Point Set Equivalence (PSE) Problem

• Point Set Equivalence (PSE) is the problem of testing whether two finite projective point sets over a finite field are equivalent via an invertible linear transformation, underpinning key concepts in algebraic geometry and coding theory.
• It is computationally equivalent to Linear Code Equivalence and is reformulated through algebraic constructions such as homogeneous coordinate rings and Artinian Gorenstein algebras.
• Polynomial-time reductions using methods like the Buchberger–Möller algorithm and linear algebra enable practical solutions for moderate cases, with significant implications for cryptanalysis and structural finite geometry.

The Point Set Equivalence (PSE) problem concerns the determination of whether two finite sets of points in a projective space over a finite field are equivalent under a linear change of coordinates. Equivalence here means the existence of an invertible linear transformation mapping one ordered set of points to the other, modulo projective scalars. This notion links deeply with linear code equivalence, polynomial isomorphism, and broader isomorphism problems in algebraic geometry. Recent results establish formally that PSE is computationally equivalent to the Linear Code Equivalence (LCE) problem, and both reduce (under mild assumptions) to explicit algebraic and polynomial isomorphism problems, linking these areas to structural questions in finite geometry and computational algebra.

1. Formal Definition and Core Problem

Let $F_q$ be a finite field, $k\geq1$ an integer, and $P^{k-1}$ the $(k-1)$-dimensional projective space over $F_q$. Given two point sets $X = \{p_1, \dots, p_n\}$ and $X' = \{p'_1, \dots, p'_n\}$ in $P^{k-1}(F_q)$, $X$ and $X'$ are equivalent ($X \sim X'$) if there exists $A \in GL_k(F_q)$ and an ordering such that $A \cdot p_i = p'_i$ for all $i = 1, \dots, n$, i.e., $X' = \{A \cdot p \mid p \in X\}$. The computational problems are:

• PSE Decision: Given $X, X'$, decide if $X \sim X'$.
• PSE Search: If so, find $A \in GL_k(F_q)$ realizing the equivalence.

These definitions establish PSE as an isomorphism-type problem in finite projective geometry.

2. Relationship to Linear Code Equivalence

The PSE problem is computationally equivalent to the Linear Code Equivalence (LCE) problem. Given a linear $[n, k]_q$ code $C \subset F_q^n$ with generator matrix $G \in F_q^{k \times n}$, two codes $C, C'$ are equivalent if:

$G' = A \cdot G \cdot D \cdot P$

where $A \in GL_k(F_q)$, $D$ is diagonal and invertible, and $P$ is a permutation matrix. If $C$ is projective (no two columns of $G$ are scalar multiples), then the columns of $G$ define a projective point set $X$. Conversely, a point set $X$ in $P^{k-1}$ defines an evaluation code $C = Ev_X(P_1)$. This yields a natural mapping $G \leftrightarrow X$.

Proposition. LCE $\equiv$ PSE under this correspondence. Left-multiplication by $A$ applies a linear automorphism in $P^{k-1}$, while right-multiplication by $D, P$ renormalizes (via scalar multiplication, permutation) the representatives in $X$.

3. Algebraic and Polynomial Formulation

PSE admits an algebraic reformulation via the homogeneous coordinate ring $R_X = F_q[x_1,\ldots,x_k]/I_X$ for the vanishing ideal $I_X$ of $X$. $R_X$ is 1-dimensional Cohen–Macaulay, with Hilbert function $H_X(i) = \dim_{F_q} (R_X)_i$. Let $r_X$ be its regularity index, i.e., the smallest $r$ with $H_X(r) = n$.

Fix a linear form $L \notin I_X$; its image $\ell \in (R_X)_1$ is a nonzerodivisor, yielding a Noether normalization $F_q[\ell] \subset R_X$. The canonical module $\omega_{R_X}$ can be embedded into $R_X$ as a homogeneous ideal $J_X$, generating the canonical ideal. The doubling $D_X = R_X/J_X$ is a finite-dimensional (Artinian), graded Gorenstein algebra of socle degree $2r_X-1$.

A linear transformation $A \in GL_k(F_q)$ induces an $F_q$-algebra isomorphism between $R_X$ and $R_{X'}$ sending $J_X$ to $J_{X'}$ and $D_X \cong D_{X'}$. Conversely, any graded $F_q$-algebra isomorphism $D_X \rightarrow D_{X'}$ induced in degree 1 corresponds to $A \in GL_k(F_q)$ sending $X \mapsto X'$. Therefore,

$X \sim X' \Longleftrightarrow D_X \cong D_{X'}$

as graded $F_q$-algebras via a degree-1 induced isomorphism.

