Polynomial Arithmetic Research Articles

An Explicit Treatment of Cubic Function Fields with Applications

AbstractWe give an explicit treatment of cubic function fields of characteristic at least five. This includes an efficient technique for converting such a field into standard form, formulae for the field discriminant and the genus, simple necessary and sufficient criteria for non-singularity of the defining curve, and a characterization of all triangular integral bases. Our main result is a description of the signature of any rational place in a cubic extension that involves only the defining curve and the order of the base field. All these quantities only require simple polynomial arithmetic as well as a few square-free polynomial factorizations and, in some cases, square and cube root extraction modulo an irreducible polynomial. We also illustrate why and how signature computation plays an important role in computing the class number of the function field. This in turn has applications to the study of zeros of zeta functions of function fields.

Toward high-performance polynomial system solvers based on triangular decompositions

This thesis is devoted to the design and implementation of polynomial system solvers based on symbolic computation. Solving systems of non-linear, algebraic or differential equations, is a fundamental problem in mathematical sciences. It has been studied for centuries and still stimulates many research developments, in particular on the front of high-performance computing. Triangular decompositions are a highly promising technique with the potential to produce high-performance polynomial system solvers. This thesis makes several contributions to this effort. We propose asymptotically fast algorithms for the core operations on which triangular decompositions rely. Complexity results and comparative implementation show that these new algorithms provide substantial performance improvements. We present a fundamental software library for polynomial arithmetic in order to support the implementation of high-performance solvers based on triangular decompositions. We investigate strategies for the integration of this library in high-level programming environments where triangular decompositions are usually implemented. We obtain a high performance library combining highly optimized C routines and solving procedures written in the Maple computer algebra system. The experimental result shows that our approaches are very effective, since our code often outperforms pre-existing solvers in a significant manner.

Reliability Analysis of Interval Parameters Bar Structures with Affine Arithmetic

By representing the uncertain parameters as interval numbers, the reliability index equations of bars structures were obtained. A modified matrix affine arithmetic polynomial evaluation method plus recursive derivative information was proposed in this paper, which keeps all powers of noise symbols without approximation. Based on the nature that affine forms and intervals variables could transform each other, affine forms of bounded uncertain variables and modified affine arithmetic including derivative information for interval univariate polynomial evaluation were introduced into modeling and calculating non-probabilistic reliability index. An extended beam example and a ten-bar truss structure example were provided to illustrate the validity and feasibility of the presented procedures.

Group Re-keying Protocol Based on Modular Polynomial Arithmetic Over Galois Field GF(2n)

Problem statement: In this study we propose a group re-keying protocol based on modular polynomial arithmetic over Galois Field GF(2n). Common secure group communications requires encryption/decryption for group re-keying process, especially when a group member is leaving the group. Approach: This study proposes secret keys multiplication protocol based on modular polynomial arithmetic (SKMP), which eliminates the need for the encryption/decryption during the group re-keying. Results: The implementation based on modular polynomial arithmetic over Galois Field GF(2n) offers fast re-keying process (about 50% faster than Secret Keys Multiplication Protocol (SKM) for 128 bit key) and compact key size representation against other secret keys multiplication protocols. With SKMP group re-keying is handled more efficiently through modular polynomial arithmetic manipulation rather than the expensive encryption/encryption which need to be done on every membership change.

A unified localization approach for machining allowance optimization of complex curved surfaces

The goal of workpiece localization is of interest to find the optimal Euclidean transformation that aligns the sampled points to the nominal CAD model to ensure sufficient stock allowance during the machining process. In this paper, a unified localization technique is developed for sculptured surface machining. This technique concerns an alignment process to satisfy a user-defined set of constraints for some specific surfaces where the machining allowance is preferentially guaranteed. The mathematical model of the constrained optimization alignment is firstly established, and is efficiently solved by a combination of the multipliers method and the BFGS algorithm to handle the large number of constraints in allowance optimization. To efficiently calculate the Euclidean oriented distance, a novel approach, which combines the robust arithmetic for multivariate Bernstein-form polynomials and Bezier surface segmentation algorithm, is presented based on recursive quadtree decomposition. Two typical sculptured surfaces are used to test the developed algorithm and comparisons between the proposed algorithm and the existing algorithms are given. Experiment results show that the proposed method is appropriate and feasible to distribute the stock allowance for proper sculptured surface machining.

