a semi circulant preconditioner for the convection diffusion equation in this paper we define and analyze a semi circulant preconditioner for the convection diffusion equation we derive analytical formulas for the eigenvalues and the eigenvectors of the preconditioned system of equations we show that for mesh peclet numbers less than 2 the rate of convergence depends only on the mesh peclet number and the direction of the convective field and not on the spatial grid ratio or the number of unknowns

analysis of three dimensional fracture mechanics problems a two scale approach using coarse generalized fem meshes this paper presents a generalized finite element method gfem based on the solution of interdependent global structural and local crack scale problems the local problems focus on the resolution of fine scale features of the solution in the vicinity of three dimensional cracks while the global problem addresses the macro scale structural behavior the local solutions are embedded into the solution space for the global problem using the partition of unity method the local problems are accurately solved using an hp gfem and thus the proposed method does not rely on analytical solutions the proposed methodology enables accurate modeling of three dimensional cracks on meshes with elements that are orders of magnitude larger than the process zone along crack fronts the boundary conditions for the local problems are provided by the coarse global mesh solution and can be of dirichlet neumann or cauchy type the effect of the type of local boundary conditions on the performance of the proposed gfem is analyzed several three dimensional fracture mechanics problems aimed at investigating the accuracy of the method and its computational performance both in terms of problem size and cpu time are presented copyright 2009 john wiley sons ltd

formulation of elastostatic hybrid trefftz stress elements abstract the hybrid trefftz stress hts element is derived from the fundamental conditions of linear elastostatics the stress and boundary displacement fields are directly approximated consistent finite element variables are so defined as to obtain boundary integral expressions for the finite element equilibrium and compatibility conditions encoded by dual transformations independent of the constitutive relations reciprocity is preserved in the finite element elasticity relations which are also found to be described by boundary integral arrays when the trefftz constraint is enforced consequent upon duality and reciprocity the resulting boundary integral solving system is symmetric and sparse the energy statements associated with the hybrid trefftz stress element are established and the conditions for the existence uniqueness multiplicity and static and kinematic admissibility of the finite element solutions are presented the hts clement is compared with the displacement frame based hybrid trefftz ht d element of jirousek it is shown that the ht d element is in fact an hybrid stress element the numerical tests performed to assess the relative performances of the hts and ht d elements show that better accuracy levels are obtained using discontinuous boundary approximation bases

low complexity multiplication in a finite field using ring representation elements of a finite field gf 2 sup m are represented as elements in a ring in which multiplication is more time efficient this leads to faster multipliers with a modest increase in the number of xor and and gates needed to construct the multiplier such multipliers are used in error control coding and cryptography we consider rings modulo trinomials and 4 term polynomials in each case we show that our multiplier is faster than multipliers over elements in a finite field defined by irreducible pentanomials these results are especially significant in the field of elliptic curve cryptography where pentanomials are used to define finite fields finally an efficient systolic implementation of a multiplier for elements in a ring defined by x sup n x 1 is presented

mastrovito multiplier for general irreducible polynomials we present a new formulation of the mastrovito multiplication matrix for the field gf 2 sup m generated by an arbitrary irreducible polynomial we study in detail several specific types of irreducible polynomials e g trinomials all one polynomials and equally spaced polynomials and obtain the time and space complexity of these designs particular examples illustrating the properties of the proposed architecture are also given the complexity results established in this paper match the best complexity results known to date the most important new result is the space complexity of the mastrovito multiplier for an equally spaced polynomial which is found as m sup 2 spl delta xor gates and m sup 2 and gates where spl delta is the spacing factor

efficient normal basis multipliers in composite fields it is well known that a class of finite fields gf 2 sup n using an optimal normal basis is most suitable for a hardware implementation of arithmetic in finite fields in this paper we introduce composite fields of some hardware applicable properties resulting from the normal basis representation and the optimal condition we also present a hardware architecture of the proposed composite fields including a bit parallel multiplier

an efficient optimal normal basis type ii multiplier this paper presents a new parallel multiplier for the galois field gf 2 sup m whose elements are represented using the optimal normal basis of type ii the proposed multiplier requires 1 5 m sup 2 m xor gates as compared to 2 m sup 2 m xor gates required by the massey omura multiplier the time complexities of the proposed and the massey omura multipliers are similar

