low complexity bit parallel architectures for polynomial basis multiplication over gf 2m representing the field elements with respect to the polynomial or standard basis we consider bit parallel architectures for multiplication over the finite field gf 2m in this effect first we derive a new formulation for polynomial basis multiplication in terms of the reduction matrix q the main advantage of this new formulation is that it can be used with any field defining irreducible polynomial using this formulation we then develop a generalized architecture for the multiplier and analyze the time and gate complexities of the proposed multiplier as a function of degree m and the reduction matrix q to the best of our knowledge this is the first time that these complexities are given in terms of q unlike most other articles on bit parallel finite field multipliers here we also consider the number of signals to be routed in hardware implementation and we show that compared to the well known mastrovito s multiplier the proposed architecture has fewer routed signals the proposed generalized architecture is further optimized for three special types of polynomials namely equally spaced polynomials trinomials and pentanomials we have obtained explicit formulas and complexities of the multipliers for these three special irreducible polynomials this makes it very easy for a designer to implement the proposed multipliers using hardware description languages like vhdl and verilog with minimum knowledge of finite field arithmetic

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

fast bit parallel binary multipliers based on type i pentanomials in this paper a fast implementation of bit parallel polynomial basis pb multipliers over the binary extension field gf 2 m generated by type i irreducible pentanomials is presented explicit expressions for the coordinates of the multipliers and a detailed example are given complexity analysis shows that the multipliers here presented have the lowest delay in comparison to similar bit parallel pb multipliers found in the literature based on this class of irreducible pentanomials in order to prove the theoretical complexities hardware implementations over xilinx fpgas have also been performed experimental results show that the approach here presented exhibits the lowest delay with a balanced area times time complexity when it is compared with similar multipliers

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

condition numbers for the outer inverse and constrained singular linear system in this paper we investigate the condition number of the outer inverse at s 2 and outer inverse at s 2 solution of constrained linear system ax b x t where a is a real m n matrix b and x are real vectors t is a subspace let and be two positive real numbers when we consider the weighted frobenius norm a b qp f on the data we get the formula of condition number for the outer inverse at s 2 solution of constrained linear system for the normwise condition number the sensitivity of the relative condition number itself is studied and the componentwise perturbation is also investigated

systolic gaussian elimination over gf p with partial pivoting a systolic architecture is proposed for the triangularization by means of the gaussian elimination algorithm of large dense n n matrices over gf p where p is a prime number the solution of large dense linear systems over gf p is the major computational step in various algorithms issuing from arithmetic number theory and computer algebra the proposed architecture implements the elimination with partial pivoting although the operation of the array remains purely systolic extension of the array to the complete solution of a linear system ax b over gf p is also considered

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

mastrovito form of non recursive karatsuba multiplier for all trinomials we present a new type of bit parallel non recursive karatsuba multiplier over gf 2 m generated by an arbitrary irreducible trinomial this design effectively exploits mastrovito approach and shifted polynomial basis spb to reduce the time complexity and karatsuba algorithm to reduce its space complexity we show that this type of multiplier is only one t x slower than the fastest bit parallel multiplier for all trinomials where t x is the delay of one 2 input xor gate meanwhile its space complexity is roughly 3 4 of those multipliers to the best of our knowledge it is the first time that our scheme has reached such a time delay bound this result outperforms previously proposed non recursive karatsuba multipliers

global well posedness and blow up of solutions for the camassa holm equations with fractional dissipation consideration in this paper is the effect of varying fractional dissipation with the dissipative operator power gamma ge 0 on the well posedness of the camassa holm equations with fractional dissipation it is shown that the zero filter limit alpha rightarrow 0 of the camassa holm equation with fractional dissipation is the fractal burgers equation it is known that in the supercritical case gamma in 0 1 the fractal burgers equation blows up in finite time in h s mathbb r with s frac 3 2 gamma it is established here that the dissipative camassa holm equation is globally well posed in the critical sobolev space h frac 3 2 gamma mathbb r with the fractional parameter gamma in frac 1 2 1 moreover it is also demonstrated that the solution of the dissipative camassa holm equation blows up in finite time in the particular supercritical case when gamma 0

