[Paper Review] Algorithms for commutative algebras over the rational numbers

H. W. Lenstra, Alice Silverberg|arXiv (Cornell University)|Sep 29, 2015

Coding theory and cryptography7 references3 citations

TL;DR

This paper presents deterministic polynomial-time algorithms for computing fundamental structures in finite-dimensional commutative algebras over ℚ, including the nilradical, prime ideals, localizations, residue fields, separable subalgebra, and primitive idempotents. It introduces a novel derivation-based method to lift idempotents against nilpotents, replacing traditional Hensel-Newton iterations, and provides an explicit formula valid in any commutative ring, enabling efficient solution of the discrete logarithm problem in the multiplicative group.

ABSTRACT

The algebras considered in this paper are commutative rings of which the additive group is a finite-dimensional vector space over the field of rational numbers. We present deterministic polynomial-time algorithms that, given such an algebra, determine its nilradical, all of its prime ideals, as well as the corresponding localizations and residue class fields, its largest separable subalgebra, and its primitive idempotents. We also solve the discrete logarithm problem in the multiplicative group of the algebra. While deterministic polynomial-time algorithms were known earlier, our approach is different from previous ones. One of our tools is a primitive element algorithm; it decides whether the algebra has a primitive element and, if so, finds one, all in polynomial time. A methodological novelty is the use of derivations to replace a Hensel-Newton iteration. It leads to an explicit formula for lifting idempotents against nilpotents that is valid in any commutative ring.

Motivation & Objective

• To develop deterministic polynomial-time algorithms for core computational problems in finite-dimensional commutative ℚ-algebras.
• To provide an alternative to Hensel-Newton iteration by using derivations for lifting idempotents.
• To compute the Jordan-Chevalley decomposition (separable and nilpotent parts) efficiently and simultaneously.
• To solve the discrete logarithm problem in the multiplicative group of the algebra.
• To determine the structure of the unit group, including generators of the kernel of group homomorphisms from ℤ^S to E^*.

Proposed method

• The paper uses a derivation-based approach to replace Hensel-Newton iteration, introducing a universal formula for lifting idempotents in any commutative ring.
• It defines a canonical ℚ-linear map δ: ℚ[X]/(g) → ℚ[X]/(g, g') whose kernel is E_sep, enabling direct computation of the separable subalgebra.
• The nilradical and separable subalgebra are computed simultaneously via a non-iterative algorithm based on Theorem 1.3, avoiding coefficient blow-up.
• The algorithm for the discrete logarithm in E^* uses reduction modulo prime ideals and logarithmic lifting in the nilradical, leveraging the Jordan-Chevalley decomposition.
• Kernel computation in the unit group is achieved by combining kernel algorithms over residue fields and the nilradical, using integer matrix methods.
• The method relies on a primitive element algorithm that decides existence and computes a primitive element in polynomial time.

Experimental results

Research questions

• RQ1 Can the nilradical and separable subalgebra of a ℚ-algebra be computed simultaneously in polynomial time without iterative methods?
• RQ2 Is there a derivation-based method to lift idempotents against nilpotents that is valid in any commutative ring and avoids Hensel-Newton iteration?
• RQ3 Can the discrete logarithm problem in the multiplicative group of a ℚ-algebra be solved in deterministic polynomial time?
• RQ4 What is the computational complexity of determining all prime ideals and their corresponding localizations in a ℚ-algebra?
• RQ5 How can the kernel of a group homomorphism from ℤ^S to E^* be computed efficiently when E is a finite-dimensional ℚ-algebra?

Key findings

• A deterministic polynomial-time algorithm computes a ℚ-basis for both E_sep and √0, along with the Jordan-Chevalley decomposition map and its inverse.
• The kernel of the derivation map δ: E → Ω_{E/ℚ} is not always equal to E_sep, as shown by a counterexample with a non-separable element in ker(d).
• An explicit formula for lifting idempotents against nilpotents is derived using derivations, valid in any commutative ring.
• The algorithm for the discrete logarithm in E^* runs in polynomial time by combining reductions modulo prime ideals and logarithmic lifting in the nilradical.
• The kernel of group homomorphisms from ℤ^S to E^* is computed as the intersection of kernels over residue fields and the nilradical, using integer matrix kernel algorithms.
• The paper provides a primitive element algorithm that determines existence and computes a primitive element in polynomial time.