Power Series Composition in Near-Linear Time (2404.05177v2)

Abstract: We present an algebraic algorithm that computes the composition of two power series in softly linear time complexity. The previous best algorithms are $\mathop{\mathrm O}(n^{1+o(1)})$ by Kedlaya and Umans (FOCS 2008) and an $\mathop{\mathrm O}(n^{1.43})$ algebraic algorithm by Neiger, Salvy, Schost and Villard (JACM 2023). Our algorithm builds upon the recent Graeffe iteration approach to manipulate rational power series introduced by Bostan and Mori (SOSA 2021).

• The paper introduces a near-linear time algorithm for power series composition with a complexity of O(M(n)log m + M(m)), significantly improving efficiency over earlier approaches.
• It extends Graeffe's iteration and utilizes the transposition principle to transform the power projection problem into a scalable computational task.
• The research offers practical insights for computer algebra, combinatorics, and cryptography, laying the groundwork for future innovations in modular composition and polynomial multiplication.

Overview of "Power Series Composition in Near-Linear Time"

This paper, authored by Yasunori Kinoshita and Baitian Li, presents a significant advancement in the computation of power series composition, delivering an algebraic algorithm with a complexity consistent with softly linear time. Notably, this algorithm surpasses previous leading methods, such as the non-algebraic approach by Kedlaya and Umans and the algebraic method by Neiger et al. Critical within fields of computer algebra, combinatorics, and cryptography, the power series composition problem involves finding the coefficients of $f(g(x)) \mod x^m$ for polynomials $f(x)$ and $g(x)$ of degrees less than $m$ and $n$, respectively.

Key Technical Contributions

1. Graeffe's Iteration Approach: The developed algorithm is an extension of the Graeffe root squaring technique, which was initially purposed for computing reciprocals of power series. The novelty lies in applying Graeffe’s method to bivariate power series $P(x)/Q(x,y)$, and transforming this into efficiently computable tasks.
2. Transposition Principle: The paper applies this principle effectively, uniquely transforming the power projection problem into the power series composition challenge with nearly identical computational demands.
3. Algorithm Design: The authors structure the algorithm to maintain the problem's size while iteratively halving and doubling respective degrees of polynomials, a method conducive to stable computational complexity.

Results and Complexity

The fundamental theorem presented ensures that for given polynomials $f(x), g(x)$, the power series composition can be realized with complexity $O(M(n)\log m + M(m))$ arithmetic operations, where $M(n)$ is a function that denotes time complexity for polynomial multiplication. This result not only optimizes regime for imminent logarithmic factors but substantially minimizes practical execution time compared to former benchmarks.

Implications and Future Prospects

The proposed algorithmic improvement is impactful for domains reliant on efficient polynomial computations, extending utility in boolean circuits and multitape Turing machines where bit complexity is pivotal. This research may also be a preamble to prospective innovations in modular composition, potentially elucidating novel pathways within computational algebra and leading to even more efficient methodologies for related algebraic computations.

Further speculation could involve enhancing the polynomial multiplication models, striving to refine function $M$ across diverse computational mediums. In parallel, the theoretical frameworks underpinning transposition principles might be explored to uncover latent efficiencies in other similar algebraic transformations.

Conclusion

Kinoshita and Li's work represents a robust analytical and technical contribution, refining the computational strategies for power series composition to near-linear time efficacy. This accomplishment, grounded in algebraic manipulations and complex analysis, exemplifies an essential optimization within computational algebra that is poised to enhance computational tools and methodologies across multiple scientific and practical applications.