on stable perturbations for outer inverses of linear operators in banach spaces abstract in this paper we investigate stable perturbations and their characterizations for various types of outer inverses such as generalized 2 3 2 4 2 5 1 2 3 1 2 4 moore penrose group drazin and generalized drazin inverses some known results are improved and extended

characterizations and representations of the p q outer generalized inverse we present a new explicit expression for the p q outer generalized inverse based on the idempotent operators p and q we derive properties of the p q outer generalized inverse and analyse relations among the various p q outer generalized inverses

on integral representation of the generalized inverse at s 2 we present a general integral representation for the generalized inverse a t s 2 which extends earlier result on the moore penrose inverse weighted moore penrose inverse and drazin inverse

bit parallel polynomial basis multiplier for new classes of finite fields in this paper three small classes of finite fields gf 2m are found for which low complexity bit parallel multipliers are proposed the proposed multipliers have lower complexities compared to those based on the irreducible pentanomials it is also shown that there does not always exist an irreducible all one polynomial equally spaced polynomial or trinomial for the new classes of fields

on the blow up rate and the blow up set of breaking waves for a shallow water equation we consider the problem of the development of singularities for classical solutions to a new periodic shallow water equation blow up can occur only in the form of wave breaking i e the solution remains bounded but its slope becomes unbounded in finite time a quite detailed description of the wave breaking phenomenon is given there is at least a point in general depending on time where the slope becomes infinite exactly at breaking time the precise blow up rate is established and for a large class of initial data we also determine the blow up set

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

perturbation theory for the generalized bott duffin inverse and its applications a perturbation theory for the bott duffin inverse a l of a with respect to a subspace l is developed the perturbation bound for the solution of the constrained system ax y b email protected l email protected l is established where email protected m n c a subspace email protected c n and email protected c n meanwhile this paper shows the generalized bott duffin condition number k g b d a email protected email protected email protected a l to be minimum in the inequality of error analysis

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

derivation of reduced test vectors for bit parallel multipliers over gf 2 m this paper presents an algebraic testing method for detecting stuck at faults in the polynomial basis pb bit parallel bp multiplier circuits over gf 2m the proposed technique derives the test vectors from the expressions of the inner product ip variables without any requirement of the atpg tool this low complexity testing method requires 2m 1 test vectors for detecting single stuck at faults in the and part and multiple stuck at faults in the exor part of the multiplier circuits the test vectors are independent of the multiplier s structure as proposed in t a gulliver et al 1991 but are dependent on m for the multiplier circuits the test set is found to be smaller in size than the atpg generated test set the test set provides 100 percent single stuck at fault coverage

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

condition number of drazin inverse and their condition numbers of singular linear systems various normwise relative condition numbers that measure the sensitivity of drazin inverse and the solution of singular linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of structured perturbation

double basis multiplicative inversion over gf 2 sup m inversion over galois fields is much more difficult than the corresponding multiplication efficient computation of inverses in gf 2 sup m is considered by solving a set of linear equations over the ground field gf 2 the proposed algorithm uses two separate bases for the representation of its input and output elements and has low computational complexity the algorithm is also suitable for hardware implementation using vlsi technologies

new low complexity bit parallel finite field multipliers using weakly dual bases new structures of bit parallel weakly dual basis wdb multipliers over the binary ground field are proposed an upper bound on the size complexity of bit parallel multiplier using an arbitrary generating polynomial is given when the generating polynomial is an irreducible trinomial x sup m x sup k 1 1 spl les k spl les m 2 the structure of the proposed bit parallel multiplier requires only m sup 2 two input and gates and at most m sup 2 1 xor gates the time delay is no greater than t sub a log sub 2 m 2 t sub x where t sub a and t sub x are the time delays of an and gate and an xor gate respectively

