Minimum Number Of Multiplications Research Articles

A lower bound for differential uniformity by multiplicative complexity & bijective functions of multiplicative complexity 1 over finite fields

The multiplicative complexity of an S-box over a finite field is the minimum number of multiplications needed to implement the S-box as an arithmetic circuit. In this paper we fully characterize bijective S-boxes with multiplicative complexity 1 up to affine equivalence over any finite field. We show that under affine equivalence in odd characteristic there are two classes of bijective functions and in even characteristic there are three classes of bijective functions with multiplicative complexity 1. Moreover, in (Jeon et al., Cryptogr. Commun., 14(4), 849-874 (2022)) A-boxes where introduced to lower bound the differential uniformity of an S-box over F2n\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{2}^{n}$$\\end{document} via its multiplicative complexity. We generalize this concept to arbitrary finite fields. In particular, we show that the differential uniformity of a (n, m)-S-box over Fq\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {F}_{q}$$\\end{document} is at least qn-l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$q^{n - l}$$\\end{document}, where ⌊n-12⌋+l\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\lfloor \\frac{n - 1}{2} \\rfloor + l$$\\end{document} is the multiplicative complexity of the S-box.

On lower bounds for optimal Jacobian accumulation

The task of finding a way to compute by using the minimal number of multiplications is generally referred to as the Optimal Jacobian Accumulation () problem. Often a special case of the problem is examined, where the accumulation of the Jacobian is considered as a transformation of the linearized directed acyclic graph (l-DAG) G into , such that is a subgraph of the complete directed bipartite graph . The transformation of the l-DAG is performed by elimination methods. The most commonly used elimination methods are vertex elimination and edge elimination. The problem of transforming an l-DAG into a bipartite graph by vertex or edge eliminations with minimal costs is generally referred as the Vertex Elimination () or Edge Elimination () problem. Both the and problems are conjectured to be NP-hard. Thus Branch and bound based algorithms in addition to greedy heuristics are used to solve these problems. For efficient application of such algorithms, good lower bounds are essential. In this paper, we develop a new approach for computing the lower bounds for the optimal solution of the and problems.

Design and FPGA implementation of multiplierless comb filter

SummaryThe main objective of this paper is to design and implement minimum multiplier, low latency structures of a comb filter. Multipliers are the most area and power consuming elements; therefore, it is desirable to realize a filter with minimum number of multipliers. In this paper, design of comb filters based on lattice wave digital filters (LWDF) structure is proposed to minimize the number of multipliers. The fundamental processing unit employed in LWDF requires only one multiplier. These lattice wave digital comb filters (LWDCFs) are realized using Richards' and transformed first‐order and second‐order all‐pass sections. The resulting structural realizations of LWDCFs exhibit properties such as low coefficient sensitivity, high dynamic range, high overflow level, and low round‐off noise. Multiplier coefficients of the proposed structures are implemented with canonic signed digit code (CSDC) technique using shift and add operations leading to multiplierless implementation. This contributes in reduction of number of addition levels which reduces the latency of the critical loop. A field programmable gate array (FPGA) platform is used for evaluation and testing of the proposed LWDCFs to acquire advantages of the parallelism, low cost, and low power consumption. The implementation of the proposed LWDCFs is accomplished on Xilinx Spartan‐6 and Virtex‐6 FPGA devices. By means of examples, it is shown that the implementations of the proposed LWDCFs attain high maximum sampling frequency, reduced hardware, and low power dissipation compared with the existing comb filter structures. Copyright © 2017 John Wiley & Sons, Ltd.

On the Implementation of Time-Multiplexed Frequency-Response Masking Filters

The complexity of narrow transition band FIR filters is high and can be reduced by using frequency-response masking (FRM) techniques. These techniques use a combination of periodic model filters and, possibly periodic, masking filters. Time-multiplexing is in general beneficial since only rarely does the technology bound maximum obtainable clock frequency and the application determined required sample rate correspond. Therefore, architectures for time-multiplexed FRM filters that benefit from the inherent sparsity of the periodic filters are introduced in this paper. We show that FRM filters not only reduce the number of multipliers needed, but also have benefits in terms of memory usage. Despite the total amount of samples to be stored is larger for FRM, it results in fewer memory resources needed in FPGAs and more energy efficient memory schemes in ASICs. In total, the power consumption is significantly reduced compared with a single-stage implementation. Furthermore, we show that the choice of the interpolation factor that gives the least complexity for the periodic model filter and subsequent masking filter(s) is a function of the time-multiplexing factor, meaning that the minimum number of multipliers not always corresponds to the minimum number of multiplications. Both single-port and dual-port memories are considered, and the involved tradeoff in number of multipliers and memory complexity is illustrated. The results show that, for FPGA implementation, the power reduction ranges from 23% to 68% for the considered examples.