The modpn library

We investigate the integration of C implementation of fast arithmetic operations into Maple, focusing on triangular decomposition algorithms. We show substantial improvements over existing Maple implementations; our code also outperforms Magma on many examples. Profiling data show that data conversion can become a bottleneck for some algorithms, leaving room for further improvements. Since the early days of computer algebra systems, their designers have investigated many aspects of this kind of software. For systems born in the 70’s and 80’s, such as AXIOM and Maple, the primary concerns were probably the expressiveness of the programming language and the convenience of the user interface; the implementation of modular methods for operations such as polynomial GCD or factorization was also among these concerns. Computer algebra systems and libraries born in the 90’s, such as Magma and NTL, have brought forward a new priority: the implementation of asymptotically fast arithmetic for polynomials and matrices. They have demonstrated that, for relatively small input data size, FFT-based polynomial operations could outperform operations based on classical quadratic algorithms or on the Karatsuba trick. This increased in a spectacular manner the range of solvable problems. Meanwhile, AXIOM and Maple remain highly attractive: the former one by its programming environment and the latter one by its users community. In previous work [6, 9, 11] we have investigated the integration of asymptotically fast arithmetic operations into AXIOM. Since AXIOM is based today on GNU Common Lisp (GCL), we realized optimized implementations of these fast routines in C and

NTRU‐based sensor network security: a low‐power hardware implementation perspective

AbstractWireless sensor network security requires the cryptography software extremely low complex and energy efficient due to the limited memory and CPU capacity in a sensor. The NTRU (Nth degree truncated polynomial ring) encrypt algorithm has been shown to provide certain advantages when designing low power and resource constrained systems, while still providing comparable security levels to higher complexity algorithms. Unlike the current works that build NTRU software in a chip, this research focuses on the hardware implementation of NTRU algorithms because hardware implementation has much higher execution speed than software implementation. In contrast to previous research, the focus is shifted away from specific optimizations but rather provides a study of many of the recommended practices and suggested optimizations with particular emphasis on polynomial arithmetic and parameter selection. Recommendations for algorithm and parameter selection are made regarding implementation in hardware with respect to the resources available. Copyright © 2008 John Wiley & Sons, Ltd.

Methods of finding multiplicative inverses in GF(2 8)

Finding the multiplicative inverse of an element in Galois Field( p), GF( p) for small values of p such as 5 or 7 is no problem. One can find the multiplicative inverse by constructing multiplication tables and establish the desired value directly. The look-up table procedure, when implemented through software, is fast and handy, and is employed in the design of S-boxes of the Rijndael encryption algorithm used in advanced encryption standard (AES). However, a second choice, if execution time is not the consideration, is the use of extended Euclid’s algorithm. The two methods of finding multiplicative inverse in GF(2 8) presented in many books are discussed in this paper. An effort is made to present the working of these methods in a detailed manner which is not found in any text. A comparison of these methods vis-à-vis time, transistor count and complexity is also made. Section 1 contains a brief introduction to the topic, Section 2 reviews multiplication in polynomial arithmetic. Multiplicative inverses in GF(2 8) are taken up in Sections 3, and 4 concludes the paper.

DSP-128: Stream Cipher Based On Discrete Log Problem And Polynomial Arithmetic

DSP-128 is a new multithreaded stream cipher based on the intractability of the Discrete Logarithm Problem (DLP) with key size of 128-bit. The design of DSP-128 is divided into three stages: Initialization Stage, Keystream Generation Stage, and Encryption Stage. The design goal of DSP-128 is to come up with a secure stream cipher with good performance for data encryption. The experimental results show that the encryption rate of DSP-128 is one time slower (running on single processor) than the widely adapted stream cipher RC4, with a higher level of security against possible cryptanalysis attacks. However, because of its multithreaded nature, DSP-128 can take the speed up advantage of multi-core processor architectures which are available widely.

An efficient sparse adaptation of the polytope method over [formula omitted] and a record-high binary bivariate factorisation

A recent bivariate factorisation algorithm appeared in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC’04. pp. 4–11] based on the use of Newton polytopes and a generalisation of Hensel lifting. Although possessing a worst-case exponential running time like the Hensel lifting algorithm, the polytope method should perform well for sparse polynomials whose Newton polytopes have very few Minkowski decompositions. A preliminary implementation in Abu-Salem et al. [Abu-Salem, F., Gao, S., Lauder, A., 2004. Factoring polynomials via polytopes. In: Proc. ISSAC’04. pp. 4–11] indeed reflects this property, but does not exploit the fact that the algorithm preserves the sparsity of the input polynomial, so that the total amount of work and space required are O ( d 4 ) and O ( d 2 ) respectively, for an input bivariate polynomial of total degree d . In this paper, we show that the polytope method can be made sensitive to the number of non-zero terms of the input polynomial, so that the input size becomes dependent on both the degree and the number of terms of the input bivariate polynomial. We describe a sparse adaptation of the polytope method over finite fields with prime order, which requires fewer bit operations and memory references given a degree d sparse polynomial whose number of terms t satisfies t < d 3 / 4 , and which is known to be the product of two sparse factors. For t < d , and using fast polynomial arithmetic over finite fields, our refinement reduces the amount of work per extension of a coprime dominating edge factorisation and the total spatial cost to O ( t λ d 2 + t 2 λ d L ( d ) + t 4 λ d ) bit operations and O ( t λ d ) bits of memory respectively, for some 1 / 2 ≤ λ < 1 , and L ( d ) = log d log log d . To the best of our knowledge, the sparse binary factorisations achieved using this adaptation are of the highest degree so far, reaching a world record degree of 20 000 for a very sparse bivariate polynomial over F 2 .

