Polynomial Isomorphism Problem

• Polynomial Isomorphism (PI) Problem is a computational challenge that determines if two tuples of multivariate polynomials are equivalent under invertible linear transformations and group actions.
• Algorithmic approaches reduce the problem to matrix isometry for quadratic cases and tensor isomorphism for higher degrees, offering randomized polynomial-time solutions under regular conditions.
• Practical applications span multivariate cryptography, code equivalence, and group isomorphism, unifying diverse isomorphism challenges in theoretical computer science and algebra.

The Polynomial Isomorphism (PI) Problem encompasses a broad class of computational equivalence problems for tuples of multivariate polynomials, notably including the Isomorphism of Polynomials with One Secret (IP1S) and Two Secrets (IP2S) problems, cubic form equivalence, and higher-order tensor isomorphism. Originally motivated by complexity theory, multivariate cryptography, and group theory, the PI problem asks whether two tuples of polynomials (or related algebraic objects) are equivalent under specified group actions, usually via invertible linear changes of variables (and sometimes value permutations). Its significance arises from both algorithmic hardness implications (e.g., connections to Graph Isomorphism, Group Isomorphism, and code equivalence) and its role as the “canonical” representative of several TI-complete isomorphism problems across algebra, combinatorics, and cryptography.

1. Formal Statement and Key Variants

Let $k$ be a field (typically finite, $k = \mathbb{F}_q$) and consider $m$ polynomials in $n$ variables:

$$\mathbf{f} = (f_1, \dots, f_m) \in (k[x_1, \dots, x_n])^m, \quad \mathbf{g} = (g_1, \dots, g_m) \in (k[x_1, \dots, x_n])^m$$

The Polynomial Isomorphism Problem asks: Find (if any) pairs of invertible matrices $A \in \mathrm{GL}_n(k)$ and $B \in \mathrm{GL}_m(k)$ such that

$\mathbf{g}(x) = B\, \mathbf{f}(A x)$

where $(A x)_i = \sum_j A_{ij} x_j$ and the group action on values (the “second secret”) is optional in applications.

Notable special cases:

• IP1S (Isomorphism with One Secret): $B = I_m$, so only transformation in the variables.
• IP2S (Two Secrets): Both $A \in \mathrm{GL}_n(k)$ and $B \in \mathrm{GL}_m(k)$.
• Cubic Form Equivalence: $m = 1$, degree $d = 3$, i.e., $f, g$ are single cubic forms.
• Higher-Degree PI: $d$-homogeneous forms, generalizing the above to tensors.

2. Algebraic Reformulation and Matrix Encodings

Quadratic case ($d=2$) yields natural translation to simultaneous matrix conjugacy via Hessian representation:

$$f_i(x) = x^\top H_i x, \quad g_i(x) = x^\top H'_i x$$

The IP1S becomes finding $A \in \mathrm{GL}_n(k)$ such that

$A^\top H_i A = H'_i \quad \forall\, i\,.$

For cubics and higher-degree forms, polynomial equivalence is captured by symmetry and structure in the corresponding order-$d$ tensor $\mathcal{T}_f$, with

$$g(x) = f(Ax) \iff \mathcal{T}_g = (A \otimes \cdots \otimes A)\,\mathcal{T}_f$$

for the group action of $\mathrm{GL}_n(k)$ on symmetric $d$-tensors.

General PI problems thus reduce to the Tensor Isomorphism (TI) problem—in particular, equivalence of symmetric or alternating tensors under group actions.

3. Algorithmic Approaches: Quadratic Forms and Cubic Tensors

Quadratic IP1S (Regular Case)

Polynomial-time algorithms hinge on reduction to module isomorphism problems and matrix factorizations (Ivanyos et al., 2017, Berthomieu et al., 2013, Plût et al., 2014):

• Regularity: Existence of a nondegenerate combination, allowing reduction to canonical form.
• Linearization: Simultaneous matrix conjugacy—solve for $Y \in k^{n \times n}$ such that $D^{-1} H_i Y = Y D^{-1} H'_i$ for all $i$, then recover $A$ via generalized polar decomposition.
• Complexity: $O(n^{2\omega})$ randomized polynomial time for quadratic IP1S (with $\omega$ the matrix multiplication exponent).

For two quadratics ($m=2$), exact solutions exploit singular pencil decomposition (Kronecker), primary factorization, and local classification of bilinear forms over finite local rings (Plût et al., 2014).

Kronecker–Weierstraß Theorem

Every singular pencil admits an orthogonal decomposition into Kronecker blocks and a regular part, computable in $O(n^4)$ time, typically $O(n^3)$.

Local Classification

Regular symmetric forms over a local ring $R = k[x]/(f(x)^{e})$ are congruent to canonical diagonal forms, enabling efficient isomorphism testing.

Cubic and Higher-Degree Tensor Isomorphism

Cubic form equivalence, via symmetric trilinear tensor encoding, is TI-complete (Grochow et al., 2020, Grochow et al., 2023). Average-case algorithms succeed for almost all cubic forms in $q^{O(n)}$ time (Grochow et al., 2020).

Linear-Length Gadget Reductions

Recent advances eliminate quadratic blow-up in reductions among TI instances, reverting to $O(n)$ blow-up and enabling tight complexity for cubic and algebra isomorphism (Grochow et al., 2023).