a new approach to subquadratic space complexity parallel multipliers for extended binary fields based on toeplitz matrix vector products and coordinate transformation techniques we present a new scheme for subquadratic space complexity parallel multiplication in gf 2n using the shifted polynomial basis both the space complexity and the asymptotic gate delay of the proposed multiplier are better than those of the best existing subquadratic space complexity parallel multipliers for example with n being a power of 2 the space complexity is about 8 percent better while the asymptotic gate delay is about 33 percent better respectively another advantage of the proposed matrix vector product approach is that it can also be used to design subquadratic space complexity polynomial dual weakly dual and triangular basis parallel multipliers to the best of our knowledge this is the first time that subquadratic space complexity parallel multipliers are proposed for dual weakly dual and triangular bases a recursive design algorithm is also proposed for efficient construction of the proposed subquadratic space complexity multipliers this design algorithm can be modified for the construction of most of the subquadratic space complexity multipliers previously reported in the literature

improved area time tradeoffs for field multiplication using optimal normal bases in this paper we propose new schemes for subquadratic arithmetic complexity multiplication in binary fields using optimal normal bases the schemes are based on a recently proposed method known as block recombination which efficiently computes the sum of two products of toeplitz matrices and vectors specifically here we take advantage of some structural properties of the matrices and vectors involved in the formulation of field multiplication using optimal normal bases this yields new space and time complexity results for corresponding bit parallel multipliers

efficient multiplication beyond optimal normal bases in cryptographic applications the use of normal bases to represent elements of the finite field gf 2 sup m is quite advantageous especially for hardware implementation in this article we consider an important field operation namely multiplication which is used in many cryptographic functions we present a class of algorithms for normal basis multiplication in gf 2 sup m our proposed multiplication algorithm for composite finite fields requires a significantly lower number of bit level operations and hence can reduce the space complexity of cryptographic systems

diagonalization of tensors with circulant structure the concepts of tensors with diagonal and circulant structure are defined and a framework is developed for the analysis of such tensors it is shown a tensor of arbitrary order which is circulant with respect to two particular modes can be diagonalized in those modes by discrete fourier transforms this property can be used in the efficient solution of linear systems involving contractive products of tensors with circulant structure tensors with circulant structure occur in models for image blurring with periodic boundary conditions it is shown that the new framework can be applied to such problems

low complexity bit parallel canonical and normal basis multipliers for a class of finite fields we present a new low complexity bit parallel canonical basis multiplier for the field gf 2m generated by an all one polynomial the proposed canonical basis multiplier requires m sup 2 1 xor gates and m sup 2 and gates we also extend this canonical basis multiplier to obtain a new bit parallel normal basis multiplier

systematic design of original and modified mastrovito multipliers for general irreducible polynomials this paper considers the design of bit parallel dedicated finite field multipliers using standard basis an explicit algorithm is proposed for efficient construction of mastrovito product matrix based on which we present a systematic design of mastrovito multiplier applicable to gf 2 sup m generated by an arbitrary irreducible polynomial this design effectively exploits the spatial correlation of elements in mastrovito product matrix to reduce the complexity using a similar methodology we propose a systematic design of modified mastrovito multiplier which is suitable for gf 2 sup m generated by high hamming weight irreducible polynomials for both original and modified mastrovito multipliers the developed multiplier architectures are highly modular which is desirable for vlsi hardware implementation applying the proposed algorithm and design approach we study the mastrovito multipliers for several special irreducible polynomials such as trinomial and equally spaced polynomial and the obtained complexity results match the best known results moreover we have discovered several new special irreducible polynomials which also lead to low complexity mastrovito multipliers

high speed architectures for multiplication using reordered normal basis normal basis has been widely used for the representation of binary field elements mainly due to its low cost squaring operation optimal normal basis type ii is a special class of normal basis exhibiting very low multiplication complexity and is considered as a safe choice for hardware implementation of cryptographic applications in this paper high speed architectures for binary field multiplication using reordered normal basis are proposed where reordered normal basis is referred to as a certain permutation of optimal normal basis type ii complexity comparison shows that the proposed architectures are faster compared to previously presented architectures in the open literature using either an optimal normal basis type ii or a reordered normal basis one advantage of the new word level architectures is that the critical path delay is a constant not a function of word size this enables the multipliers to operate at very high clock rates regardless of the field size or the number of words hardware implementation of some practical size multipliers for elliptic curve cryptography is also included

