Efficient transitive closure of sparse matrices over closed semirings

This paper surveys several alternative data structures and algorithms for multiplying sparse upper-triangular matrices over closed semirings, and evaluates their efficiency in computing transitive closures of matrices over the Boolean semiring. Two new variants are introduced that outperform previously known methods on a collection of large data-sets drawn from linguistic applications.     

The semiring of matrices over a finite chain

Let L_m denote the chain {0,1,2,…,m-1} with the usual ordering and M_n(L_m) the matrix semiring of all n×n matrices with elements in L_m. We firstly introduce some order-preserving semiring homomorphisms from M_n(L_m) to M(L_k). By using these homomorphisms, we show that a matrix over the finite chain L_m can be decomposed into the sum of some matrices over the finite chain L_k, where k<m. As a result, cut matrices decomposition theorem of a fuzzy matrix (Theorem 4 in [Z.T. Fan, Q.S. Cheng, A survey on the powers of fuzzy matrices and FBAMs, International Journal of Computational Cognition 2 (2004) 1-25 (invited paper)]) is generalized and extended. Further, we study the index and periodicity of a matrix over a finite chain and get some new results. On the other hand, we introduce a semiring embedding mapping from the semiring M_n(L_m) to the direct product of the h copies of the semiring M_n(L_k) and discuss Green’s relations on the multiplicative semigroup of the semiring M_n(L_m). We think that some results obtained in this paper is useful for the study of fuzzy matrices.     

一种通用GF(2m)模乘加速器的快速实现

在椭圆曲线密码体制(ECC)中,有限域GF(2m)上模乘运算是最基本的运算,加速模乘运算是提高ECC算法性能的关键。针对不同不可约多项式广泛应用的现状,提出了一种通用GF(2m)模乘加速器设计方案。该加速器通过指令调度的方式,能快捷地完成有限域上模乘运算。实现结果表明,该设计完全适用于智能卡等应用要求。     

Implementing Gaussian elimination on a matrix-matrix multiplication systolic array

We show that any systolic array dedicated to matrix-matrix multiplication can also execute Gaussian elimination.     

基于FPGA的Fm2域椭圆曲线点乘的快速实现

椭圆曲线点乘的实现速度决定了椭圆曲线密码算法（ECC）的实现速度。采用蒙哥马利点乘算法,其中模乘运算、模平方运算采用全并行算法,模逆运算采用费马·小定理并在实现中进行了优化,完成了椭圆曲线点乘的快速运算。采用Xilinx公司的Virtex-5器件族的XCV220T作为目标器件,完成了综合与实现。通过时序后仿真,其时钟频率可以达到40MHz,实现一次点乘运算仅需要14.9μs。     

Optimization techniques for small matrix multiplication

The complexity of matrix multiplication has attracted a lot of attention in the last forty years. In this paper, instead of considering asymptotic aspects of this problem, we are interested in reducing the cost of multiplication for matrices of small size, say up to 30. Following the previous work of Probert & Fischer, Smith, and Mezzarobba, in a similar vein, we base our approach on the previous algorithms for small matrices, due to Strassen, Winograd, Pan, Laderman, and others and show how to exploit these standard algorithms in an improved way. We illustrate the use of our results by generating multiplication codes over various rings, such as integers, polynomials, differential operators and linear recurrence operators.     

Communication efficient matrix multiplication on hypercubes

In a recent paper Fox, Otto and Hey consider matrix algorithms for hypercubes. For hypercubes allowing pipelined broadcast of messages they present a communication efficient algorithm. We present in this paper a similar algorithm that uses only nearest neighbour communication. This algorithm will therefore by very communication efficient also on hypercubes not allowing pipelined broadcast. We introduce a new algorithm that reduces the asymptotic communication cost from . This is achieved by regarding the hypercube as a set of subcubes and by using the cascade sum algorithm.     

Derandomization,witnesses for Boolean matrix multiplication and construction of perfect hash functions

Small sample spaces with almost independent random variables are applied to design efficient sequential deterministic algorithms for two problems. The first algorithm, motivated by the attempt to design efficient algorithms for the All Pairs Shortest Path problem using fast matrix multiplication, solves the problem of computingwitnesses for the Boolean product of two matrices. That is, ifA andB are twon byn matrices, andC=AB is their Boolean product, the algorithm finds for every entryC _ij =1 a witness: an indexk so thatA _ik =B _kj =1. Its running time exceeds that of computing the product of twon byn matrices with small integer entries by a polylogarithmic factor. The second algorithm is a nearly linear time deterministic procedure for constructing a perfect hash function for a givenn-subset of {1,...,m}.Part of this paper was presented at the IEEE 33rd Symposium on Foundations of Computer Science.Research supported in part by a USA-Israeli BSF grant and by the Fund for Basic Research administered by the Israel Academy of Sciences.Supported by an Alon Fellowship and by a grant from the Israel Science Foundation administered by the Israeli Academy of Sciences. Some of this work was done while the author was with the IBM Almaden Research Center.     

