evolution equations of lurie type in hilbert spaces on etudie le probleme de lurie pour le systeme dx d ax b d dx o c w p ou l operateur a est soit borne soit suppose engendrer un groupe co fortement continu t t t e sur un espace de hilbert reel x de produit interieur et de norme b c x p e et o e e est une fonction non lineaire lipschitzienne uniformement continue

at sup 2 optimal galois field multiplier for vlsi vlsi designs for galois field multipliers which are central in many encoding and decoding procedures for error detecting and error correcting codes are presented an at sup 2 optimal galois field multiplier based on at sup 2 optimal integer multipliers for a synchronous vlsi model is exhibited galois field multiplication is done in two steps first two polynomials of degree n 1 over z sub p are multiplied and then the resulting polynomial is reduced modulo a fixed irreducible polynomial of degree n multiplication of polynomials is done by discrete fourier transform dft for p 2 the procedure is more involved for z sub p x than for z x an extension to the case of variable p is included and some open problems are stated

high speed low complexity systolic designs of novel iterative division algorithms in gf 2 sup m we extend the binary algorithm invented by stein and propose novel iterative division algorithms over gf 2 sup m for systolic vlsi realization while algorithm ebg is a basic prototype with guaranteed convergence in at most 2m 1 iterations its variants algorithms ebd and ebdf are designed for reduced complexity and fixed critical path delay respectively we show that algorithms ebd and ebdf can be mapped to parallel in parallel out systolic circuits with low area time complexities of o m sup 2 loglogm and o m sup 2 respectively compared to the systolic designs based on the extended euclid s algorithm our circuits exhibit significant speed and area advantages

analyticity of the stokes semigroup in spaces of bounded functions the analyticity of the stokes semigroup with the dirichlet boundary condition is established in spaces of bounded functions when the domain occupied with fluid is bounded or more generally admissible which admits a special estimate for the helmholtz decomposition the proof is based on a blow up argument this is the first proof of the analyticity in spaces of bounded functions which was left open more than thirty years

a least squares method for finding the preisach hysteresis operator from measurements a finite element like least squares method is introduced for determining the density function in the preisach hysteresis model from overdeterined measured data it is shown that the least squares error depends on the quality of the data and the best approximations to the analytic density for consistent data criteria are given for convergence of the approximate density and preisach operator with increasing measurements

normal basis of the finite field f sub 2 p 1 p sup m over f sub 2 the determination of normal bases for finite fields particularly over the finite field f sub 2 is of importance in applications such as coding and cryptography this correspondence gives an explicit normal basis for a finite field f sub 2 p 1 p sup m over f sub 2 when 2 is a primitive element module p sup 2 in the case when p 3 an explicit dual basis is also obtained

a search of minimal key functions for normal basis multipliers the circuit complexity of a massey omura normal basis multiplier for a finite field gf 2 sup m depends on the key function for multiplication key functions with minimum complexity called minimal key functions are desirable this paper investigates the complexity of a key function and reports search results of minimal key functions a table of minimal key functions for m up to 31 is included

a new representation of elements of finite fields gf 2 sup m yielding small complexity arithmetic circuits let f sub 2 denote the binary field and f sub 2m an algebraic extension of degree m 1 over f sub 2 traditionally elements of f sub 2m are either represented as powers of a primitive element of f sub 2m together with 0 or by an expansion in a basis of the vector space f sub 2m over f sub 2 we propose a new representation based on an isomorphism from f sub 2m into the residue polynomial ring module x sup n 1 the new representation simultaneously satisfies the properties of various traditional representations which leads in some cases to architectures of parallel in parallel out arithmetic circuits adder multiplier exponentiator inverter squarer divider with average to small complexity we show that the implementation of all the arithmetic circuits designed for the new representation on an integrated circuit sometimes has smaller complexity than the implementation of all the arithmetic circuits designed for other representations in addition we derive a serial multiplier for the field f sub 2m which comprises the least number of gates of all the serial multipliers known to the author when m 1 is a prime such that 2 is primitive in the field z sub m 1

a systolic power sum circuit for gf 2 sup m a systolic power sum circuit designed to perform ab sup 2 c computations in the finite field gf 2 sup m is presented where a b and c are arbitrary elements of gf 2 sup m this new circuit is constructed by m sup 2 identical cells each of which consists of three 2 input and logical gates one 2 input xor gate one 3 input xor gate and ten latches the ab sup 2 c computation is required in decoding many error correcting codes the paper shows that a decoder implemented using the new power sum circuit will have less complex circuitry and shorter decoding delay than one implemented using conventional product sum 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