a new kurtosis matrix with statistical applications abstract the number of fourth order moments which can be obtained from a random vector rapidly increases with the vector s dimension scalar measures of multivariate kurtosis may not satisfactorily capture the fourth order structure and matrix measures of multivariate kurtosis are called for in this paper we propose a kurtosis matrix derived from the dominant eigenpair of the fourth standardized moment we show that it is the best symmetric positive semidefinite kronecker square root approximation to the fourth standardized moment additional properties are derived for realizations from garch and reversible random processes statistical applications include independent component analysis and projection pursuit the star product of matrices highlights the connection between the proposed kurtosis matrix and other kurtosis matrices which appeared in the statistical literature a simulation study assesses the practical relevance of theoretical results in the paper

the extended generalized finite element method an overview of the method and its applications an overview of the extended generalized finite element method gefm xfem with emphasis on methodological issues is presented this method enables the accurate approximation of solutions that involve jumps kinks singularities and other locally non smooth features within elements this is achieved by enriching the polynomial approximation space of the classical finite element method the gefm xfem has shown its potential in a variety of applications that involve non smooth solutions near interfaces among them are the simulation of cracks shear bands dislocations solidification and multi field problems copyright 2010 john wiley sons ltd

on the inherent space complexity of fast parallel multipliers for gf 2 sup m a lower bound to the number of and gates used in parallel multipliers for gf 2 sup m under the condition that time complexity be minimum is determined in particular the exact minimum number of and gates for primitive normal bases and optimal normal bases of type ii multipliers is evaluated this result indirectly suggests that space complexity is essentially a quadratic function of m when time complexity is kept minimum

low complexity bit parallel systolic montgomery multipliers for special classes of gf 2 sup m recently cryptographic applications based on finite fields have attracted much interest this paper presents a transformation method to implement low complexity montgomery multipliers for all one polynomials and trinomials using this method we propose a new bit parallel systolic architecture for computing multiplications over gf 2 sup m these new multipliers have a latency m 1 clock cycles and each cell incorporates at most one 2 input and gate two 2 input xor gates and four 1 bit latches moreover these new multipliers are shown to exhibit significantly lower latency and circuit complexity than the related systolic multipliers and are highly appropriate for vlsi systems because of their regular interconnection pattern modular structure and fully inherent parallelism

a word level finite field multiplier using normal basis hardware implementations of finite field arithmetic using normal basis are advantageous due to the fact that the squaring operation can be done at almost no cost in this paper a new word level finite field multiplier using normal basis is proposed the proposed architecture takes d clock cycles to compute the product bits where the value for d 1 d m can be arbitrarily selected by the designer to set the tradeoff between area and speed when there exists an optimal normal basis it is shown that the proposed design has a smaller critical path delay than other word level normal basis multipliers found in the literature while its circuit complexities are moderate and comparable to the others different word size multipliers were implemented in hardware and implementation results are also presented

a modified massey omura parallel multiplier for a class of finite fields a massey omura parallel multiplier of finite fields gf 2 sup m contains m identical blocks whose inputs are cyclically shifted versions of one another it is shown that for fields gf 2 sup m generated by irreducible all one polynomials a portion of the block is independent of the input cyclic shift hence the multiplier contains redundancy by removing the redundancy a modified parallel multiplier is presented which is modular and has a lower circuit complexity

a matrix decomposition method for optimal normal basis multiplication we introduce a matrix decomposition method and prove that multiplication in gf 2 k with a type 1 optimal normal basis for can be performed using k 2 1 xor gates irrespective of the choice of the irreducible polynomial generating the field the previous results achieved this bound only with special irreducible polynomials furthermore the decomposition method performs the multiplication operation using 1 5k k 1 xor gates for type 2a and 2b optimal normal bases which matches previous bounds

montgomery multiplier and squarer for a class of finite fields montgomery multiplication in gf 2 sup m is defined by a x b x r sup 1 x mod f x where the field is generated by a root of the irreducible polynomial f x a x and b x are two field elements in gf 2 sup m and r x is a fixed field element in gf 2 sup m in this paper first a slightly generalized montgomery multiplication algorithm in gf 2 sup m is presented then by choosing r x according to f x we show that efficient architectures of bit parallel montgomery multiplier and squarer can be obtained for the fields generated with an irreducible trinomial complexities of the montgomery multiplier and squarer in terms of gate counts and time delay of the circuits are investigated and found to be as good as or better than that of previous proposals for the same class of fields

