter sets from Table 4 using results from the existing literature.However, there are no non-abelian groups of orders 5, 9, 13, 17, or 19.Therefore non-trivial SEDFs in groups of order up to 24 are possible only for two orders, 10 and 21.
In each case there is one non-abelian group: the dihedral group D 10 of order 10 and the semi-direct product Z 7 ⋊ Z 3 of order 21.The non-trivial parameter sets for these are (10, 2, 3, 1), (21, 2, 10, 5) and (21, 6, 2, 1).The results are shown in Table 6 In this paper, we have separated non-existence results for SEDFs into those which must hold for SEDFs in any finite group, and those which have been proven to hold only in the abelian setting.It would be good to know which of the latter results also hold in the non-abelian case.
If such a result is not valid for non-abelian groups in general, is it valid for some sub-class?An example of an SEDF with parameters which cannot be realised in an abelian group but can be obtained in a non-abelian setting would be of particular interest; at present we do not have such an example.We have initiated an investigation of the questions we posed at the outset: for given parameters, which groups contain SEDFs with these parameters?What are all possible structures for such SEDFs?How many non-equivalent SEDFs exist with the same parameters?All of these questions open promising avenues for future work.
We may ask whether certain phenomena observable in the computational results for SEDFs in groups of order up to 24, are representative of a more general situation.For example, is it the case that in a group of order p 2 for p an odd prime, no SEDF with m = 2 can exist in the cyclic group of order p 2 ?Does the finite field construction using squares and non-squares yield a unique (up to equivalence) (p 2 , 2, p−1 2 , p−1 4 )-SEDF?Finally, this paper has introduced the systematic study of SEDFs in non-abelian groups, providing the first construction for an infinite family.Further investigation of non-abelian SEDFs is a natural focus for future work.

Proposition 2 . 4 .
(1) Necessary conditions for the existence of an (n, m, k, λ)-EDF in a group G of order n are (i) m ≥ 2;
comprises λ occurrences of each nonzero element of G.Remark 3.10.(i) Clearly if G is an abelian group, coEDFs and coSEDFs coincide precisely with EDFs and SEDFs.

Proposition 4 . 3 .
Let G be a group and let g ∈ G.(i) If A = {A 1 , . . ., A m } is an (n, m, k, λ)-EDF over G,then the left and right translates gA and Ag of A also form (n, m, k, λ)-EDFs over G. (ii) If A = {A 1 , . . ., A m } is an (n, m, k, λ)-SEDF over G, then the left and right translates gA and Ag of A also form (n, m, k, λ)-SEDFs over G.

Table 1 :
All admissible SEDF parameter sets for orders 1 to 64

Table 2
demonstrates the multiset of external differences for the (26, 2, 5, 1)-SEDF over D 26 ; the entry in the row labelled by x and the column labelled by y is xy −1 .All entries of A 1 A 2 5 r 10 sr 2 sr 7 sr 12 e r 8 r 3 sr 2 sr 7 sr 12 r r 9 r 4 sr sr 6 sr 11 r 2 r 10 r 5 s sr 5 sr 4 s sr 8 sr 3 r 2 r 7 r 12 sr sr 9 sr 4 r r 6 r 11 r 5 r 5 r 4 r 3 sr 8 sr 9 r 10 r 10 r 9 r 8 sr 3 sr 4 sr 2 sr 2 sr s r 11 r 12 sr 7 sr 7 sr 6 sr 5 r 6 r 7 sr 12 sr 12 sr 11 sr 10 r r 2

Table 4 :
All admissible SEDF parameters for n ≤ 24