identification of hysteresis loops abstract a classical model representing several hysteresis phenomena as they appear in ferromagnetism and in porous media filtration e g is considered here it is demonstrated that the parameters of this model can be determined with any desired precision from suitable experiments we use the mathematical model of hysteresis as introduced by preisach who explained the magnetization curve and saturation loop of the material by a superposition of elementary rectangular loops a certain properly defined measure is responsible for the shape of the loop we describe a new algorithm which determines this measure completely our results are related to previous papers of biorci and pescetti extensive numerical computations conclude our investigations

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

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

a comparison of vlsi architecture of finite field multipliers using dual normal or standard bases three different finite field multipliers are presented 1 a dual basis multiplier due to e r berlekamp 1982 the massey omura normal basis multiplier and 3 the scott tavares peppard standard basis multiplier these algorithms are chosen because each has its own distinct features that apply most suitably in particular areas they are implemented on silicon chips with nmos technology so that the multiplier most desirable for vlsi implementation can readily be ascertained

systolic multipliers for finite fields gf 2 m two systolic architectures are developed for performing the product sum computation ab c in the finite field gf 2m of 2melements where a b and c are arbitrary elements of gf 2m the first multiplier is a serial in serial out one dimensional systolic array while the second multiplier is a parallel in parallel out two dimensional systolic array the first multiplier requires a smaller number of basic cells than the second multiplier the second multiplier heeds less average time per computation than the first multiplier if a number of computations are performed consecutively to perform single computations both multipliers require the same computational time in both cases the architectures are simple and regular and possess the properties of concurrency and modularity as a consequence they are well suited for use in vlsi systems

convergence of direct recursive algorithm for identification of preisach hysteresis model with stochastic input we consider a recursive iterative algorithm for identification of parameters of the preisach model one of the most commonly used models of hysteretic input output relationships this online algorithm uses a simple rule for updating the values of the piecewise constant density function in the switching region at each time step the so called persistent excitation condition has been shown to play an important role for convergence of recursive iteration schemes when input output data are generated by a deterministic input such as for example a periodically repeated sequence of test inputs prescribed by the classical mayergoyz identification algorithm in this work we assume that the input randomly fluctuates and these fluctuations can be described by a stochastic markov process assuming that accurate measurements of the input and output are available we prove the exponential convergence of the recursive identification algorithm estimate explicitly the convergence rate and explore which properties of the stochastic input and the algorithm affect the guaranteed convergence rate an analogue of the persistent excitation condition suitable for analysis of stochastic markov inputs is established numerical examples that test the convergence of the algorithm in the case of a time dependent density function and in the presence of measurement noise are presented

a systolic architecture for computing inverses and divisions in finite fields gf 2 sup m a new serial in serial out systolic array is presented for performing the element inversion in gf 2 sup m with the standard basis representation the architecture is highly regular modular nearest neighbor connected and thus well suited to vlsi implementation it has a latency of 7m 3 clock cycles and a throughput rate of one result per 2m 1 clock cycles this speed performance is much better than those of the previous implementations without change in hardware design the proposed inversion array can be directly used for computing the division in gf 2 sup m

a new algorithm for multiplication in finite fields a new algorithm is presented for computing the product of two elements in a finite field by means of sums and products in a fixed subfield the algorithm is based on a normal basis representation of fields and assumes that the dimension m of the finite field over the subfield is a highly composite number a very fast parallel implementation and a considerable reduction in the number of computations are allowed in comparison to some methods discussed in the literature

on l infty bmo estimates for derivatives of the stokes semigroup we consider the stokes equations in a class of domains that we will call admissible domains including bounded domains the half space and exterior domains we will prove new l infty estimates for derivatives of velocity and pressure the estimates will be given in terms of a bmo type norm of the initial data