a new hardware architecture for operations in gf 2 sup n the efficient computation of the arithmetic operations in finite fields is closely related to the particular ways in which the field elements are presented the common field representations are a polynomial basis representation and a normal basis representation in this paper we introduce a nonconventional basis and present a new bit parallel multiplier which is as efficient as the modified massey omura multiplier using the type i optimal normal basis

bit parallel systolic multipliers for gf 2 sup m fields defined by all one and equally spaced polynomials two operations the cyclic shifting and the inner product are defined by the properties of irreducible all one polynomials an effective algorithm is proposed for computing multiplications over a class of fields gf 2 sup m using the two operations then two low complexity bit parallel systolic multipliers are presented based on the algorithm the first multiplier is composed of m 1 sup 2 identical cells each consisting of one 2 input and gate one 2 input xor gate and three 1 bit latches the other multiplier comprises of m 1 sup 2 identical cells and mxor gates each cell consists of one 2 input and gate one 2 input xor gate and four 1 bit latches each multiplier exhibits very low latency and propagation delay and is thus very fast moreover the architectures of the two multipliers can be applied in computing multiplications over the class of fields gf 2 sup m in which the elements are represented with the root of an irreducible equally spaced polynomial of degree m

concurrent error detection and correction in gaussian normal basis multiplier over gf 2 m fault based cryptanalysis has been developed to effectively break both private key and public key cryptosystems making robust finite field multiplication a very important research topic in recent years however no robust normal basis multiplier has been proposed in the literature therefore this investigation presents a semisystolic gaussian normal basis multiplier based on the proposed gaussian normal basis multiplier both concurrent error detection and correction capabilities can be easily achieved using time redundancy technology with no hardware modification

independent component analysis and simultaneous third order tensor diagonalization comon s 1994 well known scheme for independent component analysis ica is based on the maximal diagonalization in a least squares sense of a higher order cumulant tensor in a previous paper we proved that for fourth order cumulants the computation of an elementary jacobi rotation is equivalent to the computation of the best rank 1 approximation of a fourth order tensor in this paper we show that for third order tensors the computation of an elementary jacobi rotation is again equivalent to a best rank 1 approximation however here it is a matrix that has to be approximated this favorable computational load makes it attractive to do something third order like for fourth order cumulant tensors as well we show that simultaneous optimal diagonalization of third order tensor slices of the fourth order cumulant is a suitable strategy this simultaneous third order tensor diagonalization approach stotd is similar in spirit to the efficient jade algorithm

a generalized method for constructing subquadratic complexity gf 2 sup k multipliers we introduce a generalized method for constructing subquadratic complexity multipliers for even characteristic field extensions the construction is obtained by recursively extending short convolution algorithms and nesting them to obtain the short convolution algorithms the winograd short convolution algorithm is reintroduced and analyzed in the context of polynomial multiplication we present a recursive construction technique that extends any d point multiplier into an n d sup k point multiplier with area that is subquadratic and delay that is logarithmic in the bit length n we present a thorough analysis that establishes the exact space and time complexities of these multipliers using the recursive construction method we obtain six new constructions among which one turns out to be identical to the karatsuba multiplier all six algorithms have subquadratic space complexities and two of the algorithms have significantly better time complexities than the karatsuba algorithm

structure of parallel multipliers for a class of fields gf 2 m abstract this paper presents a configuration of parallel multipliers for gf 2 m based on canonical bases the possible parallel multipliers by the proposed configuration are limited to a class of fields gf 2 m however they can be constructed by o m 2 and gates and o m 2 eor gates with the structural modularity this is a desirable feature for the hardware implementation and their operation time is about log m t where m is the dimension of gf 2 m and t is the delay time of an eor gate in order to construct such parallel multipliers we define two types of polynomials of special form over gf 2 one is called all one polynomial denoted by aop and the other is called equally spaced polynomial denoted by esp furthermore we show a necessary and sufficient condition for esps to be irreducible over gf 2 and the uniqueness of the irreducible esps over gf 2 finally we propose the configuration of parallel multipliers for a class of fields gf 2 m based on irreducible aops and esps over gf 2

low complexity bit parallel multiplier for gf 2 sup m defined by all one polynomials using redundant representation this paper presents a new bit parallel multiplier for the finite field gf 2 sup m defined by an irreducible all one polynomial in order to reduce the complexity of the multiplier we introduce a redundant representation and use the well known multiplication method proposed by karatsuba the main idea is to combine the redundant representation and the karatsuba method to design an efficient bit parallel multiplier as a result the proposed multiplier requires about 25 percent fewer and xor gates than the previously proposed multipliers using an all one polynomial while it has almost the same time delay as the previously proposed ones

