symmetric polynomial construction

Affine equivalence in multivariate public-key cryptography exposes algebraic structure, rendering schemes vulnerable to algebraic attacks. Method: This work introduces CCZ (Carlet–Charpin–Zinoviev) equivalence—previously unexploited in post-quantum multivariate cryptography—by constructing a novel trapdoor framework based on vectorial Boolean functions under CCZ equivalence. This framework transcends the restrictive equivalence-class space imposed by affine equivalence, substantially enlarging the space of constructible secure keys. Contribution/Results: We formally prove that the framework preserves trapdoor invertibility while increasing the algebraic indistinguishability and hardness of recovering the public key. Experiments demonstrate enhanced robustness against canonical algebraic attacks, including Gröbner basis and linearization methods. To our knowledge, this is the first systematic application of CCZ equivalence to the design of multivariate cryptographic primitives, establishing a new paradigm for concealing algebraic structure and strengthening post-quantum security.

This paper addresses the explicit representation problem for subrings of multivariate polynomial rings over fields of characteristic zero: given algebraically independent generators $g_1,dots,g_n$ and an element $h in K[g_1,dots,g_n]$, compute the unique polynomial $f in K[u_1,dots,u_n]$ satisfying $h = f(g_1,dots,g_n)$. We propose the first general-purpose algorithm applicable to arbitrary algebraically independent generators—surpassing prior approaches restricted to symmetric polynomials. Our method integrates Gröbner basis techniques, variable substitution, and sparse interpolation, augmented by degree bounds and sparsity-aware optimizations. The algorithm runs in time linear in the input size and polynomial in $n$ (for fixed $deg f$), and we provide a rigorous correctness proof. Experimental evaluation confirms its efficiency and practicality on non-symmetric generator sets, demonstrating substantial speedups over existing methods.

This study addresses the construction of functions in algebraic combinatorics subject to stringent distributional constraints and the discovery of previously unknown combinatorial symmetries. To this end, we propose the SLURP framework, which integrates MapSeek-Functional and MapSeek-Symbolic approaches through alternating pseudo-label supervised learning, symbolic regression, and formal verification in Lean 4. The framework yields the first combinatorial interpretation of $q,t$-Narayana polynomials based on non-crossing partitions and provides a combinatorial proof of symmetry in previously unresolved cases by leveraging newly discovered statistics. All code and formalized results are publicly released to ensure reproducibility and rigorous verification.

For zero-dimensional polynomial systems, this paper proposes a separation-based linear-type unconditional certification method and an algorithm for constructing Rational Univariate Representations (RURs) without requiring additional structural assumptions—such as the shape lemma. The core method derives the RUR directly from a lexicographic Gröbner basis via binary elimination ideals, augmented by Gaussian elimination and ideas inspired by the FGLM algorithm. This approach eliminates reliance on geometric properties of the ideal (e.g., radicality or generic position), ensuring both formal correctness and certifiability. Experimental results demonstrate high efficiency: a Maple implementation requires only ~300 lines of code, while a naive Julia implementation achieves state-of-the-art performance, significantly outperforming existing uncertified parametric solvers.

This work addresses the lack of efficient parametric solution methods for zero-dimensional parametric polynomial systems that exhibit favorable specialization properties. It presents the first systematic study of the specialization behavior of Rational Univariate Representations (RURs) in parametric settings, establishing explicit upper bounds on the degree and height of their constituent elements. By leveraging techniques from algebraic geometry and symbolic computation, the authors develop a general RUR-based parametrization framework and introduce two efficient algorithms. The proposed approach guarantees stable specialization, provides rigorous algebraic complexity bounds, and yields a fully computable and implementable parametric solution method.

This work addresses the inefficiency of traditional methods for constructing resultant systems of polynomial systems, which typically yield an excessively large number of polynomials. The authors propose a novel approach that leverages linear combinations of the input polynomials to construct resultants, drastically reducing the system size. In the homogeneous case, they prove that the existence of a nontrivial common solution can be characterized using only ${d+n-1 \choose n-1}s - n^2 + 1$ polynomials. Moreover, when the number of variables is fixed, they provide an explicit construction of polynomial size. By integrating linear combinations, resultant theory, and coefficient matrix analysis from algebraic geometry, this method substantially improves the known upper bounds on resultant system size—even outperforming existing results in the bivariate case.

This study addresses the long-standing lack of explicit formulas and structural understanding of Chern classes expressed as symmetric polynomials across various bases of symmetric functions. By integrating multiple artificial intelligence systems with human mathematical insight, we establish a collaborative workflow that closes the research loop from experimental exploration and conjecture generation to symbolic proof. We demonstrate for the first time the feasibility of AI-augmented pure mathematical discovery, providing explicit expressions for the Chern and K-theoretic classes of $\mathrm{Sym}^d(\mathbb{C}^n)$. Furthermore, we prove refined positivity and a novel form of log-concavity for their Schur coefficients when expanded in the binomial basis, uncovering deep combinatorial structures in the rank-two case.

This work investigates the computational complexity of finding a non-vanishing point for a polynomial given by an arithmetic circuit, with a focus on the constructive aspects under the combinatorial Nullstellensatz framework. By leveraging tools from computational complexity theory, arithmetic circuit models, and probabilistic reductions, the study establishes—for the first time—that the problem remains NP-hard even when the individual degree of every variable is at most two. Consequently, under the standard complexity assumption that RP ≠ NP, no randomized polynomial-time algorithm can find such a non-zero evaluation point with constant success probability. This result underscores the inherent difficulty of constructively realizing the combinatorial Nullstellensatz, revealing fundamental limitations in efficiently producing witnesses guaranteed to exist by this algebraic principle.

This work investigates the algebraic–combinatorial mechanisms underlying graph isomorphism discrimination and introduces the framework of “separating modules,” a polynomial vector space grounded in the representation theory of symmetric groups. By employing complexity measures such as support size, symmetric circuit size, and multiplicity, it establishes equivalences with subgraph counting (support size \(k\) corresponds to order \(O(k)\)) and the Weisfeiler–Leman algorithm (circuit size \(n^{\Theta(k)}\) corresponds to \(\Theta(k)\)-WL). The central contribution provides the first intrinsic characterization of multiplicity separation: two graphs are distinguishable if and only if their automorphism groups have distinct cycle indices. Furthermore, the paper demonstrates that the multiplicity barrier is strictly stronger than the occurrence barrier and connects invariant polynomials to the graph reconstruction conjecture and finite-type invariants.

This work addresses the maximization of multilinear polynomials over \( n \) binary variables, a problem that encompasses numerous NP-hard special cases, including unconstrained quadratic binary optimization. The authors introduce a general variable elimination framework that, for the first time, explicitly constructs an equivalent multilinear polynomial after elimination and enables efficient solution through recursive application. Built upon elementary algebraic operations, this approach unifies and extends all previously known tractable cases—such as those with bounded treewidth or various acyclic structures—yielding a broader and more computationally efficient algorithmic framework.