Improvement of the estimates of the computational complexity for monomials and sets of powers in Bellman’s and Knuth’s problems

We study two generalizations of the classical problem of fast exponentiation, namely: Bellman’s problem on computational complexity (the minimal number of multiplications) based only on the variables of a normalized monomial of m variables and Knuth’s problem on the complexity of the simultaneous calculation of a system of m powers of one variable. Some results for these problems are surveyed in the paper. The asymptotic complexity bounds for Bellman’s and Knuth’s problems are improved for the cases when m and complexity behave similarly (have the same growth order). The upper and lower complexity bounds for almost all sets of exponents for Bellman’s and Knuth’s problems that are established provide the complexity growth asymptotics for a wide range of relations between parameters (the number of variables or the computed exponents, the maximal power, and the product of all powers). Moreover, they guarantee the ratio of the upper and lower bounds not exceeding 5/3 for all relations between the parameters.

Minimum multiplicative complexity algorithm for computing a single component of the discrete Fourier transform

A new algorithm for computing a single component of the discrete Fourier transform is presented. The algorithm, which is optimum in the sense that it requires the minimum number of multiplications, is based on the cyclotomic basis decomposition.

Ant Colony Optimization With Combining Gaussian Eliminations for Matrix Multiplication

One of the main unsolved problems in computer algebra is to determine the minimal number of multiplications which is necessary to compute the product of two matrices. For practical value, the small format is of special interest. This leads to a combinatorial optimization problem which is unlikely solved in polynomial time. In this paper, we present a method called combining Gaussian eliminations to reduce the number of variables in this optimization problem and use heuristic ant colony algorithm to solve the problem. The results of experiments on 2 × 2 case show that our algorithm achieves significant performance gains. Extending this algorithm from 2 × 2 case to 3 × 3 case is also discussed. Index Terms—Ant colony optimization (ACO), evolutionary algorithms, Gaussian eliminations, matrix multiplication, multiplicative complexity, Strassen's algorithm.

Multiplication of polynomials modulo [formula omitted

Let n , ℓ be positive integers with ℓ ≤ 2 n − 1 . Let R be an arbitrary nontrivial ring, not necessarily commutative and not necessarily having a multiplicative identity and R [ x ] be the polynomial ring over R . In this paper, we give improved upper bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most ( n − 1 ) modulo x n over R . Moreover, we introduce a new complexity notion, the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most ( n − 1 ) modulo x ℓ over R . This new complexity notion provides improved bounds on the minimum number of multiplications needed to multiply two arbitrary polynomials of degree at most ( n − 1 ) modulo x n over R .

Relation between two measures of the computation complexity for systems of monomials

For a class of matrices defining exponents of variables in a system of monomials, a nontrivial lower bound of complexity is found (where the complexity is defined as the minimum number of multiplications required to compute the system starting from variables). An example of a sequence of matrices (systems of monomials, respectively) is also given so that the usage of inverse values of variables (in addition to the variables themselves) makes the complexity asymptotically two times less.

Formula omitted]-infinity structures related to bi-Koszul algebras

Let A be a bi-Koszul algebra. We describe all possible A ∞ -algebra structures on the Ext-algebra E ( A ) and prove that E ( A ) must be [ m 2 , m 3 ] -finitely generated. We give an equivalent description of the condition for a connected graded algebra to be a bi-Koszul algebra in the language of A ∞ -algebras. We discuss decomposition in the case that E ( A ) is endowed with minimal number of multiplications.

Wave digital filters with minimum multiplier for discrete Hilbert transformer realization

This paper describes the design of wave digital filters for discrete Hilbert transformer realization. A method for designing real and complex half-band wave digital filters (WDFs) with a minimum number of multipliers is afforded where the multiplier coefficients for the adaptors are calculated directly from the digital domain transfer functions. This technique uses the equations derived for a class of infinite impulse response (IIR) filter and hence avoids the conventional technique of synthesizing a WDF from an analogue reference filter. The selection of the adaptor configuration is determined from the multiplier coefficient values. A minimum multiplier realization of real half-band WDFs can be obtained to meet the low-pass digital filter magnitude specifications. By applying a transformation equation, a complex half-band WDF suitable for the realization of the discrete Hilbert transformer with a minimum number of multiplier is obtained.

Efficient and secure cryptographic systems based on addition chains: Hardware design vs. software/hardware co-design

Most of cryptographic systems are based on modular exponentiation, wherein the operands are considerably large. Generally, modular exponentiation is implemented using a chain of modular multiplications. One way of improving the throughput of a cryptographic system implementation is reducing the number of the required modular multiplications. Given the exponent, finding the minimal number of multiplications that are necessary to reach the exponentiation results is a hard problem. We take advantage of an evolutionary stochastic process, commonly called genetic algorithms, to compute minimal addition chains, allow us to reduce drastically the number of the required modular multiplications. In this paper, we investigate two different ways to implement modular exponentiation based on the evolved addition chain: the first approach consists of implementing all the exponentiation operation on hardware and the second approach combines software and hardware. While the first approach attempts to optimise the encryption/decryption throughput regardless of the cost of system implementation, the second attempts to reach a balance between the throughput and the cost of the implementation.

