Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients

Abstract: Sparse polynomial multiplication is a fundamental problem in computer algebra and the theory of computation, and the development of a quasi-linear time output-sensitive multiplication algorithm has been posed as an open challenge. In this paper, a counterexample is provided to a previously claimed solution to this open problem for integer coefficients. By employing the existing quasi-linear modular-black-box interpolation algorithm, we are able to provide an algorithm with quasi-linear bit complexity for the integer coefficients setting. Furthermore, in the case of coefficients over a finite field, we obtain an algorithm whose bit complexity is linear in the number of terms, the logarithm of the degree, and the logarithm of the size of the finite field.

• The paper introduces an algorithm achieving quasi-linear bit complexity for multiplying sparse polynomials with integer coefficients via modular-black-box interpolation.
• It refutes previous probabilistic claims by providing a counterexample, emphasizing the importance of output-sensitive complexity analysis.
• The approach extends to multivariate and finite field cases with formal verification in Lean 4, offering significant practical and theoretical improvements.

Quasi-linear Complexity Algorithms for Sparse Polynomial Multiplication with Integer Coefficients

Introduction and Context

The multiplication of sparse polynomials with integer coefficients is a central problem within symbolic computation, computer algebra, and complexity theory. Classical algorithms targeting dense representations, including Karatsuba, Toom-Cook, and Schönhage-Strassen, deliver quasi-linear time complexity in the degree and size of the input. However, many applications operate with sparse polynomials, for which the coefficient vector's support is substantially smaller than its degree. This presents unique algorithmic challenges: output-sensitive complexity becomes essential since the output's sparsity may grow exponentially relative to the input, and the bit size of the result is dictated not merely by sparsity but also by degree and coefficient height.

A longstanding open question posited whether there exists a quasi-linear bit complexity, output-sensitive algorithm for sparse integer polynomial multiplication—i.e., one with complexity $\widetilde{O}(T\log(DH))$, where $T$ is a bound on the number of terms in the product, $D$ is the degree bound, and $H$ the maximum coefficient magnitude.

This work demonstrates both the intricacy of the problem—by refuting prior claims of resolution—and the construction of an algorithm that achieves the desired complexity bounds via modular-black-box (MBB) interpolation for integer polynomials. The results generalize to polynomials over large-characteristic finite fields, where the bit complexity is shown to be $\widetilde{O}(T \log D \log q)$.

Counterexample to Prior Claims

The paper establishes, via explicit counterexample, the failure of a key probabilistic lemma underpinning the earlier claim in "Essentially optimal sparse polynomial multiplication" [Giorgi et al., ISSAC 2020]. Specifically, Lemma 4.7(ii) is invalidated as its collision-counting argument underestimates the likelihood that cancellation of terms splits across modular images, so that the sparsity of modular products exceeds the true output sparsity (demonstrated through manipulation of $F=(1-X^N)^4$ and $G=(1+X^N)^4$ for large $N$ over appropriate primes). This nullifies the previously claimed proof of output-sensitive quasi-linear time complexity in the integer case.

Integer Sparse Multiplication via Modular Black-Box Interpolation

The paper proposes an alternative approach based on modular-black-box sparse interpolation—specifically, Theorem 3.4 from "Sparse Polynomial Interpolation and Division in Soft-linear Time" [Giorgi et al., ISSAC 2022]—yielding an algorithm with bit complexity $O(T \log(DH))$ provided the output sparsity $T$ is known. This algorithm operates in univariate and, via Kronecker substitution, in the multivariate case.

Key features:

• The algorithm treats polynomial multiplication as an instance of sparse interpolation, answering black-box evaluation queries by evaluating the pointwise product $T$0 modulo random primes and power-of-two modulus.
• Output-sparsity $T$1 must be supplied, but when it is unknown, a doubling strategy plus probabilistic verification yields the same complexity with high probability (Las Vegas algorithm via product verification following [GIORGI202398-JSC]).
• The modular evaluation and batched root-of-unity interpolation enable quasi-linear bit complexity, matching the size of the output.

Multivariate Generalization

Via Kronecker substitution, the result extends to polynomials in $T$2 variables. Degrees scale as $T$3, but the resulting bit complexity scales as $T$4.

Verification and Unknown Sparsity

In the unknown output-sparsity regime, the algorithm repeatedly guesses sparsity bounds of the form $T$5, verifies candidate multiplications with probabilistic, output-sensitive modular verification [GIORGI202398-JSC], and thus achieves expected complexity matching the known-sparsity case up to logarithmic factors.

Finite Field Multiplication

A parallel result is established for multiplication over finite fields $T$6, with characteristic exceeding the degree bound. The finite field framework employs sparse interpolation techniques leveraging derivative images and prime-isolation:

• For a product $T$7 with degree bound $T$8 and supplied output-sparsity $T$9, the algorithm uses reductions modulo $D$0 for sampled primes $D$1, derivative images to generate candidate exponents, and cyclic images to filter candidates.
• The cost is $D$2 bit operations, assuming arithmetic over $D$3 is at unit cost.
• This approach implies a near-optimal bit complexity for sparse products over large characteristic finite fields.

The algorithm's probabilistic correctness is controlled via union bounds over the candidate isolation and filtering stages. Both derivative and candidate filtering steps are efficiently implementable due to efficient FFT-based dense cyclic multiplication [Cantor-Kaltofen 1991].

Lean Formalization

A significant aspect of the paper is the formalization of the main theorems, algebraic invariants, cost bounds, and algorithmic wrappers in Lean 4, facilitated by the MechMath Agent Team (MMAT) system. The Lean development encapsulates all structural correctness claims, with formalized versions of the supplied-output-sparsity algorithm and the finite-field route. Algorithmic black-box theorems and deep bit-complexity or number-theoretic estimates are packaged as named axioms, with careful boundary management of internal/external proof obligations.

Theoretical and Practical Implications

The paper corrects the literature by explicitly identifying a flaw in prior probabilistic analysis for sparse multiplication with integer coefficients. More importantly, it establishes that (Monte Carlo) algorithms achieving bit complexity quasi-linear in the true output size exist for both integer and finite-field coefficients, thus matching the information-theoretic lower bounds up to soft-O(logarithmic) factors.

From a practical point of view:

• These algorithms can be implemented to improve the performance of sparse arithmetic routines in computer algebra systems, addressing realistic example classes where both input and output sparsity are high but degree and coefficient sizes are large.
• The methodology is amenable to parallelization, as the modular black-box evaluations are independent and can be processed concurrently.
• The extension to multivariate polynomials preserves the output-sensitive guarantees, a significant requirement in the manipulation of high-dimensional symbolic polynomials.

Theoretically, this work unlocks new directions for further derandomization, deterministic complexity upper bounds, and more refined analyses regarding the average-case behavior of sparse products in both algebraic and complexity-theoretic settings.

Conclusion

This paper both corrects a prior literature error and introduces algorithms for sparse integer (and large-characteristic finite field) polynomial multiplication attaining quasi-linear bit complexity in the output size. The modular-black-box interpolation paradigm, together with rigorous product verification, ensures correctness and efficiency. Formalization in Lean highlights advances in the machine certification of algebraic algorithms. Further work may address removing randomness, extending to arbitrary coefficient domains, and eliminating the remaining logarithmic factors in complexity for some regimes.

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Reference: "Quasi-linear Time Multiplication of Sparse Polynomials with Integer Coefficients" (2606.12100)