Complexity Table for Main Cases

Problem	Reduction Target	Randomized Time	Deterministic / Worst Case
Quadratic IP1S (regular)	Matrix Isometry	$O(n^{2\omega})$	$O(n^{3})$
Two quadratics (IP2S)	Local algebra	$\tilde O(n^4)$	$\tilde O(n^4)$
Cubic Form (TI-complete)	3-tensor TI	$q^{O(n)}$	$q^{O(n^{1.8}\log q)}$
Algebra Isomorphism	3-tensor TI	$q^{O(n)}$	$q^{O(n^{1.8}\log q)}$

4. Applications and Connections

Multivariate Cryptography

PI problems serve as hardness assumptions for MQ schemes (notably Patarin’s authentication protocols) (Ivanyos et al., 2017). The insecurity of IP1S/IP2S for random instances informs both construction and cryptanalysis.

Group and Code Isomorphism

Reductions show equivalence between PI, linear code equivalence (LCE), point set equivalence (PSE), and Gorenstein algebra isomorphism via Macaulay inverse systems (Kreuzer, 10 Nov 2025). For iso-dual codes, the chain of equivalences reduces LCE directly to cubic PI.

Polynomial Identity Testing and Noncommutative Rank

Skew-symmetric and Hermitian matrix tuple isometry, as analyzed via *-algebras (Ivanyos et al., 2017), yields two families of singularity witnesses—a next step after the resolution of the noncommutative rank problem.

Complexity-Theoretic Connections

TI-completeness unifies many previously distinct isomorphism problems: cubic-form equivalence, algebra isomorphism, group isomorphism, and tensor isomorphism, all have polynomial-time reductions between them (Grochow et al., 2020, Grochow et al., 2023).

If Graph Isomorphism is in P, then cubic-form equivalence and algebra isomorphism admit $q^{O(n)}$ algorithms.

5. Structural and Algorithmic Reductions

Partition Removal Gadget

The linear-length gadget of (Grochow et al., 2023) transforms TI problems with direct-sum restrictions (partitioned actions) to plain TI with only $O(\text{length})$ blow-up per mode. The construction augments tensors with identity blocks and “cancellation” slices, enforcing block-diagonal preservation and enabling reduction among all five natural TI group actions.

Automorphism and Counting-to-Decision Reductions

Automorphism gadgets allow polynomial-time reduction from search and counting versions of isomorphism to the decision version in the Cayley-table model for $p$-groups of class $2$ and exponent $p$. By iterative fixing of monomial support and rank-isolation, the full isomorphism or automorphism group can be recovered using a TI oracle in total $2^n q^{O(n)}$ time (Grochow et al., 2023).

Algebraic Gorenstein Criteria

Doubling of coordinate rings, canonical ideals, and Macaulay inverse systems establish equivalence between geometric, algebraic, and polynomial isomorphism settings (Kreuzer, 10 Nov 2025, Isaev, 2015). The criteria for isomorphism of Artinian Gorenstein algebras involve affine equivalence of hypersurfaces determined by the graph of nil-polynomials, strictly extending inverse system methods (Isaev, 2015).

6. Limitations, Open Problems, and Generalizations

• Characteristic Two: Extension of classification and efficient IP1S/IP2S algorithms remains partly unresolved for quadratic pencils in characteristic $2$ due to alternating bilinear forms and trace-lifting intricacies (Plût et al., 2014).
• Multiple Quadratic Equations $(m \geq 3)$: The geometry of the determinant variety complicates the extension of polynomial-time solutions for IP1S, though tensor-encoding and TI reductions advance progress.
• Exceptional Cases: For field isomorphism problems (distinct from PI but closely related in Galois-theoretic settings), recent work on quartic fields illustrates rigorously explicated exceptional uniqueness results (Pincus et al., 14 Jun 2024).
• TI-Completeness by Construction: Equivalence of symmetric and alternating trilinear forms, cubic form equivalence, and algebra isomorphism can all be forced to respect block structures via explicit gadget embeddings, confirming TI-completeness in each (Grochow et al., 2020, Grochow et al., 2023).

7. Impact, Unification, and Outlook

The PI problem, through TI-completeness, unites major computational equivalence and isomorphism questions across mathematics, theoretical computer science, and cryptography. Advancements in reduction frameworks, structure-theoretic algorithms, and gadget constructions yield tight complexity boundaries and facilitate transfer of algorithmic techniques among group, code, polynomial, and tensor isomorphism spheres.

The polynomial isomorphism landscape now encompasses both theoretical complexity (under reductions from Graph Isomorphism and related problems) and practical algorithmics (as in cryptography, codes, and canonical forms). Improvements in matrix-algebraic and tensor methods are likely to further impact both the worst-case complexity of isomorphism problems and the design of secure multivariate cryptosystems.

The unification of isomorphism problems motivates future research in both structural algorithmics (local and global invariants, automorphism group enumeration) and deeper complexity-theoretic classification, with ongoing investigation into characteristic-dependent phenomena, average-case versus worst-case hardness, and connections to algebraic geometry (e.g., Gorenstein rings, hypersurfaces, elliptic and hyperelliptic curve isomorphism criteria).