Transversal Difference Numbers in Finite Abelian Quotients

Abstract: Given (H\leq G) finite abelian groups, a transversal (T\subseteq G) for (G/H) has fixed size (|G/H|), but its ambient difference support (D(T)=T-T) can vary with the embedding of (H) in (G). We call $δ(G,H)=\min_T |D(T)|$ the transversal difference number of the pair ((G,H)). This invariant is related to finite abelian factorisation, tiling complements, and small-sumset questions, and is motivated by recent work regarding ambient Galois labels in CRT transforms for cyclotomic-subfield homomorphic encryption. We prove various results regarding this invariant, including a general lower bound $δ(G,H)\geq 2|G/H|-m(G,H),$ where (m(G,H)) is the largest order of a subgroup of (G) disjoint from (H). The bound is sharp for cyclic quotients, and Kneser's theorem gives a cross-transversal estimate leading to exact product families with one nonsplit cyclic coordinate and arbitrary split factors. These results isolate the first genuinely new residual obstruction, namely the same-prime square plane [ G=(\mathbb Z/p^2\mathbb Z)^2,\qquad H=pG. ] For odd (p), this case is the technical core of the paper. Here transversals are graphs of functions (\mathbb F_p^2\to \mathbb F_p^2), and (D(T)) decomposes into carry-corrected finite-field derivative images. We conjecture that [ δ(G,H)=(2p-1)^2 ] for all odd primes (p), prove the unconditional lower bound (3p^2-p-1), and give small-prime, probabilistic, and fixed-polynomial evidence for the conjecture.

• The paper introduces the invariant δ(G, H) as the minimal size of the ambient difference support for any transversal of G/H, providing sharp lower bounds.
• It employs combinatorial and algebraic methods, using tools like Kneser’s theorem and singleton reduction to analyze cyclic and product cases.
• The study reveals new non-cyclic obstructions with practical implications for efficient fully homomorphic encryption and additive combinatorics.

Transversal Difference Numbers in Finite Abelian Quotients: An Expert Assessment

Introduction and Context

The paper centers on the transversal difference number $\delta(G, H)$ for a pair $(G, H)$ where $H \leq G$ are finite abelian groups. For a transversal $T$ of the quotient $G/H$, the object of study is the cardinality of its “ambient difference support” $D(T) = T - T$ in $G$. The minimal such support—in other words, $\delta(G, H) = \min_{T} |D(T)|$ as $T$ ranges over all transversals for $G/H$—is a new group-theoretic invariant relating to tiling, factorization, and additive combinatorics. This invariant is sharply sensitive not just to the isomorphism class of $(G, H)$0 but to the embedding $(G, H)$1 itself.

A strong motivation is provided through applications in computational number theory and cryptography, especially the structure arising in devising efficient fully homomorphic encryption (FHE) protocols over cyclotomic and abelian rings, where the reduction of automorphism/key-switching cost maps precisely to the transversal difference problem.

Structural Theoretical Framework

A transversal for $(G, H)$2 with minimal $(G, H)$3 is characterized combinatorially and algebraically. The authors relate the support-minimization to classical factorization theory, where the critical object is the presence or absence of subgroup complements to $(G, H)$4 in $(G, H)$5. If $(G, H)$6 admits a subgroup complement $(G, H)$7, then $(G, H)$8 achieves the lower bound $(G, H)$9.

The paper introduces an important parameter $H \leq G$0—the maximum order of a subgroup of $H \leq G$1 disjoint from $H \leq G$2—noting that this parameter can vary substantially depending on the pair $H \leq G$3. The analysis employs classical results such as Kneser’s theorem for bounding sumsets/difference sets in abelian groups, combined with a reduction to primitive quotients (via singleton quotient directions of transversals, see Proposition~\ref{prop:primitive-quotient-reduction}).

The sharp lower bound proved is:

$H \leq G$4

where the extreme cases (when $H \leq G$5 or $H \leq G$6) are characterized as split and residual pairs, respectively.