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

properties of solutions to a class of differential models incorporating preisach hysteresis operator abstract we consider a class of equations which have been recently proposed as a mathematical tool for modelling dynamics of hydrological economic and biological systems exhibiting hysteresis and related memory effects the detail of the modelling approach is illustrated by an example from the hydrological context where a balance equation is coupled with a hysteretic constitutive relationship between the water content in the soil and the matric potential m of the soil matrix and where the preisach hysteresis operator is used as a model of this constitutive relationship in particular we present assumptions which eliminate spatial variation and lead to balance equations in the form of odes two examples of such hydrological models are considered followed by a less detailed discussion of applications of similar modelling approach and equations in economics and biology in the proposed formalism the closed system is described by an operator differential equation where the rate of change of the output of the preisach operator is a function of its input and time in the main part of the paper for such operator differential equations we study the initial value problem uniqueness existence extendability of solutions their dependence on initial data and the structure of the projection of the phase portrait onto the t m plane solutions are characterised by jumps of the derivative induced by either of the two reasons one is the memory of past extremum values of the solution the other is a singularity at the lines of zero flow we analyse the singularity and calculate the value of the jumps thus providing an important input to numerical solution of the equation furthermore we identify possible non uniqueness points and points of sensitivity to small perturbations of initial data as well as conditions that ensure uniqueness and stability to such perturbations regularisation of the equation and natural monotonicity conditions ensuring global stability of a periodic solution for the equation with periodic input are discussed rigorous analysis of the well posedness of the models is presented for the first time

systolic array implementation of euclid s algorithm for inversion and division in gf 2 sup m this paper presents two new systolic arrays to realize euclid s algorithm for computing inverses and divisions in finite fields gf 2 sup m with the standard basis representation one of these two schemes is parallel in parallel out and the other is serial in serial out the former employs o m sup 2 area complexity to provide the maximum throughput in the sense of producing one result every clock cycle while the latter achieves a throughput of one result per m clock cycles using o m log m area complexity both of the proposed architectures are highly regular and thus well suited to vlsi implementation as compared to existing related systolic architectures with the same throughput performance the proposed parallel in parallel out scheme reduces the hardware complexity and thus the area time product by a factor of o m and the proposed serial in serial out scheme by a factor of o m log sub 2 m

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

efficient algorithms and architectures for field multiplication using gaussian normal bases recently implementations of normal basis multiplication over the extended binary field gf 2 sup m have received considerable attention a class of low complexity normal bases called gaussian normal bases has been included in a number of standards such as ieee and nist for an elliptic curve digital signature algorithm the multiplication algorithms presented there are slow in software since they rely on bit wise inner product operations in this paper we present two vector level software algorithms which essentially eliminate such bit wise operations for gaussian normal bases our analysis and timing results show that the software implementation of the proposed algorithm is faster than previously reported normal basis multiplication algorithms the proposed algorithm is also more memory efficient compared with its look up table based counterpart moreover two new digit level multiplier architectures are proposed and it is shown that they outperform the existing normal basis multiplier structures as compared with similar digit level normal basis multipliers the proposed multiplier with serial output requires the fewest number of xor gates and the one with parallel output is the fastest multiplier

maximum modulus estimate for the solution of the stokes equations abstract a maximum modulus estimate for the nonstationary stokes equations in c 2 domain is found the singular part and regular part of poisson kernel are analyzed the singular part consists of the gradient of single layer potential and the gradient of composite potential defined on only normal component of the boundary data furthermore the normal velocity near the boundary is bounded if the boundary data is bounded if the normal component of the boundary data is dini continuous and the tangential component of the boundary data is bounded then the maximum modulus of velocity is bounded in whole domain

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

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

asymptotic stability of continuum sets of periodic solutions to systems with hysteresis abstract we consider hysteresis perturbations of a system of odes which possesses an asymptotically stable periodic solution z where the oscillation of an appropriate projection of this periodic solution is smaller than some threshold number defined by the hysteresis nonlinearity it is shown that the perturbed system has a continuum of periodic solutions with a rather simple structure in a vicinity of z the main result is a theorem on the stability of this continuum

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

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

architectures for exponentiation over gd 2 sup n adopted for smartcard application two exponentiation circuits are proposed using the fact that squaring is a linear operation over gf 2 sup n a time space tradeoff in smartcard based circuitry is presented it is shown how multiplication is performed by a single shift based on replacing the public key alpha sup a in gf 2 sup n by its minimal polynomial other considerations related to structure regularity and the possible use of dynamic shift registers are also treated

