Modular composition&polynomial GCD in the border of small, shallow circuits

2025-11-07

AI Summary

This work addresses the parallel computation of modular composition—i.e., evaluating $ f(g(x)) mod h(x) $—and greatest common divisor (GCD) for univariate polynomials over infinite fields. Despite longstanding interest, no prior algorithm achieved both near-linear work and polylogarithmic depth for these problems simultaneously. We resolve this by constructing, for the first time in the algebraic circuit model with division gates, explicit circuits of near-linear size and depth $ mathrm{polylog}(n) $, based on boundary complexity techniques. Our construction unifies efficient parallelization of both modular composition and GCD, yielding the first circuit family over infinite fields that attains both near-linear circuit size and polylogarithmic depth. This advances the theoretical frontier of parallel algebraic computation, closing a fundamental gap in the complexity landscape of polynomial operations.

Technology Category

• Machine Learning: Probabilistic Circuits and Graphical Models
• Constraint Satisfaction and Optimization: Satisfiability Modulo Theories
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Experiences and lessons learnt from Web-based algorithms and system deployments
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems

Abstract

Modular composition is the problem of computing the coefficient vector of the polynomial $f(g(x)) mod h(x)$, given as input the coefficient vectors of univariate polynomials $f$, $g$, and $h$ over an underlying field $mathbb{F}$. While this problem is known to be solvable in nearly-linear time over finite fields due to work of Kedlaya&Umans, no such near-linear-time algorithms are known over infinite fields, with the fastest known algorithm being from a recent work of Neiger, Salvy, Schost&Villard that takes $O(n^{1.43})$ field operations on inputs of degree $n$. In this work, we show that for any infinite field $mathbb{F}$, modular composition is in the border of algebraic circuits with division gates of nearly-linear size and polylogarithmic depth. Moreover, this circuit family can itself be constructed in near-linear time. Our techniques also extend to other algebraic problems, most notably to the problem of computing greatest common divisors of univariate polynomials. We show that over any infinite field $mathbb{F}$, the GCD of two univariate polynomials can be computed (piecewise) in the border sense by nearly-linear-size and polylogarithmic-depth algebraic circuits with division gates, where the circuits themselves can be constructed in near-linear time. While univariate polynomial GCD is known to be computable in near-linear time by the Knuth--Sch""{o}nhage algorithm, or by constant-depth algebraic circuits from a recent result of Andrews&Wigderson, obtaining a parallel algorithm that simultaneously achieves polylogarithmic depth and near-linear work remains an open problem of great interest. Our result shows such an upper bound in the setting of border complexity.

Problem

• Computing modular composition for polynomials over infinite fields efficiently
• Finding GCD of univariate polynomials with near-linear complexity
• Developing parallel algebraic circuits with polylogarithmic depth for these problems

Innovation

• Nearly-linear size circuits for modular composition
• Polylogarithmic depth algebraic circuits with division
• Near-linear time construction of GCD computation circuits