Intersection of finitely generated congruences over term algebra

We show that it is decidable for any given ground term rewrite systems R and S if there is a ground term rewrite system U such that ↔_U^*=↔_R^*∩↔_S^*. If the answer is yes, then we can effectively construct such a ground term rewrite system U. In other words, for any given finitely generated congruences ρ and τ over the term algebra, it is decidable if ρ∩τ is a finitely generated congruence. If the answer is yes, then we can effectively construct a ground term rewrite system U such that ↔_U^*=ρ∩τ.     

基于重编码的快速标量乘算法

采用回溯法设计出一种重编码算法。该算法只需对标量序列进行一次变换、至多四个中间变量,以及只需基于比特位比较赋值操作,效率更高,利于硬件实现标量乘法,并证明了所得结果具有正则序列的性质。该算法应用到计算数字签名中常用的gP+hQ时,得到g、h的具有最小联合重量序列。     

External matrix multiplication and all-pairs shortest path

Algorithms are presented for external matrix multiplication and for all-pairs shortest path computation. In comparison with earlier algorithms, the amount of I/O is reduced by a constant factor. The all-pairs shortest path algorithm even performs fewer internal operations, making the algorithm practically interesting.     

Edwards曲线上的快速标量乘法算法

在特征不等于2的域上,将椭圆曲线转换为与其双有理等价的Edwards曲线,可以有效提高ECC的软、硬件实现速度。首先简化了Edwards曲线上倍点的计算公式,然后根据连续倍点2mP(m=2,3,…)的坐标具有统一表示形式的特征,基于递归技术提出了一种计算2mP的连续倍点算法(CDA)。通过算法的复杂性分析与实例计算表明：CDA可使Edwards曲线上标量运算的速度提高10%以上。     

快速傅里叶变换乘法的性能研究

大数相乘是密码学的一种关键运算,其性能影响许多密码算法,如RSA、ElGamal等公钥密码运算的性能。对常见的大数乘法算法进行了实验、分析和比较,特别针对快速傅里叶变换（Fast Fourier Transform,FFT）算法,分析了其在大数乘法中的应用,并与其他常见大数算法的效率进行了比较,归纳了快速傅里叶变换的优势范围与劣势范围。同时,由于快速傅里叶变换计算过程中有误差,当数据位足够多时,可能导致计算结果不正确,因此进一步分析了傅里叶快速变换计算正确的数据位上限,这些工作对于快速乘法算法的正确选择有重要的实际意义。     

Lower bounds for the multiplicative complexity of matrix multiplication

We prove a lower bound of km + mn + k—m + n— 3 for the multiplicative complexity of the multiplication of -matrices with -matrices using the substitution method. Received: July 8, 1997.     

基于折半运算的快速双基数标量乘算法

为了提高椭圆曲线标量乘法效率,对二元域上椭圆曲线的基于双基数的标量乘法进行改进。在底层域推导出直接计算3^kP的快速算法,该算法只需一次求逆;新设计的以1/2和3为基的双基数编码可结合高效的直接计算3^kP和折半运算,基于该双基数编码的标量乘算法只涉及到点加运算、折半运算、三倍点和直接计算3^kP,底层域运算复杂性得到降低,在NIST推荐的椭圆曲线上比Dimitrov算法效率提高70%以上,比Wong方法提高10%以上。     

基于CUDA的快速大整数乘法

针对快速傅里叶变换下的快速大整数乘法,给出了一种基于CUDA架构的GPU并行化加速的实现方法。通过分析整数快速乘法中的每一步骤,分别给出各步骤的并行化实现方法,并采用数据压缩等策略,对算法进行优化。实验表明该方法有效地提高了算法效率,随着数据规模的增长,可获得18倍以上的加速比。     

基于素数域上复合运算的快速标量乘算法

底层有限域上点群运算是影响椭圆曲线密码效率的主要因素, 利用混合坐标下快速复合运算2P+Q代替传统的点加运算作为基本计算单元, 对NAF标量乘算法进行改进, 改进后算法与基于最优坐标下的NAF标量乘算法相比, 效率提高7%。通过预计算对标量k进行分段编码, 提出基于复合运算的分段并行标量乘快速算法, 在基点和标量长固定的情况下, 该算法与原有NAF算法相比计算效率提高了46. 5%, 而且改进后算法仅需存储三个预计算点坐标, 存储空间小。     

商务安全中随机窗宽的椭圆标量乘快速方法

ECC在商务电子签名等安全过程得以广泛应用,为提高弱计算环境中椭圆曲线标量点乘效率及增强标量点乘的安全性,分析了加法链方法和NAF方法的加速性能优势,提出了基于深度优先加法链及无约束窗口宽度的w-CNAF算法。理论分析结果和实验数据表明,该算法能有效降低标量的平均汉明重量,节省标量点乘的计算成本,其自嵌入了窗口宽度随机化机制,且针对能量分析攻击表现出较好的免疫能力,总体上具备很好的实用性。