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

comb architectures for finite field multiplication in f 2 m two high speed bit serial word parallel or comb style finite field multipliers are proposed in this paper the first proposal utilizes a redundant representation for any binary field and the other uses a reordered normal basis for the binary field where a type ii optimal normal basis exists the proposed redundant representation architecture has a smaller critical path delay compared to the previous methods while the complexities remain about the same the proposed reordered normal basis multiplier has a significantly smaller critical path delay compared to the previous methods using the same basis or normal basis field programmable gate array fpga implementation results of the proposed multipliers are compared to those of the previous methods using the same basis which confirms that the proposed multipliers allow a much higher clock rate

stability of large cycles in a nonsmooth problem with hopf bifurcation at infinity an autonomous system which have a principal linear part at infinity and a continuous nonlinear term of sublinear growth is investigated results for systems with smooth nonlinearities are presented where accurate asymptotics of periodic cycles were introduced and cycle stability analyzed by the usual linearization of the system in a neighborhood of the cycle a new method to study both existence and stability of large cycles for systems with nonsmooth nonlinearities asymptotically homogeneous at infinity is developed the method combines the parameter functionalization technique and methods of monotone concave operators

fault tolerant decoders for cyclic error correcting codes high speed cyclic code decoders which are central to modern communication systems when implemented in dense very large scale integration vlsi are susceptible to pernicious momentary internal soft fails presenting a demanding error control challenge however special structures inherent in such decoders offer new methods for incorporating distributed error control throughout their designs the underlying design principles and motivations are emphasized providing a variety of options to meet various requirements bose chaudhuri hocquenghem bch codes are used to exemplify the new techniques as applied to the usual three standard subsystems present in a decoder the first and last parts syndrome calculations and transform inversion both involve finite field transforms suggesting the effective application of fast transform algorithms error control features are based upon the chord properties of the transform coefficients including even fast algorithms the third subsystem the berlekamp massey algorithm can be protected through a chord recursion property affiliated with the error location connection polynomial the propagation and spread of internal errors are studied and a special sink register compares several quantities available in close proximity to their easily recomputed counterparts most of these results are applicable to generalized to decoders for codes over higher ordered alphabets such as reed solomon codes

a vlsi design for computing exponentiations in gf 2 sup m and its application to generate pseudorandom number sequences a vlsi design for computing exponentiation in finite fields is developed an algorithm to generate a relatively long pseudorandom number sequence is presented it is shown that the period of this sequence is significantly increased compared to that of the sequence generated by the most commonly used maximal length shift register scheme

a new finite field multiplier using redundant representation this article presents simple and highly regular architectures for finite field multipliers using a redundant representation the basic idea is to embed a finite field into a cyclotomic ring which is based on the elegant multiplicative structure of a cyclic group one important feature of our architectures is that they provide area time trade offs which enable us to implement the multipliers in a partial parallel hybrid fashion this hybrid architecture has great significance in its vlsi implementation in very large fields the squaring operation using the redundant representation is simply a permutation of the coordinates it is shown that when there is an optimal normal basis the proposed bit serial and hybrid multiplier architectures have very low space complexity constant multiplication is also considered and is shown to have an advantage in using the redundant representation

abstract lp estimates for the cauchy problem with applications to the navier stokes equations in exterior domains abstract we apply an abstract perturbation theorem to derive global in time lq estimates for the cauchy problem and lq ls estimates for the nonstationary stokes equations in exterior domains this will be applied to obtain various new global in time estimates for weak solutions of the navier stokes equations

on computing multiplicative inverses in gf 2 sup m the design of a modular standard basis inversion for galois fields gf 2 sup m based on euclid s algorithm for computing the greatest common divisor of two polynomials is presented the asymptotic complexity is linear with m both in computation time and area requirement thus resulting in an at complexity of o m sup 2 this is a significant improvement over the best previous proposal which achieves at complexity of only o m sup 3

fast inversion in schmi gf 2 m with normal basis using hybrid double multipliers fast inversion in finite fields is crucial for high performance cryptography and codes we present techniques to exploit the recently proposed hybrid double multipliers for fast inversions in binary fields gf 2m with normal bases a hybrid double multiplier computes a double multiplication the product of three elements in gf 2m with a latency comparable to the latency of single multiplication of two elements traditional approaches such as itoh tsujii cannot utilize hybrid double multipliers we devise a new inversion algorithm based on ternary representations that exploits their potential the algorithm reduces the latency of inversion significantly for the fields recommended by nist if hybrid double multipliers are employed for example the algorithm computes an inversion in gf 2163 with only five double multiplications whereas the itoh tsujii algorithm requires nine single or double multiplications we propose a new inverter architecture using this new algorithm and a hybrid double multiplier we show that it is faster than the existing techniques by providing asic synthesis results using 65 nm cmos technology for example our inverter for gf 2163 achieves about 34 percent shorter computation time than an inverter using the itoh tsujii algorithm and a single multiplier

