A New Reduction Method from Multivariate Polynomials to Univariate Polynomials

TL;DR

The paper tackles efficient multiplication of multivariate polynomials by transforming them reversibly into univariate polynomials, performing fast univariate multiplication, and precisely recovering the multivariate product. It introduces three reduction strategies—Iterative Kronecker Substitution (IKS), CRT-based reduction, and a Hybrid reduction that combines both—aimed at minimizing the derived univariate degree $d_{h_x}$. Degree bounds and complexity analyses are provided for each method, along with a unifying comparison showing significant potential gains (e.g., reductions to around $3\%$ of standard Kronecker substitution in some cases). The results enable leveraging optimized univariate multiplication libraries and have practical impact for symbolic computation tasks such as Gröbner bases and post-quantum cryptography, while leaving open the problem of identifying the globally optimal substitution sequence across general cases.

Abstract

Polynomial multiplication is a fundamental problem in symbolic computation. There are efficient methods for the multiplication of two univariate polynomials. However, there is rarely efficiently nontrivial method for the multiplication of two multivariate polynomials. Therefore, we consider a new multiplication mechanism that involves a) reversibly reducing multivariate polynomials into univariate polynomials, b) calculating the product of the derived univariate polynomials by the Toom-Cook or FFT algorithm, and c) correctly recovering the product of multivariate polynomials from the product of two univariate polynomials. This work focuses on step a), expecting the degrees of the derived univariate polynomials to be as small as possible. We propose iterative Kronecker substitution, where smaller substitution exponents are selected instead of standard Kronecker substitution. We also apply the Chinese remainder theorem to polynomial reduction and find its advantages in some cases. Afterwards, we provide a hybrid reduction combining the advantages of both reduction methods. Moreover, we compare these reduction methods in terms of lower and upper bounds of the degree of the product of two derived univariate polynomials, and their computational complexities. With randomly generated multivariate polynomials, experiments show that the degree of the product of two univariate polynomials derived from the hybrid reduction can be reduced even to approximately 3% that resulting from the standard Kronecker substitution, implying an efficient subsequent multiplication of two univariate polynomials.

Figures (1)

• Figure 1: Ratios ${d_{h_x}^{\textrm{IKS}}}/{d_{h_x}^{\textrm{SKS}}}$ and ${d_{h_x}^{\textrm{HR}}}/{d_{h_x}^{\textrm{SKS}}}$ at different tuples of degrees in the partially random case

Theorems & Definitions (7)

• Proposition 1
• Theorem 1
• Theorem 2
• Proposition 2
• Example 1
• Lemma 1: Chinese remainder theorem
• Lemma 2: CRT addition