programmable cellular automata based montgomery hardware architecture abstract this study presents efficient hardware architectures for montgomery multiplication and squaring based on programmable cellular automata pca multiplication and squaring are the key operation in implementing circuits for various applications of error control coding such as reed solomon code thus we employ montgomery multiplication and squaring algorithms and construct simple hardware architectures based on pca in gf 2 m our architectures are highly optimized and have low complexity based on irreducible all one polynomial aop

bit parallel finite field multipliers for irreducible trinomials a new formulation for the canonical basis multiplication in the finite fields gf 2 sup m based on the use of a triangular basis and on the decomposition of a product matrix is presented from this algorithm a new method for multiplication named transpositional applicable to general irreducible polynomials is deduced the transpositional method is based on the computation of 1 cycles and 2 cycles given by a permutation defined by the coordinate of the product to be computed and by the cardinality of the field gf 2 sup m the obtained cycles define groups corresponding to subexpressions that can be shared among the different product coordinates this new multiplication method is applied to five types of irreducible trinomials these polynomials have been widely studied due to their low complexity implementations the theoretical complexity analysis of the corresponding bit parallel multipliers shows that the space complexities of our multipliers match the best results known to date for similar canonical gf 2 sup m multipliers the most important new result is the reduction in two of the five studied trinomials of the time complexity with respect to the best known results

splitting method and hybrid trefftz formulation for multisite damage analysis in two dimensional domains abstract the aim of this paper is to present a new strategy for multiple site damage analysis by combining the splitting method and hybrid trefftz stress formulation in a two dimensional domain the first is a decomposition method originally conceived to obtain the solution of a cracked solid accordingly the given problem is split into three subproblems an uncracked global problem p g 0 a local problem p l k considering a single crack located in a reduced domain and a global problem p g k aiming to account for the relative interaction effects among all the cracks most subproblems are independent and the solution results from a non iterative strategy which is one of the main benefits of this method the hybrid trefftz stress formulation is a non conventional finite element method where stresses and displacements are approximated independently in the domain and boundary of the element respectively the stress fields are hereby approximated by a basis function that solves the navier equation and includes analytical fracture mechanics functions so that the values of stress intensity factors are computed from the solution of the linear system the hybrid trefftz formulation is used in the local problem p l k of the splitting method therefore improving the performance of the method and providing accurate solutions with very coarse meshes in conclusion it is shown through the numerical examples that the splitting method and hybrid trefftz formulation framework hereby proposed allows conducing accurate analysis once exploring a parallel processing taking advantage of the independence of each subproblem the framework can be extended to consider different scenarios of multisite cracked solids with a low computation cost

factorization strategies for third order tensors abstract operations with tensors or multiway arrays have become increasingly prevalent in recent years traditionally tensors are represented or decomposed as a sum of rank 1 outer products using either the candecomp parafac cp or the tucker models or some variation thereof such decompositions are motivated by specific applications where the goal is to find an approximate such representation for a given multiway array the specifics of the approximate representation such as how many terms to use in the sum orthogonality constraints etc depend on the application in this paper we explore an alternate representation of tensors which shows promise with respect to the tensor approximation problem reminiscent of matrix factorizations we present a new factorization of a tensor as a product of tensors to derive the new factorization we define a closed multiplication operation between tensors a major motivation for considering this new type of tensor multiplication is to devise new types of factorizations for tensors which can then be used in applications specifically this new multiplication allows us to introduce concepts such as tensor transpose inverse and identity which lead to the notion of an orthogonal tensor the multiplication also gives rise to a linear operator and the null space of the resulting operator is identified we extend the concept of outer products of vectors to outer products of matrices all derivations are presented for third order tensors however they can be easily extended to the order p p 3 case we conclude with an application in image deblurring

vlsi architectures for computing multiplications and inverses in gf 2 m finite field arithmetic logic is central in the implementation of reed solomon coders and in some cryptographic algorithms there is a need for good multiplication and inversion algorithms that can be easily realized on vlsi chips massey and omura 1 recently developed a new multiplication algorithm for galois fields based on a normal basis representation in this paper a pipeline structure is developed to realize the massey omura multiplier in the finite field gf 2m with the simple squaring property of the normal basis representation used together with this multiplier a pipeline architecture is also developed for computing inverse elements in gf 2m the designs developed for the massey omura multiplier and the computation of inverse elements are regular simple expandable and therefore naturally suitable for vlsi implementation

