Hermite Normal Form Research Articles

An [formula omitted]-adic algorithm for bivariate Gröbner bases

Let A be a domain, with m⊆A a maximal ideal, and let F⊆A[x,y] be any finite generating set of an ideal with finitely many roots (in an algebraic closure of the fraction field K of A). We present a randomized m-adic algorithm to recover the lexicographic Gröbner basis G of 〈F〉⊆K[x,y], or of its primary component at the origin. We observe that previous results of Lazard's that use Hermite normal forms to compute Gröbner bases for ideals with two generators can be generalized to a generating set F of cardinality greater than two. We use this result to bound the size of the coefficients of G, and to control the probability of choosing a good maximal ideal m⊆A. We give a complete cost analysis over number fields (K=Q(α)) and function fields (▪), and we obtain a complexity that is less than cubic in terms of the dimension of K/〈G〉 and softly linear in the size of its coefficients.

Extracting Topological Orders of Generalized Pauli Stabilizer Codes in Two Dimensions

In this paper, we introduce an algorithm for extracting topological data from translation invariant generalized Pauli stabilizer codes in two-dimensional systems, focusing on the analysis of anyon excitations and string operators. The algorithm applies to Zd qudits, including instances where d is a nonprime number. This capability allows the identification of topological orders that differ from the Zd toric codes. It extends our understanding beyond the established theorem that Pauli stabilizer codes for Zp qudits (with p being a prime) are equivalent to finite copies of Zp toric codes and trivial stabilizers. The algorithm is designed to determine all anyons and their string operators, enabling the computation of their fusion rules, topological spins, and braiding statistics. The method converts the identification of topological orders into computational tasks, including Gaussian elimination, the Hermite normal form, and the Smith normal form of truncated Laurent polynomials. Furthermore, the algorithm provides a systematic approach for studying quantum error-correcting codes. We apply it to various codes, such as self-dual CSS quantum codes modified from the two-dimensional honeycomb color code and non-CSS quantum codes that contain the double semion topological order or the six-semion topological order. Published by the American Physical Society 2024

Crystal-Structure Matches in Solid-Solid Phase Transitions.

The exploration of solid-solid phase transition suffers from the uncertainty of how atoms in two crystal structures match. We devised a theoretical framework to describe and classify crystal-structure matches (CSM). Such description fully exploits the translational and rotational symmetries and is independent of the choice of supercells. This is enabled by the use of the Hermite normal form, an analog of reduced echelon form for integer matrices. With its help, exhausting all CSMs is made possible, which goes beyond the conventional optimization schemes. In an example study of the martensitic transformation of steel, our enumeration algorithm finds many candidate CSMs with lower strains than known mechanisms. Two long-sought CSMs accounting for the most commonly observed Kurdjumov-Sachs orientation relationship and the Nishiyama-Wassermann orientation relationship are unveiled. Given the comprehensiveness and efficiency, our enumeration scheme provide a promising strategy for solid-solid phase transition mechanism research.

A Cubic Algorithm for Computing the Hermite Normal Form of a Nonsingular Integer Matrix

A Las Vegas randomized algorithm is given to compute the Hermite normal form of a nonsingular integer matrix A of dimension n . The algorithm uses quadratic integer multiplication and cubic matrix multiplication and has running time bounded by O(n 3 (log n + log ||A||) 2 (log n ) 2 ) bit operations, where || A ||= max ij | A ij | denotes the largest entry of A in absolute value. A variant of the algorithm that uses pseudo-linear integer multiplication is given that has running time (n 3 log || A ||) 1+ o (1) bit operations, where the exponent “ + o (1)” captures additional factors c 1 (log n ) c2 (loglog|| A ||) c3 for positive real constants c 1 ,c 2 ,c 3 .

