Single-factor lifting and factorization of polynomials over local fields

Let f(x)$f\left(x\right)$ be a separable polynomial over a local field. The Montes algorithm computes certain approximations to the different irreducible factors of f(x)$f\left(x\right)$, with strong arithmetic properties. In this paper, we develop an algorithm to improve any one of these approximations, till a prescribed precision is attained. The most natural application of this “single-factor lifting” routine is to combine it with the Montes algorithm to provide a fast polynomial factorization algorithm. Moreover, the single-factor lifting algorithm may be applied as well to accelerate the computational resolution of several global arithmetic problems in which the improvement of an approximation to a single local irreducible factor of a polynomial is required.     

Approximate factorization of multivariate polynomials using singular value decomposition

We describe the design, implementation and experimental evaluation of new algorithms for computing the approximate factorization of multivariate polynomials with complex coefficients that contain numerical noise. Our algorithms are based on a generalization of the differential forms introduced by W. Ruppert and S. Gao to many variables, and use singular value decomposition or structured total least squares approximation and Gauss–Newton optimization to numerically compute the approximate multivariate factors. We demonstrate on a large set of benchmark polynomials that our algorithms efficiently yield approximate factorizations within the coefficient noise even when the relative error in the input is substantial (10⁻³).     

Generalized Lyapunov equation and factorization of matrix polynomials

This paper presents an algorithm for the construction of a solution of the generalized Lyapunov equation. It is proved that the polynomial matrix factorization relative to the imaginary axis may be reduced to the successive solution of Lyapunov equations, i.e. the factorization is reduced to the solution of a sequence of generalized Lyapunov equations, not to the solution of generalized Riccati equation.     

Factoring polynomials over global fields II

In this paper we describe software for an efficient factorization of polynomials over global fields F. The algorithm for function fields was recently incorporated into our system KANT. The method is based on a generic algorithm developed by the second author in an earlier paper in this journal. Besides algorithmic aspects not contained in that paper we give details about the current implementation and about some complexity issues as well as a few illustrative examples. Also, a generalization of the application of LLL reduction for factoring polynomials over arbitrary global fields is developed.     

基于有限域上Chebyshev多项式的密钥协商方案

利用传统RSA算法和有限域上离散对数问题,提出一种新的基于混沌映射的密钥协商方案。该方案基于有限域上Chebyshev多项式良好的半群特性,运用RSA算法巧妙地隐藏通信双方产生的有限域上的Cheby-shev多项式值,从而避免了以往的种种主动攻击,保证了密钥协商的安全;同时,该密钥协商方案还实现了身份认证功能。理论分析和软件实现证明了该方案的可行性、正确性和安全性。     

On factorization forests of finite height

Simon (Theoret. Comput. Sci. 72 (1990) 65–94) has proved that every morphism from a free semigroup to a finite semigroup S admits a Ramseyan factorization forest of height at most 9|S|. In this paper, we prove the same result of Simon with an improved bound of 7|S|. We provide a simple algorithm for constructing a factorization forest. In addition, we show that the algorithm cannot be improved significantly. We give examples of semigroup morphism such that any Ramseyan factorization forest for the morphism would require a height not less than |S|.     

New algorithms for generating Conway polynomials over finite fields

Arithmetic in a finite field of prime characteristic p normally employs an irreducible polynomial in $\text{Z}{}_{\mathrm{p}}\text{[X]}$. A particular class of irreducible polynomials, generally known as Conway polynomials, provides a means for representing several finite fields of characteristic p in a compatible manner. Conway polynomials are used in computational algebra systems such as GAP and Magma to represent finite fields. The generation of the Conway polynomial for a particular finite field has previously been done by an often expensive brute force search. We present two new algorithms for generating Conway polynomials that avoid the brute force search. We have implemented one of these algorithms in Magma and present numerous new Conway polynomials generated by our implementation.     

Consensus networks over finite fields

This work studies consensus strategies for networks of agents with limited memory, computation, and communication capabilities. We assume that agents can process only values from a finite alphabet, and we adopt the framework of finite fields, where the alphabet consists of the integers {0,…,p−1}$\{0,\dots ,p-1\}$, for some prime number p$p$, and operations are performed modulo p$p$. Thus, we define a new class of consensus dynamics, which can be exploited in certain applications such as pose estimation in capacity and memory constrained sensor networks. For consensus networks over finite fields, we provide necessary and sufficient conditions on the network topology and weights to ensure convergence. We show that consensus networks over finite fields converge in finite time, a feature that can be hardly achieved over the field of real numbers. For the design of finite-field consensus networks, we propose a general design method, with high computational complexity, and a network composition rule to generate large consensus networks from smaller components. Finally, we discuss the application of finite-field consensus networks to distributed averaging and pose estimation in sensor networks.     

Linear structures of symmetric functions over finite fields

It is shown that nonlinear symmetric functions over finite fields GF(p) have no linear structures other than equal component vectors.     

Towards toric absolute factorization

This article presents an algorithmic approach to study and compute the absolute factorization of a bivariate polynomial, taking into account the geometry of its monomials. It is based on algebraic criterions inherited from algebraic interpolation and toric geometry.     