subquadratic computational complexity schemes for extended binary field multiplication using optimal normal bases based on a recently proposed toeplitz matrix vector product approach a subquadratic computational complexity scheme is presented for multiplications in binary extended finite fields using type i and ii optimal normal bases

the splitting method as a tool for multiple damage analysis the paper presents a reliable and efficient method for solving problems on a large set of domains with different geometries in applications these geometries might be idealizations of a mechanical structure with multiple damage the method is designed and used to analyse large three dimensional problems the present paper formulates the method and presents the theory for a simple model problem i e the laplace equation in two dimensions in the numerical implementation of the splitting scheme the p version of the finite element method and strongly graded meshes are used this approach provides high accuracy and error control the exponential rate of convergence of the method is shown and practically exemplified a numerical example shows the effects of various choices of parameters on the accuracy of the method a complex problem related to aircraft safety which was solved using the present method is used to exemplify the generality and the efficiency of the proposed method

parallel multipliers based on special irreducible pentanomials the state of the art galois field gf 2 sup m multipliers offer advantageous space and time complexities when the field is generated by so special irreducible polynomial to date the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial esp unfortunately there exist only a few irreducible esps in the range of interest for most of the applications e g error correcting codes computer algebra and elliptic curve cryptography furthermore it is not always possible to find an irreducible trinomial of degree m in this range for those cases where neither an irreducible trinomial nor an irreducible esp exists the use of irreducible pentanomials has been suggested irreducible pentanomials are abundant and there are several eligible candidates for a given m we promote the use of two special types of irreducible pentanomials we propose new mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials and give rigorous analyses of their space and time complexity

mastrovito multiplier for all trinomials an efficient algorithm for the multiplication in gf 2 sup m was introduced by mastrovito the space complexity of the mastrovito multiplier for the irreducible trinomial x sup m x 1 was given as m sup 2 1 xor and m sup 2 and gales in this paper we describe an architecture based on a new formulation of the multiplication matrix and show that the mastrovito multiplier for the generating trinomial x sup m x sup n 1 where m spl ne 2n also requires m sup 2 1 xor and m sup 2 and gates however m sup 2 x sup m 2 xor gates are sufficient when the generating trinomial is of the form x sup m x sup m 2 1 for an even m we also calculate the time complexity of the proposed mastrovito multiplier and give design examples for the irreducible trinomials x sup 7 x sup 4 1 and x sup 6 x sup 3 1

a new construction of massey omura parallel multiplier over gf 2 sup m the massey omura multiplier of gf 2 sup m uses a normal basis and its bit parallel version is usually implemented using m identical combinational logic blocks whose inputs are cyclically shifted from one another in the past it was shown that for a class of finite fields defined by irreducible all one polynomials the parallel massey omura multiplier had redundancy and a modified architecture of lower circuit complexity was proposed in this article it is shown that not only does this type of multiplier contain redundancy in that special class of finite fields but it also has redundancy in fields gf 2 sup m defined by any irreducible polynomial by removing the redundancy we propose a new architecture for the normal basis parallel multiplier which is applicable to any arbitrary finite field and has significantly lower circuit complexity compared to the original massey omura normal basis parallel multiplier the proposed multiplier structure is also modular and hence suitable for vlsi realization when applied to fields defined by the irreducible all one polynomials the multiplier s circuit complexity matches the best result available in the open literature

hardware and software normal basis arithmetic for pairing based cryptography in characteristic three although identity based cryptography offers a number of functional advantages over conventional public key methods the computational costs are significantly greater the dominant part of this cost is the tate pairing which in characteristic three is best computed using the algorithm of duursma and lee however in hardware and constrained environments this algorithm is unattractive since it requires online computation of cube roots or enough storage space to precompute required results we examine the use of normal basis arithmetic in characteristic three in an attempt to get the best of both worlds an efficient method for computing the tate pairing that requires no precomputation and that may also be implemented in hardware to accelerate devices such as smart cards