Explicit Results and Sharpness

The authors obtain a vast suite of exact formulas and bounds:

• Cyclic Quotients: For $H \leq G$7 cyclic of order $H \leq G$8, $H \leq G$9 (Theorem~\ref{thm:cyclic-quotient-formula}), with explicit computation in all cyclic cases (Corollary~\ref{cor:cyclic-fixed-stage-formula}).
• Product Phenomena: In direct products, elementary multiplicativity bounds are established, but the authors show these are non-optimal in general, and give precise characterizations when products yield optimality.
• Singleton Reduction & Chain Bounds: Reduction via singleton directions yields exact recursive/multiplicative identities for the difference supports upon passing to primitive quotients. Chain/projection and product formulas are derived with Kneser-sharpened bounds.

New Obstructions and Open Phenomena

A central highlight is the identification of non-cyclic residual obstructions—that is, situations where the lower bound is not achieved. Especially noteworthy is their analysis of same-prime square moduli, e.g., $T$0, $T$1 for odd $T$2. Here, the natural “coordinate box” construction gives a transversal $T$3 with $T$4, and the authors conjecture that this is minimal:

$T$5

The paper proves an unconditional lower bound $T$6 (Theorem~\ref{thm:unconditional-square-plane-bound}), demonstrating a gap between what general combinatorial methods can achieve and the conjectured optimum. For rank-two $T$7-groups, they show the involvement of parity yields further obstructions.

Their methods involve an analysis of the combinatorial structure of difference supports using carry-corrected finite-field derivatives. The structure means transversals correspond to graphs of functions $T$8, and the intricate sumset behavior in affine planes over $T$9 is intimately linked with minimal possible difference support.

Asymptotic and Experimental Evidence

The authors deploy both probabilistic and computational methods to support their conjectures:

• Probabilistic Analysis: For a random lifting (i.e., $G/H$0 chosen uniformly), they show that with high probability, the resulting transversal achieves the conjectured minimal difference support as $G/H$1.
• Fixed-Polynomial Lifting: For any fixed polynomial map $G/H$2, the authors show that for sufficiently large $G/H$3, the support $G/H$4 cannot fall below the box value.
• Exact Verification for Small Primes: Using exhaustive computation for small $G/H$5, the conjecture is confirmed in low-dimensional cases; in rank-two for $G/H$6 and $G/H$7.

Additive Combinatorics Interface

The paper’s approach and open problems link the transversal difference minimization directly to additive combinatorial invariants—Kemperman’s theory, small sumset analysis, and incidence geometry. Notably, the problems relate to the structure of value sets of polynomial/rational maps, connect with sum-product phenomena over finite fields, and suggest further applications of techniques like the polynomial method and point-line/point-plane incidence bounds.

Implications and Future Directions

Practical Implications: Minimal $G/H$8 directly impacts the cost and complexity of Galois-theoretic bookkeeping in FHE systems where operations like bootstrapping, ciphertext slot packing, and SIMD computation rely on the structure of automorphism supports. Reducing the size of the difference support may decrease key-switching costs and inform CRT basis design.

Theoretical Implications: The work uncovers new obstructions in additive combinatorics not previously isolated by classical factorization or difference set theory. The detailed analysis presents challenges for established small-doubling paradigms in non-cyclic contexts and identifies tight lower and upper bounds only for a restricted class of groups. The odd square-plane and higher-rank, same-prime settings expose unexplored combinatorial and group-theoretic territory.

Future Developments:

• Resolution of the Square-Plane Conjecture: The odd-prime, rank-two case remains open, and its solution would clarify the extent to which product and cyclic phenomena are atypical.
• Generalization to Higher-Rank Quotients: Conjectures are posited for the structure of minimal difference supports in $G/H$9.
• Sharper Incidence/Expansion Techniques: The possibility of extending the analytical toolkit with sum-product, incidence, or growth methods is discussed.
• Primitive Quotients and Residual Geometries: The full classification of primitive (i.e., no nonzero singleton directions) quotient pairs with minimal difference support remains unsolved.

Conclusion

The study systematically introduces and analyzes the transversal difference number, situating it at the nexus of algebraic and combinatorial group theory, and opening new avenues both for abstract additive combinatorics and for concrete cryptographic algorithm design. The combinatorial techniques (singleton reduction, primitive quotient passage, Kneser-type bounds) are developed to a level where the first genuinely new residual obstructions—particularly, same-prime, rank-two cases—are both structurally isolated and partially untangled. The interplay between probabilistic, computational, and algebraic techniques highlights the depth of the underlying combinatorics and suggests substantial further work is required, especially in rank-two and higher, to fully describe the landscape of transversal difference minimization in finite abelian quotients.