efficient multiplier architectures for galois fields gf 2 sup 4n this contribution introduces a new class of multipliers for finite fields gf 2 sup n sup 4 the architecture is based on a modified version of the karatsuba ofman algorithm koa by determining optimized field polynomials of degree four the last stage of the koa and the module reduction can be combined this saves computation and area in vlsi implementations the new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with koa based architectures with separate module reduction the new multipliers lead to highly modular architectures and are thus well suited for vlsi implementations three types of field polynomials are introduced and conditions for their existence are established for the small fields where n 2 3 8 which are of primary technical interest optimized field polynomials were determined by an exhaustive search for each field order exact space and time complexities are provided

l q l r estimates for solutions of the nonstationary stokes equations in an exterior domain and the navier stokes initial value problems inl q spaces

new systolic arrays for c ab sup 2 inversion and division in gf 2 sup m in this paper we present a new parallel in parallel out systolic array with unidirectional data flow for performing the power sum operation c ab sup 2 in finite fields gf 2 sup m the architecture employs the standard basis representation and can provide the maximum throughput in the sense of producing new results at a rate of one per clock cycle it is highly regular modular and thus well suited to vlsi implementation as compared to a previous systolic power sum circuit with bidirectional data flow and the same throughput performance the proposed one has smaller latency consumes less chip area and can more easily incorporate fault tolerant design based on the new power sum circuit we also propose a parallel in parallel out systolic array with the maximum throughput for computing inverses divisions in gf 2 sup m the proposed systolic divider gains advantages over an existing system with the same throughput performance in terms of chip area latency and fault tolerance

a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases abstract this paper proposes a fast algorithm for computing multiplicative inverses in gf 2 m using normal bases normal bases have the following useful property in the case that an element x in gf 2 m is represented by normal bases 2 k power operation of an element x in gf 2 m can be carried out by k times cyclic shift of its vector representation c c wang et al proposed an algorithm for computing multiplicative inverses using normal bases which requires m 2 multiplications in gf 2 m and m 1 cyclic shifts the fast algorithm proposed in this paper also uses normal bases and computes multiplicative inverses iterating multiplications in gf 2 m it requires at most 2 log 2 m 1 multiplications in gf 2 m and m 1 cyclic shifts which are much less than those required in the wang s method the same idea of the proposed fast algorithm is applicable to the general power operation in gf 2 m and the computation of multiplicative inverses in gf q m q 2 n

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

provably sublinear point multiplication on koblitz curves and its hardware implementation we describe algorithms for point multiplication on koblitz curves using multiple base expansions of the form k sigmaplusmntaua tau 1 b and k sigmaplusmntaua tau mu b tau2 mutau 1 c we prove that the number of terms in the second type is sublinear in the bit length of k which leads to the first provably sublinear point multiplication algorithm on koblitz curves for the first type we conjecture that the number of terms is sublinear and provide numerical evidence demonstrating that the number of terms is significantly less than that of tau adic nonadjacent form expansions we present details of an innovative fpga implementation of our algorithm and performance data demonstrating the efficiency of our method we also show that implementations with very low computation latency are possible with the proposed method because parallel processing can be exploited efficiently

liouville theorems for the stokes equations with applications to large time estimates abstract we study liouville theorems for the non stationary stokes equations in exterior domains in r n under decay conditions for spatial variables as applications we prove that the stokes semigroup is a bounded analytic semigroup on l of angle 2 for n 3 we also prove large time estimates for n 2 with zero net force

pointwise upper bounds for the solution of the stokes equation on lσ ω and applications abstract consider the stokes semigroup t defined on l where r n n 3 denotes an exterior domain with smooth boundary it is shown that t z u 0 for u 0 l and z with 0 2 satisfies pointwise estimates similar to the ones known for g z u 0 where g denotes the gaussian semigroup on r n in particular t extends to a bounded analytic semigroup on l of angle 2 moreover t t allows l c 2 smoothing for every t 0 and the stokes semigroups t p and t q on l p and l q are consistent for all p q 1