on the convergence of splittings for semidefinite linear systems abstract recently lee et al young ju lee jinbiao wu jinchao xu ludmil zikatanov on the convergence of iterative methods for semidefinite linear systems siam j matrix anal appl 28 2006 634 641 introduce new criteria for the semi convergence of general iterative methods for semidefinite linear systems based on matrix splitting the new conditions generalize the classical notion of p regularity introduced by keller h b keller on the solution of singular and semidefinite linear systems by iterations siam j numer anal 2 1965 281 290 in view of their results we consider here stipulations on a splitting a m n which lead to fixed point systems such that the iterative scheme converges to a weighted moore penrose solution to the system ax b our results extend the result of lee et al to a more general case and we also show that it requires less restrictions on the splittings than keller s p regularity condition to ensure the convergence of iterative scheme

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

schmi gf 2 n shifted polynomial basis multipliers based on subquadratic toeplitz matrix vector product approach for all irreducible pentanomials besides karatsuba s algorithm optimal toeplitz matrix vector product tmvp formulae is another approach to design gf 2 n subquadratic multipliers however when gf 2 n elements are represented using a polynomial basis or its generalization shifted polynomial basis this approach is currently appliable only to fields gf 2 n generated by an irreducible trinomial or a special type of irreducible pentanomials but not to a general irreducible pentanomial the reason is that no transformation matrix which transforms the mastrovito matrix into a toeplitz matrix has been found in this article we propose such a transformation matrix and its inverse matrix for an arbitrary irreducible pentanomial

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

subquadratic space complexity binary field multiplier using double polynomial representation this paper deals with binary field multiplication we use the bivariate representation of binary field called double polynomial system dps presented in this concept generalizes the composite field representation to every finite field as shown in the main interest of dps representation is that it enables to use lagrange approach for multiplication and in the best case fast fourier transform approach which optimizes lagrange approach we use here a different strategy from to perform reduction and we also propose in this paper some new approaches for constructing dps we focus on dps which provides a simpler and more efficient method for coefficient reduction this enables us to avoid a multiplication required in the montgomery reduction approach of and thus to improve the complexity of the dps multiplier the resulting algorithm proposed in the present paper is subquadratic in space o n1 31 and logarithmic in time the space complexity is 33 percent better than in and 18 percent faster it is asymptotically more efficient than the best known method specifiably more efficient than when n 3 000 furthermore our proposal is available for every n and not only for n a power of two or three

bit serial and bit parallel montgomery multiplication and squaring over gf 2 m multiplication and squaring are main finite field operations in cryptographic computations and designing efficient multipliers and squarers affect the performance of cryptosystems in this paper we consider the montgomery multiplication in the binary extension fields and study different structures of bit serial and bit parallel multipliers for each of these structures we study the role of the montgomery factor and then by using appropriate factors propose new architectures specifically we propose two bit serial multipliers for general irreducible polynomials and then derive bit parallel montgomery multipliers for two important classes of irreducible polynomials in this regard first we consider trinomials and provide a way for finding efficient montgomery factors which results in a low time complexity then we consider type ii irreducible pentanomials and design two bit parallel multipliers which are comparable to the best finite field multipliers reported in the literature moreover we consider squaring using this family of irreducible polynomials and show that this operation can be performed very fast with the time complexity of two xor gates

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

well posedness and blow up solutions for an integrable nonlinearly dispersive model wave equation abstract we establish local well posedness in the sobolev space hs with any s 3 2 for an integrable nonlinearly dispersive wave equation arising as a model for shallow water waves known as the camassa holm equation however unlike the more familiar korteweg devries model we demonstrate conditions on the initial data that lead to finite time blow up of certain solutions

a characterization and representation of the generalized inverse a 2 t s and its applications abstract this paper presents an explicit expression for the generalized inverse a 2 t s based on this we established the characterization the representation theorem and the limiting process for a 2 t s as an application we estimate the error bound of the iterative method for approximating a 2 t s