基于NTL算法库的多元多项式因式分解高效实现

针对多元多项式因式分解困难问题,利用现有因式分解算法,提出了一种基于任意精度计算函数库NTL的高效多元多项式因式分解实现方法HPFMP。介绍了NTL算法库,讨论了如何运用该算法库实现高效的数论与计算代数计算;充分利用具备高效、任意精度大整数、实数的计算数论与计算代数的NTL算法库实现了多元多项式因式分解;与现有代数系统Maple11进行了对比测试,实验结果表明,该实现方法具有更高的效率。     

Efficient bit-parallel multipliers over finite fields GF(2)

Hardware implementation of multiplication in finite field GF(2^m) based on sparse polynomials is found to be advantageous in terms of space-complexity as well as the time-complexity. In this paper, we present a new permutation method to construct the irreducible like-trinomials of the form (x + 1)^m + (x + 1)ⁿ + 1 for the implementation of efficient bit-parallel multipliers. For implementing the multiplications based on such polynomials, we have defined a like-polynomial basis (LPB) as an alternative to the original polynomial basis of GF(2^m). We have shown further that the modular arithmetic for the binary field based on like-trinomials is equivalent to the arithmetic for the field based on trinomials. In order to design multipliers for composite fields, we have found another permutation polynomial to convert irreducible polynomials into like-trinomials of the forms (x² + x + 1)^m + (x² + x + 1)ⁿ + 1, (x² + x)^m + (x² + x)ⁿ + 1 and (x⁴ + x + 1)^m + (x⁴ + x + 1)ⁿ + 1. The proposed bit-parallel multiplier over GF(2^4m) is found to offer a saving of about 33% multiplications and 42.8% additions over the corresponding existing architectures.     

Optimal Deterministic Broadcasting in Known Topology Radio Networks

We consider deterministic broadcasting in radio networks whose nodes have full topological information about the network. The aim is to design a polynomial algorithm, which, given a graph G with source s, produces a fast broadcast scheme in the radio network represented by G. The problem of finding a fastest broadcast scheme for a given graph is NP-hard, hence it is only possible to get an approximation algorithm. We give a deterministic polynomial algorithm which produces a broadcast scheme of length , for every n-node graph of diameter D, thus improving a result of Gąsieniec et al. (PODC 2005) [17] and solving a problem stated there. Unless the inclusion NP BPTIME( holds, the length of a polynomially constructible deterministic broadcast scheme is optimal.A preliminary version of this paper (with a weaker result) appeared in the Proc. 7th International Workshop on Approximation Algorithms for Combinatorial Optimization Problems (APPROX’2004), August 2004, Harvard University, Cambridge, USA, LNCS 3122, 171–182. Research of the second author supported in part by NSERC discovery grant and by the Research Chair in Distributed Computing of the Université du Québec en Outaouais. Part of this work was done during the second author’s visit at the Max-Planck-Institut für Informatik.     

矩阵的Doolittle递归分解算法及符号程序设计

将矩阵An×n的Doolittle分解推广到Am×n上，并在常规的迭代算法上加以创新，给出了递归的分解算法．在实现算法的过程中，对数据进行了巧妙处理，使中间数据及最终计算结果都具有分数形式，提高了结果的精确度，而且更符合人们阅读的习惯．经过运行测试，算法设计合理，程序运行高效准确．程序是对MathSoft公司的交互式的数学文字软件Mathcad的矩阵分解的数值计算扩充到符号运算．     

Performance of parallel Cholesky factorization algorithms using BLAS

This paper considers four parallel Cholesky factorization algorithms, including SPOTRF from the February 1992 release of LAPACK, each of which call parallel Level 2 or 3 BLAS, or both. A fifth parallel Cholesky algorithm that calls serial Level 3 BLAS is also described. The efficiency of these five algorithms on the CRAY-2, CRAY Y-MP/832, Hitachi Data Systems EX 80, and IBM 3090-600J is evaluated and compared with a vendor-optimized parallel Cholesky factorization algorithm. The fifth parallel Cholesky algorithm that calls serial Level 3 BLAS provided the best performance of all algorithms that called BLAS routines. In fact, this algorithm outperformed the Cray-optimized libsci routine (SPOTRF) by 13–44%;, depending on the problem size and the number of processors used.This work was supported by grants from IMSL, Inc., and Hitachi Data Systems. The first version of this paper was presented as a poster session at Supercomputing '90, New York City, November 1990.     

非负张量分解的快速算法*

作为非负矩阵分解的多线性推广,非负张量分解已被成功地应用在信号处理、计算机视觉、数据挖掘和神经科学等领域中.提出了非负张量分解的一种快速算法.首先,将大的张量数据视做多元连续函数的离散化,对其进行采样得到一个小张量;其次,对小张量执行非负分解,可得到它的重构张量;然后,对于采样后的重构张量,使用二维线性插值方法对原始张量进行重构;最后,实验结果表明快速张量分解算法的有效性.