a novel architecture for galois fields gf 2 m multipliers based on mastrovito scheme in this paper a new gf 2m multiplier for standard basis representation is developed the proposed multiplier implements the mastrovito multiplication scheme and can be designed for every field gf 2m a minimum area implementation of the first block of mastrovito multiplier and a high speed delay driven tree architecture for the second block of the mastrovito multiplier are employed in the new circuit multiplier complexity and delay are analytically evaluated for many polynomial classes timing and area occupation performances of the proposed multiplier are also calculated for many fields used in reed solomon codes applications and compared with those of previously proposed solutions the comparison shows that the proposed multiplier outperforms previous architectures for every considered gf 2m field the effectiveness of the proposed solution in a real application is verified by implementing in a 0 25 mum cmos technology the key equation solving block of a 255 239 reed solomon decoder the use of the proposed multiplier in this application results in a substantial speed improvement without any penalty in the silicon area occupation

hybrid trefftz plane elasticity elements with p method capabilities a family of p method plane elasticity elements is derived based on the hybrid trefftz formulation 1 exact solutions of the lame navier equations are used for the intra element displacement field together with an independent displacement frame function field along the element boundary the final unknowns are the parameters of the frame function field consisting of the usual degrees of freedom at corner nodes and an optional number of hierarchic degrees of freedom associated with the mid side nodes since the element matrices do not involve integration over the element area the elements have a polygonal contour with an optional number of curved sides the quadrilateral element has the same external appearance as the conventional p method plane elasticity element 2 3 but unlike in the conventional p method approach suitable special purpose trefftz functions are generally used to handle the singularity and or stress concentration problems rather than a local mesh refinement the practical efficiency of the new elements is assessed through a series of examples

new architectures for digit level single hybrid double hybrid triple field multiplications and exponentiation using gaussian normal bases gaussian normal bases gnbs are special set of normal bases nbs which yield low complexity gf left 2 m right arithmetic operations in this paper we present new architectures for the digit level single hybrid double and hybrid triple multiplication of gf left 2 m right elements based on the gnb representation for odd values of m 1 the proposed fully serial in single multipliers perform multiplication of two field elements and offer high throughput when the data path capacity for entering inputs is limited the proposed hybrid double and hybrid triple digit level gnb multipliers perform respectively two and three field multiplications using the same latency required for a single digit level multiplier at the expense of increased area in addition we present a new eight ary field exponentiation architecture which does not require precomputed or stored intermediate values

modular construction of low complexity parallel multipliers for a class of finite fields gf 2 sup m structures for parallel multipliers of a class of fields gf 2 sup m based on irreducible all one polynomials aop and equally spaced polynomials esp are presented the structures are simple and modular which is important for hardware realization relationships between an irreducible aop and the corresponding irreducible esps have been exploited to construct esp based multipliers of large fields by a regular expansion of the basic modules of the aop based multiplier of a small field some features of the structures also enable a fast implementation of squaring and multiplication algorithms and therefore make fast exponentiation and inversion possible it is shown that if for a certain degree an irreducible aop as well as an irreducible esp exist then from the complexity point of view it is advantageous to use the esp based parallel multiplier

a novel fast tensor based preconditioner for image restoration image restoration is one of the main parts of image processing mathematically this problem can be modeled as a large scale structured ill posed linear system ill posedness of this problem results in low convergence rate of iterative solvers for speeding up the convergence preconditioning usually is used despite the existing preconditioners for image restoration which are constructed based on approximations of the blurring matrix in this paper we propose a novel preconditioner with a different viewpoint here we show that image restoration problem can be modeled as a tensor contractive linear equation this modeling enables us to propose a new preconditioner based on an approximation of the blurring tensor operator due to the particular structure of the blurring tensor for zero boundaries we show that the truncated higher order singular value decomposition of the blurring tensor is obtained very fast and so could be used as a preconditioner experimental results confirm the efficiency of this new preconditioner in image restoration and its outperformance in comparison with the other well known preconditioners

a novel approach for bit serial ab2 multiplication in finite fields gf 2m this paper presents a new inner product ab 2 multiplication algorithm and effective hardware architecture for exponentiation in finite fields gf 2 m exponentiation is more efficiently implemented by applying ab 2 multiplication repeatedly rather than ab multiplication thus efficient ab 2 multiplication algorithms and simple architectures are the key to implementing exponentiation accordingly this paper proposes an efficient inner product multiplication algorithm based on an irreducible all one polynomial aop and simple architecture which has the same hardware equipment as fenn s ab multiplier the proposed bit serial multiplication algorithm and architecture are highly regular and simpler than those of previous works