a note on condition numbers for generalized inverse at s 2 and constrained linear systems abstract in this paper we use the schur decomposition to measure the sensitivity of the general inverse a t s 2 and the constrained singular linear system ax b with regard to 2 norm and f norm rather than pq norm in 12 where p and q are nonsingular matrices the explicit forms of the condition numbers approximate the sensitivity of a t s 2 and the constrained singular linear system pretty well furthermore since schur decomposition is well posed the evaluation of the sensitivity can be numerically easy and stable

gf 2 sup m multiplication and division over the dual basis in this paper an algorithm for gf 2 sup m multiplication division is presented and a new more generalized definition of duality is proposed from these the bit serial berlekamp multiplier is derived and shown to be a specific case of a more general class of multipliers furthermore it is shown that hardware efficient bit parallel dual basis multipliers can also be designed these multipliers have a regular structure are easily extended to different gf 2 sup m and hence suitable for vlsi implementations as in the bit serial case these bit parallel multipliers can also be hardwired to carry out constant multiplication these constant multipliers have reduced hardware requirements and are also simple to design in addition the multiplication division algorithm also allows a bit serial systolic finite field divider to be designed this divider is modular independent of the defining irreducible polynomial for the field easily expanded to different gf 2 sup m and its longest delay path is independent of m

efficient bit serial multiplication and the discrete time wiener hopf equation over finite fields discrete time wiener hopf equations dtwhes over finite fields are considered it is shown that solving the dtwhe is equivalent to performing division over finite fields the proof provides a new interpretation of the relationship between bit serial multiplication and dtwhes the interpretation enables bit serial multiplication over gf 2 sup m to be understood more easily as an example bit serial multiplication methods for multiplying any two elements that can be done without performing any transformation or with only a simple transformation of the bases are presented

a new architecture for a parallel finite field multiplier with low complexity based on composite fields a bit parallel structure for a multiplier with low complexity in galois fields is introduced the multiplier operates over composite fields gf 2 sup n sup m with k nm the karatsuba ofman algorithm a karatsuba and y ofmanis 1963 is investigated and applied to the multiplication of polynomials over gf 2 sup n it is shown that this operation has a complexity of order o k sup log23 under certain constraints regarding k a complete set of primitive field polynomials for composite fields is provided which perform module reduction with low complexity as a result multipliers for fields gf 2 sup k up to k 32 with low gate counts and low delays are listed the architectures are highly modular and thus well suited for vlsi implementation

the representation and approximation for the generalized inverse at s 2 we present a unified representation theorem for the generalized inverses a t s 2 which are 2 or outer inverses having the prescribed range t and null space s specific expressions and computational procedures for the generalized inverses a t s 2 can be uniformly derived

perturbation bounds for the generalized inverses at s 2 with application to constrained linear system in this paper the more general perturbation problem of the generalized inverse a t s 2 is considered we derive its perturbation bounds when the subspaces t and s also have perturbation and in special case when t and s have no perturbation we get some results on the condition numbers with respect to a t s 2 and constrained linear system ax b x t which are similar to the regular inverse a 1 and nonsingular linear system ax b

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

high speed hybrid double multiplication architectures using new serial out bit level mastrovito multipliers the serial out bit level multiplication scheme is characterized by an important latency feature it has an ability to sequentially generate an output bit of the multiplication result in each clock cycle however the computational complexity of the existing serial out bit level multipliers in gf 2 m using normal basis representation limits its usefulness in many applications hence an optimized serial out bit level multiplier using polynomial basis representation is needed in this paper we propose new serial out bit level mastrovito multiplier schemes we show that in terms of the time complexities the proposed multiplier schemes outperform the existing serial out bit level schemes available in the literature in addition using the proposed multiplier schemes we present new hybrid double multiplication architectures to the best of our knowledge this is the first time such a hybrid multiplier structure using the polynomial basis is proposed prototypes of the presented serial out bit level schemes and the proposed hybrid double multiplication architectures 10 schemes in total are implemented over both gf 2 163 and gf 2 233 and experimental results are presented