On a Java computer algebra system, its performance and applications

This paper considers Java as an implementation language for a starting part of a computer algebra library. It describes a design of basic arithmetic and multivariate polynomial interfaces and classes which are then employed in advanced parallel and distributed Groebner base algorithms and applications. The library is type-safe due to its design with Java’s generic type parameters and thread-safe using Java’s concurrent programming facilities. We report on the performance of the polynomial arithmetic and on applications built upon the core library.

A maple library for high performance sparse polynomial arithmetic

We have developed a library for sparse polynomial arithmetic to supplement Maple's internal routines. Along the way we identified three common sources of inefficiency in existing implementations.

Optimal localization of free-form shaped parts in precision inspection

Optimal localization of free-form shaped parts is a key issue in precision inspection. Aiming at the problems on estimate of the initial transformation and the calculation of closest points, an effective and exact localization method is presented. First, this method establishes the corresponding rela-tionships between two free-form surfaces with constraints of angle and distance by means of the surface features of the Gaussian curvature and the mean curvature. Then the rough localization is realized. Using the robust arithmetic for multi-variate Bernstein-form polynomials, a novel algorithm to cal-culate closest point is proposed. An improved iterative algo-rithm is used in the exact adjustment of localization to ensure the global optimization. Experiment results demonstrate the feasibility of the proposed method.

Modular Forms of Systems of k-valued Functions of the Algebra of Logic

Methods of realization of the k-valued functions of the algebra of logic by the modular forms of arithmetic polynomials based on “weighing” by the numbers k i (i = 0, 1, 2, ...) were considered. The modular polynomial and matrix (number-theoretic) transformations were examined and extended to the case of systems of k-valued functions. A new principle of designing the modular form of one arithmetic polynomial to realize systems of k-valued functions in terms of the Chinese remainder theorem was proposed. The results obtained provide advantages in terms of complexity of the analytical description and realization of the k-valued functions.

Arithmetic of singular moduli and class polynomials

We investigate divisibility properties of the traces and Hecke traces of singular moduli. In particular we prove that, if p is prime, these traces satisfy many congruences modulo powers of p which are described in terms of the factorization of p in imaginary quadratic fields. We also study generalizations of Lehner’s classical congruences j(z)|Up ≡ 744 (mod p )( wherep 11 and j(z) is the usual modular invariant), and we investigate connections between class polynomials and supersingular polynomials in characteristic p.

Challenges Concerning Symbolic Computations on Grids

Challenges Concerning Symbolic Computations on Grids

Fast and numerically stable algorithms for discrete cosine transforms

In this paper, we derive fast and numerically stable algorithms for discrete cosine transforms (DCT) of radix-2 length which are based on real factorizations of the corresponding cosine matrices into products of sparse, (almost) orthogonal matrices of simple structure. These algorithms are completely recursive, are simple to implement and use only permutations, scaling with 2 , butterfly operations, and plane rotations/rotation–reflections. Our algorithms have low arithmetic costs which compare with known fast DCT algorithms. Further, a detailed analysis of the roundoff errors for the presented DCT algorithms shows their excellent numerical stability which outperforms a real fast DCT algorithm based on polynomial arithmetic.

On the equivalence and rewriting of aggregate queries

We introduce a first-order language with real polynomial arithmetic and aggregation operators (count, iterated sum and multiply), which is well suited for the definition of aggregate queries involving complex statistical functions. It offers a good trade-off between expressive power and complexity, with a tractable data complexity. Interestingly, some fundamental properties of first-order with real arithmetic are preserved in the presence of aggregates. In particular, there is an effective quantifier elimination for formulae with aggregation. We then consider the problem of querying data that has already been aggregated in aggregate views, and focus on queries with an aggregation over a conjunctive query (namely single-block aggregate group-by queries without having clause). Our main conceptual contribution is the introduction of a new equivalence relation among conjunctive queries, the isomorphism modulo a product. We prove that the equivalence of aggregate queries such as for instance averages reduces to it. Deciding if two queries are isomorphic modulo a product is shown to be NP-complete. We then analyze the equivalence problem in the case of aggregate conjunctive queries with comparisons. We introduce the concept of cross isomorphic linear expansions, which generalizes isomorphim modulo a product, and we show that equivalence reduces to it and that it can be decided in PSPACE. Finally, we show that the problem of complete rewriting of count queries using count views is NP-complete, and we introduce new rewriting techniques based on the isomorphism modulo a product. to recover the values of counts by complex arithmetical computation from the views.

Family of Ternary Arithmetic Polynomial Expansions Based on New Transforms

New classes of Linearly Independent Ternary Arithmetic (LITA) transforms being the bases of ternary arithmetic polynomial expansions are introduced here. Recursive equations defining the LITA transforms and the corresponding butterfly diagrams are shown. Various properties and relations between introduced classes of new transforms are discussed. Computational costs to calculate LITA transforms and applications of corresponding polynomial expansions in logic design are also discussed.

New frameworks for Montgomery’s modular multiplication method

We present frameworks for fast modular multiplication based on a modification of Montgomery’s original method. For (fixed) large integers, our algorithms may be significantly faster than conventional methods. Our techniques may also be extended to modular polynomial arithmetic.