A provably secure collusion-resistant identity-based proxy re-encryption scheme based on NTRU

Proxy re-encryption (PRE) technology realizes the transformation of decryption right from a delegator to a delegatee. In response to the rapid development of quantum attack technology, scholars have proposed the lattice-based PRE schemes, and the underlying encryption scheme of most lattice-based PRE schemes is the dual cryptosystem, which has the defects of large storage overhead, computational overhead and low efficiency. To address these deficiencies, we present a novel identity-based proxy re-encryption (IB-PRE) scheme based on NTRU. By means of the underlying efficient NTRU scheme, the proposed scheme reduces the storage overhead and computational overhead on key size and computational complexity, and improves efficiency. Our scheme possesses desired properties such as multi-hop property, unidirectionality and collusion-resistance. Meanwhile, based on the decisional Hermite normal form of ring learning with errors (D-RLWEHNF) hard problem, it is carefully designed and demonstrated to be secure under selective identity and chosen plaintext attacks in the standard model.

Two-sided Galois duals of multi-twisted codes

Galois duals of Multi-twisted (MT) codes are considered in this study. We describe a MT code mathcal {C} as a module over a principal ideal domain. Hence, mathcal {C} has a generator polynomial matrix (GPM) textbf{G} that satisfies an identical equation. We prove a GPM formula for the Euclidean dual mathcal {C}^perp using the identical equation of the Hermite normal form of textbf{G}. Next, we aim to replace the Euclidean dual with the Galois dual. The Galois inner product is an asymmetric form, so we distinguish between the right and left Galois duals. We show that the right and left Galois duals of a MT code are MT as well but with possibly different shift constants. Some interconnected identities for the right and left Galois duals of a linear code are established and we also introduce the two-sided Galois dual. We use a condition that makes the two-sided Galois dual of a MT code MT, then we describe its GPM. Two special cases are also studied, one when the right and left Galois duals trivially intersect and the other when they coincide. A necessary and sufficient condition is established for the equality of the right and left Galois duals of any linear code.

Lattice-based group signatures with forward security for anonymous authentication

Group signatures allow users to sign messages on behalf of a group without revealing authority is capable of identifying the user who generated it. However, the exposure of the user's signing key will severely damage the group signature scheme. In order to reduce the loss caused by signing key leakage, Song proposed the first forward-secure group signature. If a group signing key is revealed at the current time period, the previous signing key will not be affected. This means that the attacker cannot forge group signatures regarding messages signed in the past. To resist quantum attacks, many lattice-based forward-secure group signatures have been proposed. However, their key-update algorithm is expensive since they require some costly computations such as the Hermite normal form (HNF) operations and conversion from a full-rank set of lattice vectors into a basis.In this paper, we propose the group signature with forward security from lattice. In comparison with previous works, we have several advantages: Firstly, our scheme is more effective since we only need to sample some vectors independently from a discrete Gaussian during the key-update algorithm. Secondly, the derived secret key size is linear instead of quadratic with the lattice dimensions, which is more friendly towards lightweight applications. Anonymous authentication plays an increasingly critical role in protecting privacy and security in the environment where private information could be collected for intelligent analysis. Our work contributes to the anonymous authentication in the post-quantum setting, which has wide potential applications in the IoT environment.

Incremental Interval Assignment by Integer Linear Algebra with Improvements

Interval Assignment (IA) is the problem of selecting the number of mesh edges (intervals) for each curve for conforming quad and hex meshing. The intervals x is fundamentally integer-valued. Many other approaches perform numerical optimization then convert a floating-point solution into an integer solution, which is slow and error prone. We avoid such steps: we start integer, and stay integer. Incremental Interval Assignment (IIA) uses integer linear algebra (Hermite normal form) to find an initial solution to the meshing constraints, satisfying the integer matrix equation Ax=b. Solving for reduced row echelon form provides integer vectors spanning the nullspace of A. We add vectors from the nullspace to improve the initial solution, maintaining Ax=b. Heuristics find good integer linear combinations of nullspace vectors that provide strict improvement towards variable bounds or goals. IIA always produces an integer solution if one exists. In practice we usually achieve solutions close to the user goals, but there is no guarantee that the solution is optimal, nor even satisfies variable bounds, e.g. has positive intervals. We describe several algorithmic changes since first publication that tend to improve the final solution. The software is freely available.