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

low complexity bit parallel multipliers for a class of finite fields new implementations of bit parallel multipliers for a class of finite fields are proposed the class of finite fields is constructed with irreducible aops all one polynomials and esps equally spaced polynomials the size and time complexities of our proposed multipliers are lower than or equal to those of the previously proposed multipliers of the same class

global conservative solutions of the camassa holm equation this paper develops a new approach in the analysis of the camassa holm equation by introducing a new set of independent and dependent variables the equation is transformed into a semilinear system whose solutions are obtained as fixed points of a contractive transformation these new variables resolve all singularities due to possible wave breaking returning to the original variables we obtain a semigroup of global solutions depending continuously on the initial data our solutions are conservative in the sense that the total energy equals a constant for almost every time

condition numbers and perturbation of the weighted moore penrose inverse and weighted linear least squares problem it is well known that the normwise relative condition numbers measure the sensitivity of matrix inversion and the solution of nonsingular linear systems here we consider the condition number formulas for the weighted moore penrose inverse of a rectangular matrix and give explicit expressions for the weighted condition numbers of the singular linear systems ax b these explicit expressions extend the earlier work of several authors as we know the computed weighted least squares solution x of min email protected email protected m in the presence of the round off error satisfies the perturbed equation min email protected a e y b f m finally an upper bound of email protected n is derived for the case where the weighted matrix and vector norms and the assumption rank a e rank a are used

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

condition numbers and their condition numbers abstract various normwise relative condition numbers that measure the sensitivity of matrix inversion and the solution of linear systems are characterized new results are derived for the cases where two common noninduced matrix norms are used and where different vector norms are used for the domain and range of the matrix condition numbers that respect the structure of symmetric problems are also analyzed the sensitivity of the condition number itself is then investigated and we obtain sharp examples of demmel s general result that for certain problems in numerical analysis the condition number of the condition number is the condition number finally upper bounds are derived for the sensitivity of componentwise condition numbers

displacement structure of the generalized inverse a t s 2 it is well known that matrices with a uv displacement structure possess generalized inverse with a vu displacement structure estimation for the displacement rank of a t s 2 u va t s 2 are presented where a t s 2 is the 2 inverse of a we extend the results due to heinig and hellinger in 1994

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

condition number related with generalized inverse a t s 2 and constrained linear systems in this paper we introduce a pq norm and discuss the condition number of the generalized inverse at s 2 we get some results on the condition numbers with respect to at s 2 and constrained linear systems ax b x t which are similar to the regular inverse a 1 and nonsingular linear systems ax b finally we investigate the structured perturbation of the generalized inverse at s 2

condition number related to the outer inverse of a complex matrix in this paper we obtain the formula for computing the condition number of a complex matrix which is related to the outer generalized inverse of a given matrix we use the schur decomposition of a matrix we characterize the spectral norm and the frobenius norm of the relative condition number of the generalized inverse and the level 2 condition number of the generalized inverse the sensitivity for the generalized drazin inverse solution of linear systems is presented we also present the structured perturbation of the generalized inverse

equally spaced polynomials dual bases and multiplication in f 2 sup n a proposed multiplier for finite fields given by equally spaced polynomials is based on the transformation from the polynomial basis to its dual basis combined with multiplication by a constant we classify the constants that are optimal regarding the cost of this operation and investigate the cost of the inverse transformation

architecture for a low complexity rate adaptive reed solomon encoder multiple error correcting reed solomon rs codes have many practical applications the complexity of an rs encoder depends on multiplications in the finite field over which the code is defined we consider a triangular basis for representing the field elements and present an architecture for a rate adaptive rs encoder using a triangular basis multiplication algorithm the architecture supports pipeline and bit serial operations and has a low circuit complexity