Through Macaulay inverse systems, $D_X$ is canonically associated to a homogeneous polynomial $\Phi_X \in F_q[\pi_1, \dots, \pi_k]_{2r_X-1}$. The resulting Polynomial Isomorphism (PI) problem is: given homogeneous polynomials $\Phi, \Psi$ of degree $d$ in $k$ variables, decide or find $A \in GL_k(F_q)$ with $\Phi = A^{-1} * \Psi$, where $(A*\Phi)(\pi) = \Phi(A\cdot\pi)$.

4. Algorithmic Reductions and Complexity

The reductions from PSE to LCE, then to Artinian Gorenstein algebra isomorphism, and finally to the PI problem, are all polynomial-time under moderate regularity assumptions. Specifically:

• Computing $I_X$, $r_X$, a Gröbner basis, canonical ideal $J_X$, and doubling $D_X$ can be performed in polynomial time (in $n, k, \log q$) using the Buchberger–Möller algorithm and linear algebra over $F_q[\ell]$.
• The dimension of $D_X = n \cdot 2r_X$, and generators of the lifted ideal $\widehat{J}_X$ (degree $\leq 2r_X-1$) can be found in polynomial time.
• The Macaulay inverse polynomial $\Phi_X$ is computable in polynomial time if $r_X$ is bounded, e.g., $r_X \leq 3$ for codes in general position with rate $\geq 1/2$.
• However, general PI problem algorithms are exponential in $k$ or use graph-isomorphism–type heuristics; specialized cubics ($d=3$) admit faster (IP1S) solving for moderate $k$.

Table: Structural Reductions in the PSE Problem

Problem/Structure	Associated Algebraic Object	Complexity (when regularity moderate)
Point Set Equivalence (PSE)	Projective point sets in $P^{k-1}(F_q)$	Poly($n,k, \log q$)
Linear Code Equivalence (LCE)	Generator matrices $G, G'$	Poly($n,k, \log q$)
Artinian Gorenstein algebra isomorphism	$D_X = R_X/J_X$	Poly($n,k, \log q$)
Polynomial Isomorphism (PI)	Homogeneous $\Phi \in F_q[\pi]^d$	Poly($n,k, \log q$) if $d$ constant

This reduction chain reveals a deep algebraic and computational link between geometric, coding-theoretic, and polynomial isomorphism problems.

5. Special Cases: Iso-dual Codes and Self-Associated Point Sets

A specialized regime arises for indecomposable iso-dual codes and their associated self-associated, arithmetically Gorenstein point sets. For a projective $[2k, k]_q$ code $C$ with $C \sim C^\perp$ (iso-dual), the associated $X \subset P^{k-1}$ of size $2k$ is self-associated (Gale duality), arithmetically Gorenstein, and has $r_X = 3$. The Hilbert function difference is $\Delta_X: 1, k-1, k-1, 1$; $\omega_{R_X} \cong R_X(-2)$.

For such $X$, the Artinian reduction $\widetilde{R} = R_X/(\ell)$ has socle degree $3$; its Macaulay inverse system is generated by a cubic $\overline{\Phi} \in F_q[\pi_1,\ldots,\pi_k]_3$. The PSE (and LCE) search problem thus reduces in polynomial time (via linear algebra in $O(k^3)$ dimensions) to the PI search problem for cubics. Furthermore, existing IP1S cryptanalytic methods are effective for these instances, providing practical solving for $k$ up to 30–50.

6. Comparison to Other Isomorphism Problems

While PSE in projective space over finite fields reduces to polynomial and algebraic isomorphism, analogous reductions exist in graph theory and geometric data analysis. For example, in the context of graph isomorphism, the isomorphism problem can be recast as a point set registration problem in $\mathbb{R}^d$ via simplex embedding and sampling (Oktar, 2021). Here, perfect rigid registration (up to symmetry group) corresponds to graph isomorphism, and the problem admits a finite enumeration (searching over $d!$ permutations) but does not yield a polynomial-time algorithm for large $d$.

A notable distinction is that while the graph embedding approach hinges on geometric registration and orthogonal invariance, the projective PSE problem is fundamentally algebraic and invariant under general linear group actions.

7. Implications and Computational Significance

The reduction of PSE to explicit algebraic and polynomial isomorphism problems establishes an intrinsic computational equivalence with LCE and connects geometric and combinatorial isomorphism problems through canonical algebraic constructions. Under regularity assumptions, these reductions are polynomial-time, expanding the practical tractability of PSE in several coding-theoretic and algebraic settings. For certain code families (notably, indecomposable iso-dual codes), these techniques yield efficient algorithms for equivalence testing and illuminate connections to cryptanalytical polynomial isomorphism attacks.

A plausible implication is that further development of specialized isomorphism solvers and canonical form algorithms for the Artinian Gorenstein regime may advance both code equivalence testing and broader applications in mathematical cryptography and computational algebraic geometry.