A Short Survey on Arithmetic Transforms and the Arithmetic Hartley Transform

Arithmetic complexity has a main role in the performance of algorithms for spectrum evaluation. Arithmetic transform theory offers a method for computing trigonometrical transforms with minimal number of multiplications. In this paper, the proposed algorithms for the arithmetic Fourier transform are surveyed. A new arithmetic transform for computing the discrete Hartley transform is introduced: the Arithmetic Hartley transform. The interpolation process as the key to the arithmetic transform theory is also examined.

Method of paired transforms for reconstruction of images from projections: discrete model

In this paper, an effective method of discrete image reconstruction from its projections is introduced. The method is based on the vector and paired representations of the two-dimensional (2D) image with respect to the 2D discrete Fourier transform. Such representations yield algorithms for image reconstruction by a minimal number of attenuation measurements in certain projections. The proposed algorithms are described in detail for an N x N image, when N = 2(r), r > 1. The inverse formulas for image reconstruction are given. The efficiency of the algorithms is expressed in the fact that they require a minimal number of multiplications, or can be implemented without such at all. The problem of discrete image reconstruction is also considered in three-dimensional (3D) space, namely on the 3D torus, where the reconstruction is performed by means of the nonlinear projections that are integral over 3D spirals on the torus.

Use Of The Remez Algorithm For Designing FRM Based FIRr Filters

A very efficient technique for drastically reducing the number of multipliers and adders in narrow transition-band linear-phase finite impulse response (FIR) filters is to use the one-stage or multistage frequency-response masking (FRM) approach as originally introduced by Lim. In the original synthesis techniques developed by Lim and Lian, the subfilters in the overall approach were designed using time-consuming linear programming. In order to perform the overall synthesis faster, this paper shows how these subfilters can be designed with the aid of the the Remez multiple exchange algorithm, the most powerful technique for designing arbitrary-magnitude linear-phase FIR filters in the minimax sense. In addition to speeding up the overall procedure, the use of the Remez algorithm enables one to generate a very fast MATLAB program for the overall synthesis so that after being given the filter specifications as well as the number of stages, the program automatically provides the solution with the minimum number of multipliers and adders required in the overall implementation. This is possible because the MATLAB Remez routine is directly available and thus can be used for this purpose after appropriate modifications.

A pipelined architecture for the multidimensional DFT

This paper presents an efficient pipelined architecture for the N/sup m/-point m-dimensional discrete Fourier transform (DFT). By using a two-level index mapping scheme that is different from the conventional decimation-in-time (DIT) or decimation-infrequency (DIF) algorithms, the conventional pipelined architecture for the one-dimensional (1-D) fast Fourier transform (FFT) can be efficiently used for the computation of higher dimensional DFTs. Compared with systolic architectures, the proposed scheme is area-efficient since the computational elements (CEs) use the minimum number of multipliers, and the number of CEs increases only linearly with respect to the dimension m. It can be easily extended to the N/sup m/-point m-dimensional DFT with large m and/or N, and it is more flexible since the throughput can be easily varied to accommodate various area/throughput requirements.

Realization of Multidimensional Digital Transfer Functionswith Separable Numerator or Denominator Polynomials

In this paper a straightforward z-domain procedure for realizing multidimensional transfer functions with either numerator or denominator polynomial separable is presented. This procedure yields structures with minimum number of multipliers. It is shown that the number of delay elements can be reduced by selecting the optimum values for certain parameters. To facilitate this theorems that are applicable for some special cases are presented.

Polynomial residue complex signal processing

The polynomial residue number system (PRNS) is a system in which the product of two polynomials can take place in parallel and with the minimum number of multiplications. The system is an extension of the quadratic residue number system (QRNS) which has been successfully used in complex digital signal processing. It is shown that an N-point complex linear convolution can be computed with 4N real multiplications when using the PRNS instead of 22 real multiplications when using the QRNS. The savings in the number of multiplications occur if some restrictions are placed on the modular ring used for performing the complex residue number system operations.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Discrete spectral peak estimation in incoherent backscatter heterodyne lidar. II. Correlogram accumulation

For pt.I see ibid., vol.31, no.1, p.16-27 (1993). The correlogram, an alternative to formation of the periodogram from the time series data sample. is described. The correlogram consists of a set of autocorrelation estimates for some number of lags obtained directly from the data sample. Correlogram accumulation entails adding or averaging the sets of autocorrelation estimates prior to the spectral estimation stage. The correlogram is formed in the time domain directly from the data, using a minimal number of multiplications and additions so that it may be formed and accumulated at the speed that the data are acquired. >

New canonical state-variable structure and its application in digital filtering

A new state-variable structure with minimum number of multipliers is introduced. It is obtained by the application of the bilinear transformation to the direct form structure. It is also shown that the roundoff-noise gain is invariant under LP-LP transformation for the new structure.