on the weak solutions to a shallow water equation we obtain the existence of global in time weak solutions to the cauchy problem for a one dimensional shallow water equation that is formally integrable and can be obtained by approximating directly the hamiltonian for euler s equation in the shallow water regime the solution is obtained as a limit of viscous approximation the key elements in our analysis are some new a priori one sided supernorm and space time higher norm estimates on the first order derivatives 2000 john wiley sons inc

global well posedness of the critical burgers equation in critical besov spaces abstract we make use of the method of modulus of continuity a kiselev f nazarov r shterenberg blow up and regularity for fractal burgers equation dyn partial differ equ 5 2008 211 240 and fourier localization technique h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 h abidi t hmidi on the global well posedness of the critical quasi geostrophic equation siam j math anal 40 1 2008 167 185 to prove the global well posedness of the critical burgers equation t u u x u u 0 in critical besov spaces b p 1 1 p r with p 1 where

fault detection architectures for field multiplication using polynomial bases in many cryptographic schemes the most time consuming basic arithmetic operation is the finite field multiplication and its hardware implementation for bit parallel operation may require millions of logic gates some of these gates may become faulty in the field due to natural causes or malicious attacks which may lead to the generation of erroneous outputs by the multiplier in this paper we propose new architectures to detect erroneous outputs caused by certain types of faults in bit parallel and bit serial polynomial basis multipliers over finite fields of characteristic two in particular parity prediction schemes are developed for detecting errors due to single and certain multiple stuck at faults although the issue of detecting soft errors in registers is not considered the proposed schemes have the advantage that they can be used with any irreducible binary polynomial chosen to define the finite field

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

fast parallel gf 2 m polynomial multiplication for all degrees numerous works have addressed efficient parallel gf 2m multiplication based on polynomial basis or some of its variants for those field degrees where neither irreducible trinomials nor equally spaced polynomials epss exist the best area time performance has been achieved for special type irreducible pentanomials which however do not exist for all degrees in other words no multiplier architecture has been proposed so far achieving the best performance and at the same time being general enough to support any field degrees in this paper we propose a new representation based on what we called generalized polynomial bases gpbs covering polynomial bases and the so called shifted polynomial bases spbs as special cases in order to study the new representation we introduce a novel formulation for polynomial basis and its variants which is able to express concisely all implementation aspects of interest i e gate count subexpression sharing and time delay the methodology enabled by the new formulation is completely general and repetitive in its application allowing the development of an ad hoc software tool to derive proofs for area complexity and time delays automatically as the central contribution of this paper we introduce some new types of irreducible pentanomials and an associated gpb based on the above formulation we prove that carefully chosen gpbs yield multiplier architectures matching or even outperforming the best special type pentanomials from both the area and time point of view most importantly the proposed gpb architectures require pentanomials existing for all degrees of practical interest a list of suitable irreducible pentanomials for all degrees less than 1 000 is given in the appendix fig 5 and tables 4 11 are provided in a separate file containing the body of appendix which can be found on the computer society digital library at http doi ieeecomputersociety org 10 1109 tc 2012 63

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

on the perturbation and subproper splittings for the generalized inverse at s 2 of rectangular matrix a abstract in this paper the perturbation and subproper splittings for the generalized inverse at s 2 the unique matrix x such that xax x r x t and n x s are considered we present lower and upper bounds for the perturbation of at s 2 convergence of subproper splittings for computing the special solution at s 2 b of restricted rectangular linear system ax b x t are studied for the solution at s 2 b we develop a characterization therefore we give a unified treatment of the related problems considered in literature by ben israel berman hanke neumann plemmons etc

fast bit parallel gf 2 sup n multiplier for all trinomials based on a new representation of gf 2 sup n we present two multipliers for all irreducible trinomials space complexities of the multipliers match the best results the time complexity of one multiplier is t sub a 1 log sub 2 n t sub x for all irreducible trinomials where t sub a and t sub x are the delay of one 2 input and and xor gates respectively

on the p q outer generalized inverse in banach algebra abstract in this note the p q outer generalized inverse in banach algebra is considered a new representation based on the 1 5 inverse is presented also we construct an iterative method for computing the p q outer generalized inverse in banach algebra