Fast verification and public key storage optimization for unstructured lattice-based signatures

A recent work of Sipasseuth, Plantard and Susilo proposed to accelerate lattice-based signature verifications and compress public key storage at the cost of a precomputation on a public key. This first approach, which focused on a restricted type of key, did not include most NIST candidates or most lattice representations in general. In this work, we first present a way to improve even further both their verification speed and their public key compression capability by using a generator of numbers that better suit the method needs. We then also generalize their framework to apply to q-ary lattice schemes as well as classical lattices using Hermite Normal Form, improving their security and applicable scope, thus exhibiting potential trade-offs to accelerate lattice-based signature verification in general and compression of the public key on the verifier side for unstructured lattices.

On the determination of dense coincidence site lattice planes.

The concept of coincidence site lattice (CSL) is used in descriptions of geometry of some intercrystalline boundaries. In face-centered cubic and body-centered cubic metals, atoms are located at the nodes of lattices, and results concerning lattice nodes are applicable to atomic sites. One of the criteria for special boundary configurations is that the boundary passes through a plane with a high density of coinciding atomic sites. Hence, there is an issue of identification of such planes. This paper describes a simple and reliable method for determining the planes with high densities of coinciding lattice nodes. The key elements of the procedure are the Hermite normal form of an integer matrix and Niggli reduction of the CSL basis. In its general form, the method is applicable to arbitrary three-dimensional lattices possessing a common three-dimensional sublattice. The densest and second-densest planes are determined for low-Σ CSLs of cubic lattices.

Random Integer Lattice Generation via the Hermite Normal Form.

Lattices used in cryptography are integer lattices. Defining and generating a “random integer lattice” are interesting topics. A generation algorithm for a random integer lattice can be used to serve as a random input of all the lattice algorithms. In this paper, we recall the definition of the random integer lattice given by G. Hu et al. and present an improved generation algorithm for it via the Hermite normal form. It can be proven that with probability ≥0.99, this algorithm outputs an n-dim random integer lattice within operations.

A Note on Decomposable and Reducible Integer Matrices

We propose necessary and sufficient conditions for an integer matrix to be decomposable in terms of its Hermite normal form. Specifically, to each integer matrix, we associate a symmetric integer matrix whose reducibility can be efficiently determined by elementary linear algebra techniques, and which completely determines the decomposability of the first one.

Borel–de Siebenthal Theory for Affine Reflection Systems

We develop a Borel-de Siebenthal theory for affine reflection systems by classifying their maximal closed subroot systems. Affine reflection systems (introduced by Loos and Neher) provide a unifying framework for root systems of finite-dimensional semi-simple Lie algebras, affine and toroidal Lie algebras, and extended affine Lie algebras. In the special case of nullity $k$ toroidal Lie algebras, we obtain a one-to-one correspondence between maximal closed subroot systems with full gradient and triples $(q,(b_i),H)$, where $q$ is a prime number, $(b_i)$ is a $n$-tuple of integers in the interval $[0,q-1]$ and $H$ is a $(k\times k)$ Hermite normal form matrix with determinant $q$. This generalizes the $k=1$ result of Dyer and Lehrer in the setting of affine Lie algebras.

Representations of torsion-free arithmetic matroids

We study the representability problem for torsion-free arithmetic matroids. After introducing a “strong gcd property” and a new operation called “reduction”, we describe and implement an algorithm to compute all essential representations, up to equivalence. As a consequence, we obtain an upper bound to the number of equivalence classes of representations. In order to rule out equivalent representations, we describe an efficient way to compute a normal form of integer matrices, up to left-multiplication by invertible matrices and change of sign of the columns (we call it the “signed Hermite normal form”). Finally, as an application of our algorithms, we disprove two conjectures about the poset of layers and the independence poset of a toric arrangement.

On solvability of the matrix equation AXB = C over a principal ideal domain

In this paper we present conditions of solvability of the matrix equation AXB = B over a principal ideal domain. The necessary and sufficient conditions of solvability of equation AXB = B in term of the Smith normal forms and in term of the Hermi-te normal forms of matrices constructed in a certain way by using the coefficients of this equation are proposed. If a solution of this equation exists we propose the method for its construction.

