On the Lattice Distortion Problem

We introduce and study the \emph{Lattice Distortion Problem} (LDP). LDP asks how “similar” two lattices are. I.e., what is the minimal distortion of a linear bijection between the two lattices? LDP generalizes the Lattice Isomorphism Problem (the lattice analogue of Graph Isomorphism), which simply asks whether the minimal distortion is one. As our first contribution, we show that the distortion between any two lattices is approximated up to a $n^{O(\log n)}$ factor by a simple function of their successive minima. Our methods are constructive, allowing us to compute low-distortion mappings that are within a $2^{O(n \log \log n/\log n)}$ factor of optimal in polynomial time and within a $n^{O(\log n)}$ factor of optimal in singly exponential time. Our algorithms rely on a notion of basis reduction introduced by Seysen (Combinatorica 1993), which we show is intimately related to lattice distortion. Lastly, we show that LDP is NP-hard to approximate to within any constant factor (under randomized reductions), by a reduction from the Shortest Vector Problem.

💡 Research Summary

The paper introduces the Lattice Distortion Problem (LDP), which asks for the smallest possible distortion of a linear bijection between two n‑dimensional lattices L₁ and L₂. Distortion is measured by the condition number κ(T)=‖T‖·‖T⁻¹‖ of a linear map T that sends L₁ onto L₂; κ(T)=1 exactly characterizes lattice isomorphism (the Lattice Isomorphism Problem, LIP). The authors first derive a simple lower bound on the optimal distortion using successive minima λᵢ(L). Defining M(L₁,L₂)=max_i λᵢ(L₂)/λᵢ(L₁), they show that D(L₁,L₂)≥M(L₁,L₂)·M(L₂,L₁).

To obtain an upper bound, they employ Seysen’s condition number S(B)=max_i‖b_i‖·‖b_i*‖ for a basis B and its dual B*. The lattice‑wide parameter S(L)=min_B S(B) satisfies Seysen’s theorem S(L)≤n^{O(log n)}. For any two bases B₁ of L₁ and B₂ of L₂ (ordered by length), they prove
‖B₂B₁⁻¹‖·‖(B₂B₁⁻¹)⁻¹‖ ≤ n²·S(B₁)²·S(B₂)²·M(L₁,L₂)·M(L₂,L₁).
Consequently, the optimal distortion satisfies
D(L₁,L₂) ≤ n^{O(log n)}·M(L₁,L₂)·M(L₂,L₁).
Thus the distortion can be approximated within a quasipolynomial factor using only the successive minima.

Algorithmically, the paper combines Seysen’s reduction with slide‑reduction (Gama–Nguyen) to construct bases with small S(B). For any parameter k with log n ≤ k ≤ n, a basis with S(B) ≤ k^{O(n/k+log k)} can be found in time 2^{O(k)}. This yields a family of algorithms for γ‑LDP: choosing larger k gives a near‑optimal approximation (≈2^{O(n log log n / log n)}) in exponential time, while smaller k yields a polynomial‑time algorithm achieving an n^{O(log n)} approximation. Hence the authors provide a clear time‑approximation trade‑off for LDP.

On the hardness side, they reduce the Gap Shortest Vector Problem (GapSVP) to GapLDP. By embedding an SVP instance into a pair of lattices, they ensure that the optimal distortion reflects the length of the shortest vector. This reduction shows that for any constant γ ≥ 1, γ‑GapLDP is NP‑hard under randomized polynomial‑time reductions. This contrasts with LIP, which is not known to be NP‑hard, and aligns LDP with known hardness results for approximate graph isomorphism.

The paper also demonstrates that HKZ‑reduced bases, despite being “short”, may miss the optimal distortion by an exponential factor, highlighting the necessity of Seysen‑reduced bases for this problem.

Finally, the authors discuss open questions: tightening Seysen’s bound (perhaps to polynomial), achieving sub‑quasipolynomial approximations in polynomial time, and better characterizing near‑optimal distortion maps. Overall, the work bridges lattice geometry, basis reduction techniques, and computational complexity to give a comprehensive understanding of the Lattice Distortion Problem.