Core Concepts

We present a simple and efficient algorithm that computes the composition of two power series in near-linear time complexity, improving upon the previous best algorithms.

Abstract

The paper presents an efficient algorithm for computing the composition of two power series, f(g(x)) mod xn, where f(x) and g(x) are polynomials in A[x] of degrees less than m and n, respectively.
Key highlights:
The previous best algorithms for general power series composition had a time complexity of O(n^(1+o(1))) or O(n^1.43), whereas the new algorithm achieves a time complexity of O(M(n) log m + M(m)), where M(n) is the time complexity of polynomial multiplication.
The algorithm builds upon the Graeffe iteration approach introduced by Bostan and Mori to manipulate rational power series, reducing the problem to a series of polynomial multiplications with exponentially decreasing degrees.
The algorithm also has consequences for other computation models, such as boolean circuits and multitape Turing machines, depending on the fast arithmetic available in the underlying ring A.
The paper also presents an efficient algorithm for the transposed problem, power projection, and shows how the transposition principle can be used to derive the power series composition algorithm.

Stats

The previous best algorithms for general power series composition had a time complexity of O(n^(1+o(1))) or O(n^1.43).
The new algorithm achieves a time complexity of O(M(n) log m + M(m)), where M(n) is the time complexity of polynomial multiplication.

Quotes

"We present a simple algorithm that reduces the complexity of power series composition to near-linear time."
"Our algorithm also has consequences for other computation models measured by bit complexity, such as boolean circuits and multitape Turing machines."

Key Insights Distilled From

by Yasunori Kin... at **arxiv.org** 04-09-2024

Deeper Inquiries

The techniques used in the algorithm for power series composition can be extended to handle more general forms of power series composition, even when the input polynomials have coefficients in a non-commutative ring. One approach to extend these techniques is to adapt the algorithm to work with non-commutative rings by considering the non-commutative multiplication of polynomials. This adaptation would involve modifying the polynomial multiplication steps in the algorithm to account for the non-commutativity of the ring elements. By incorporating the appropriate algebraic operations for non-commutative rings, the algorithm can be adjusted to handle power series composition with coefficients in such rings.

The development of a near-linear time algorithm for power series composition has significant practical implications across various fields. In combinatorics, where power series are commonly used to represent generating functions for combinatorial structures, the efficiency of this algorithm can lead to faster computations of combinatorial quantities. This can streamline the analysis of combinatorial problems and facilitate the study of complex structures in combinatorics.
In cryptography, power series composition plays a crucial role in cryptographic protocols and algorithms. The near-linear time complexity of the algorithm can enhance the efficiency of cryptographic computations involving power series, leading to faster encryption and decryption processes. This can improve the overall performance of cryptographic systems and make them more resilient against attacks.
Overall, the near-linear time algorithm for power series composition opens up new possibilities for accelerating computations in combinatorics and cryptography, ultimately advancing research and applications in these areas.

While the near-linear time algorithm for power series composition represents a significant advancement in computational efficiency, there are fundamental limitations and lower bounds on the complexity of this problem that the algorithm approaches. The algorithm achieves a time complexity of O(M(n) log m + M(m)), where M(n) denotes the complexity of polynomial multiplication. This complexity is close to linear in the size of the input, making it highly efficient compared to previous algorithms.
However, there may exist inherent lower bounds on the complexity of power series composition that cannot be surpassed. These lower bounds could be related to the intrinsic computational complexity of manipulating power series and performing polynomial multiplications. Further improvements beyond the near-linear time complexity may be challenging to achieve without fundamentally altering the underlying mathematical principles governing power series composition. Nonetheless, ongoing research and advancements in algebraic algorithms may lead to refinements and optimizations that push the boundaries of computational efficiency in power series composition.

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