perform cyclotomic coset analysis

This paper systematically characterizes the structure of $q$-cyclotomic cosets modulo $n$ over the finite field $mathbb{F}_q$, thereby strengthening the theoretical foundation of cyclic codes. Methodologically, it establishes, for arbitrary $q$ and $n$, a unified explicit formula for canonical representatives of cyclotomic cosets and a closed-form expression for their exact sizes; introduces the novel *2-adic cyclotomic system* framework to uncover deep structural regularities; and constructs a direct mapping from cyclotomic information to key cyclic code parameters—including generator polynomials and self-duality properties. Main contributions include: an optimized Graner-type factorization formula yielding fine-grained factorizations of $X^n - 1$ and the $n$-th cyclotomic polynomial $Phi_n(X)$ over $mathbb{F}_q$; a complete classification of all generator polynomials of cyclic codes of length $n$ over $mathbb{F}_q$; and necessary and sufficient conditions for self-dual cyclic codes, along with their precise enumeration.

This study investigates affine-invariant codes whose defining sets consist of descendants from a single cyclotomic coset, aiming to precisely characterize the structure of these descendant sets and thereby analyze the dimension and minimum distance of both the code and its dual. By employing combinatorial methods, the authors provide the first exact computation of the size of such descendant sets. Leveraging tools from finite field theory and coding theory, they derive explicit dimension formulas for this class of affine-invariant codes—which includes narrow-sense primitive BCH codes—and establish an improved lower bound on the minimum distance of their duals. This work advances the theoretical understanding of BCH code parameters and achieves precise structural characterization of their performance in specific cases.

This paper addresses the structural characterization of cyclotomic cosets over finite fields and the binomial irreducible factorization of $X^n - 1$. We introduce the notion of *arithmetic cyclotomic cosets* and establish, for the first time, a *multiple arithmetic progression representation theory* for cyclotomic cosets: every $q$-cyclotomic coset modulo $n$ admits a unique decomposition into disjoint arithmetic progressions, and this decomposition is in one-to-one correspondence with the complete factorization of $X^n - 1$ into irreducible binomials over $mathbb{F}_q[X]$. Building upon this theory, we provide an explicit construction algorithm, necessary and sufficient criteria for such factorizations—including a generalization to extension fields $mathbb{F}_{q^m}$—and a novel efficient algorithm for computing minimal representatives of cyclotomic cosets. Our results unify combinatorial coset analysis with algebraic polynomial factorization, yielding new tools for cyclic code design and finite-field polynomial decomposition.

Solving multi-modular integer constraint systems—comprising polynomial equalities and inequalities under distinct moduli—is notoriously difficult in cryptographic protocol verification; existing SMT solvers fail to exploit their inherent algebraic structure. Method: This paper introduces the first resolution-based decision procedure tailored for multi-modular reasoning. Contributions/Results: (1) Constraints are partitioned by modulus, and novel algebraic lifting/reduction mechanisms enable information sharing across modular subsystems; (2) Weighted Gröbner basis theory is integrated into the SMT framework for precise multi-modular algebraic reasoning—the first such incorporation; (3) A modular, embeddable solving pipeline is constructed. Evaluated on Montgomery multiplication and zero-knowledge proof implementation verification, our method substantially outperforms state-of-the-art SMT solvers: solution success rate improves by 42%, and average verification time decreases by a factor of 5.8.

This paper addresses the problem of determining whether a given algebraic variety (V) arises as the Zariski closure of an orbit of a point under the action of an (s)-generated commutative matrix group. To resolve this, the authors first formulate and solve the “decidability” problem for such orbit closures, establishing a unified framework integrating commutative algebra, structural theory of matrix groups, lattice theory, and algebraic-geometric analysis of orbit closures. They devise a decision algorithm that, given (V) and (s), determines in PSPACE whether (V) equals the orbit closure of some point under an (s)-generated commutative linear algebraic group. Moreover, they prove that this problem is PSPACE-complete—establishing a tight complexity characterization. The main contribution is the first proof of computability for this geometric decision problem, together with an optimal complexity bound, thereby filling a fundamental theoretical gap in the structural decidability of orbit closures under algebraic group actions.

This work addresses the problem of efficiently determining whether a finite algebraic structure defined by two $n \times n$ operation tables forms a ring or a field. We present the first deterministic $O(n^2)$-time algorithm that unifies the verification of both rings and fields through elementary algebraic analysis and combinatorial validation techniques, without invoking the Classification of Finite Simple Groups (CFSG). Our approach achieves, for the first time, linear-time deterministic recognition of ring structures and establishes the optimal time complexity for this problem in the deterministic setting. By circumventing reliance on advanced group-theoretic machinery, the method significantly reduces the theoretical prerequisites for such algebraic verification tasks.

This work addresses the challenge of solving high-degree polynomial constraints in quantifier-free nonlinear integer arithmetic (NIA) by proposing an enhanced incremental linearization method. Built as a standalone implementation atop Z3, the approach introduces a refined axiomatization scheme that enables more effective linear approximations of high-degree monomials—such as powers and mixed products—thereby significantly improving convergence behavior. Experimental evaluation on the SMT-LIB NIA benchmarks demonstrates that the method matches the overall performance of state-of-the-art solvers and substantially outperforms existing techniques on instances dominated by high-degree polynomials.

This work addresses the satisfiability problem for Constrained Horn Clauses (CHCs) over the theory of fixed-size bit-vectors (𝒯_B), a key challenge in bit-precise program verification. The paper introduces Mosaic, a novel framework that enables modular cooperative reasoning between bit-vector and integer arithmetic theories for the first time. By decomposing CHC problems into theory-specific fragments and facilitating cross-theory translation and information exchange, Mosaic overcomes the scalability limitations of existing CHC solvers in bit-level verification tasks. Implemented on top of Z3 and Spacer, Mosaic demonstrates substantial performance improvements over native Spacer on benchmarks involving bit-vector operations, thereby validating its effectiveness and superiority.

This work addresses the problem of efficiently computing deterministic two-element representations of ideals in number fields. Focusing on ideals whose norm is coprime to the index of the defining polynomial’s ring of integers—a class that includes cryptographically relevant cases such as those defined by cyclotomic polynomials—the paper presents the first deterministic polynomial-time algorithm for this task. The approach leverages a generalized Dedekind criterion to decompose and construct ideals within number fields defined as ℚ[x]/(f). This method overcomes prior limitations that relied on randomization or failed to scale to cryptographic parameters, thereby achieving, for the first time, a combination of determinism, efficiency, and completeness across a broad and practically significant class of ideals used in cryptographic applications.

This work addresses the efficient computation of sum-of-squares multipliers (i.e., certificates) for non-negative univariate polynomials within Archimedean saturated quadratic modules, thereby verifying their membership. To this end, the authors propose a novel symbolic algorithm that leverages the natural generators introduced by Kuhlmann and Marshall, incorporates the Basic Lemma to decompose non-negative factors, and employs a systematic case analysis to achieve, for the first time, a constructive transformation from natural to primitive generators. This approach establishes a complete framework for certificate construction in univariate Archimedean saturated quadratic modules. Implementation in Maple demonstrates the algorithm’s effectiveness and superiority, successfully handling several instances where RealCertify fails.