symplectic structures their backlund transformations and hereditary symmetries abstract it is shown that compatible symplectic structures lead in a natural way to hereditary symmetries we recall that a hereditary symmetry is an operator valued function which immediately yields a hierarchy of evolution equations each having infinitely many commuting symmetries all generated by this hereditary symmetry furthermore this hereditary symmetry usually describes completely the soliton structure and the conservation laws of these equations this result then provide us with a method for constructing hereditary symmetries and hence exactly solvable evolution equations in addition we show how symplectic structures transform under backlund transformations this leads to a method for generating a whole class of symplectic structures from a given one several examples and applications are given illustrating the above results also the connection of our results with those of gelfand and dikii and of magri is briefly pointed out

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

perturbation bounds for the generalized inverse at s 2 with respect to the frobenius norm abstract in this paper we present the perturbation bounds and the relative error bound of the generalized inverse a t s 2 with respect to the frobenius norm and apply the results to the relative errors of the solution of the general restricted linear equation

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

full rank representations of 2 4 2 3 inverses and successive matrix squaring algorithm abstract we present the full rank representations of 2 4 and 2 3 inverses with given rank as well as with prescribed range and null space as particular cases of the full rank representation of outer inverses as a consequence two applications of the successive matrix squaring sms algorithm from p s stanimirovic d s cvetkovic ilic successive matrix squaring algorithm for computing outer inverses appl math comput 203 2008 19 29 are defined using the full rank representations of 2 4 and 2 3 inverses the first application is used to approximate 2 4 inverses the second application after appropriate modifications of the sms iterative procedure computes 2 3 inverses of a given matrix presented numerical examples clarify the purpose of the introduced methods

relationship between gf 2 m montgomery and shifted polynomial basis multiplication algorithms applying the matrix vector product idea of the mastrovito multiplier to the gf 2m montgomery multiplication algorithm we present a new parallel multiplier for irreducible trinomials this multiplier and the corresponding shifted polynomial basis spb multiplier have the same circuit structure for the same set of parameters furthermore by establishing isomorphisms between the montgomery and the spb constructions of gf 2m we show that the montgomery algorithm can be used to perform the spb multiplication without any changes and vice versa

condition number of bott duffin inverse and their condition numbers various normwise relative condition numbers that measure the sensitivity of bott duffin inverse and the solution of constrained linear systems are characterized the sensitivity of condition number itself is then investigated finally upper bounds are derived for the sensitivity of componentwise condition numbers

efficient bit parallel multiplier for irreducible pentanomials using a shifted polynomial basis in this paper we present a bit parallel multiplier for gf 2m defined by an irreducible pentanomial xm xk 3 xk 2 xk 1 1 where 1lesk 1 lesk2 lesk3 lesm 2 in order to design an efficient bit parallel multiplier we introduce a shifted polynomial basis and modify a reduction matrix presented by reyhani masoleh and hasan as a result the time complexity of the proposed multiplier is ta 3 log2 m 1 tx where ta and tx are the delay of one and and one xor gate respectively this result matches or outperforms the previously known results on the other hand the proposed multiplier has the same space complexity as the previously known multipliers except for special types of irreducible pentanomials note that its hardware architecture is similar to that presented by reyhani masoleh and hasan

bit parallel finite field multiplier and squarer using polynomial basis bit parallel finite field multiplication using polynomial basis can be realized in two steps polynomial multiplication and reduction modulo the irreducible polynomial in this article we present an upper complexity bound for the modular polynomial reduction when the field is generated with an irreducible trinomial closed form expressions for the coefficients of the product are derived in term of the coefficients of the multiplicands the complexity of the multiplier architectures and their critical path length are evaluated and they are comparable to the previous proposals for the same class of fields an analytical form for bit parallel squaring operation is also presented the complexities for bit parallel squarer are also derived when an irreducible trinomial is used consequently it is argued that to solve multiplicative inverse using polynomial basis can be at least as good as using a normal basis