Fast Triangularization of Ideal Latttice Basis

To improve the efficiency of the triangularization of ideal lattice basis, a fast algorithm for triangularizing an ideal lattice basis is proposed by studying the polynomial structure, which runs in time O(n3log2B), where n is the dimension of the lattice, B is the infinity norm of lattice basis. Based on the algorithm, a deterministic algorithm for computing the Smith Normal Form (SNF) of ideal lattice is given, which has the same time complexity and thus is faster than any previously known algorithms. Moreover, for a special class of ideal lattices, a method to transform such triangular bases into Hermite Normal Form (HNF) faster than previous algorithms will be present.

Securely and Efficiently Computing the Hermite Normal Form of Integer Matrices via Cloud Computing

The prevalence of cloud computing greatly promotes the traditional computing paradigm. With the assistance of a cloud, the light-weight device can achieve computation-intensive tasks which may not be done on its own. While, computing the Hermite Normal Form (HNF), which is a standard form of integer matrices, not only is inescapable when solving the linear system of equations over integers, but also has lots of applications in many other fields, such as integer programming and lattice-based cryptography. However, the fast blowup phenomenon of intermediate numbers makes the HNF computation time-consuming. In this paper, we initialize the study of the cloud-assisted HNF computation and design an efficient outsourcing algorithm that enables the resource-constrained client to securely delegate this heavy computation to a resource-abundant yet maybe untrusted cloud server. The main idea involved in our algorithm is a novel matrix encryption method based on random permutation, unimodular matrix transformation and triangular matrix transformation, which makes our algorithm protect the client’s input/output information with the one-way notion and enable the client to detect the cloud’s deception with the optimal probability 1. Besides, rigorous theoretical analysis and extensive experimental evaluation validate the efficiency and the practical performance of our design, and the substantial client-side savings are remarkable as the problem size increases.

Calcul matriciel généralisé sur les domaines de Prüfer

In this paper, we first present an algorithm for computing the Hermite normal form of pseudo-matrices over Prüfer domains. This algorithm allows us to provide constructive proofs of the main theoretical results on finitely presented modules over Prüfer domains and to discuss the resolution of linear systems. In some sense, we generalize the methodology developed by Henri Cohen for Dedekind domains. Finally, we present some results over Prüfer domains of dimension one about the Smith normal form.

Improved Combinatorial Algorithms for the Inhomogeneous Short Integer Solution Problem

The paper is about algorithms for the inhomogeneous short integer solution problem: given $$(\mathbf A , \mathbf s )$$ to find a short vector $$\mathbf{x }$$ such that $$\mathbf A \mathbf{x }\equiv \mathbf s \pmod {q}$$ . We consider algorithms for this problem due to Camion and Patarin; Wagner; Schroeppel and Shamir; Minder and Sinclair; Howgrave–Graham and Joux (HGJ); Becker, Coron and Joux (BCJ). Our main results include: applying the Hermite normal form (HNF) to get faster algorithms; a heuristic analysis of the HGJ and BCJ algorithms in the case of density greater than one; an improved cryptanalysis of the SWIFFT hash function; a new method that exploits symmetries to speed up algorithms for Ring-SIS in some cases.

A polynomial-time algorithm to compute generalized Hermite normal forms of matrices over [formula omitted

In this paper, we give the first polynomial time algorithm to compute the generalized Hermite normal form for a matrix F over Z[x], or equivalently, the reduced Gröbner basis of the Z[x]-module generated by the column vectors of F. The algorithm has polynomial bit size computational complexities and is also shown to be practically more efficient than existing algorithms. The algorithm is based on three key ingredients. First, an F4 style algorithm to compute the Gröbner basis is adopted, where a novel prolongation is designed such that the sizes of coefficient matrices under consideration are nicely controlled. Second, the complexity bound of the algorithm is achieved by a nice estimation for the degree and height bounds of the polynomials in the generalized Hermite normal form. Third, fast algorithms to compute Hermite normal forms of matrices over Z are used as the computational tool.