constructing solutions for the prescribed scalar curvature problem via local pohozaev identities abstract this paper deals with the following prescribed scalar curvature problem u q y y u n 2 n 2 u 0 y y y r 2 r n 2 where q y is nonnegative and bounded by combining a finite reduction argument and local pohozaev type of identities we prove that if n 5 and q r y has a stable critical point r 0 y 0 with r 0 0 and q r 0 y 0 0 then the above problem has infinitely many solutions whose energy can be made arbitrarily large here instead of estimating directly the derivatives of the reduced functional we apply some local pohozaev identities to locate the concentration points of the bump solutions moreover the concentration points of the bump solutions include a saddle point of q y

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

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 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

low complexity multiplier architectures for single and hybrid double multiplications in gaussian normal bases the extensive rise in the number of resource constrained wireless devices and the needs for secure communications with the servers imply fast and efficient cryptographic computations for both parties efficient hardware implementation of arithmetic operations over finite field using gaussian normal basis is attractive for public key cryptography as it provides free squarings in this paper we first present two low complexity digit level multiplier architectures it is shown that the proposed multipliers outperform the existing gaussian normal basis gnb multiplier structures available in the literature then for the first time using these two architectures we propose a new digit level hybrid multiplier which performs two successive multiplications with the same latency as the one for one multiplication we have studied the efficiency of the proposed hybrid architecture in terms of area and time delay for different digit sizes the main advantage of this new hybrid architecture is to speed up exponentiation and point multiplication whenever double multiplication is required and the traditional schemes fail due to the data dependencies we have investigated the applicability of the proposed hybrid structure to reduce the latency of exponentiation based cryptosystems our analysis and timing results show that the expected acceleration in double exponentiation is considerable prototypes of the presented low complexity multiplier architectures and the proposed hybrid architecture are implemented and experimental results are presented

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

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 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

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

non compactness and multiplicity results for the yamabe problem on sn the aim of this paper is to show the existence of metrics ge on sn where ge is a perturbation of the standard metric g0 for which the yamabe problem possesses a sequence of solutions unbounded in l sn the metrics ge that we find are of class ck on sn with k n 34 we also prove some new multiplicity results

doubly stochastic quadratic operators and birkhoff s problem abstract in the present we introduce a concept of doubly stochastic quadratic operator we study necessary and sufficient conditions for doubly stochasticity of operator moreover we study analogue of birkhoff s theorem for the class of doubly stochastic operators

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

prescribing scalar curvature on s3 s4 and related problems abstract we show that for the prescribing scalar curvature problem on sn n 3 4 we can perturb in an explicit way any given positive continuous function in any neighborhood of any given point on sn such that for the perturbed function there exist many solutions the related critical exponent equations u k x u n 2 n 2 in rn n 3 4 with k x being asymptotically periodic in one of the variables are also studied and infinitely many positive solutions modulo translations by its periods are obtained under some additional mild hypotheses on k x some a priori estimates to solutions of the prescribing scalar curvature problem on s3 are as given

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

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

g decompositions of matrices and related problems i in the present paper we introduce a notion of g decompositions of matrices main result of the paper is that a symmetric matrix am has a g decomposition in the class of stochastic resp substochastic matrices if and only if am belongs to the set um resp um to prove the main result we study extremal points and geometrical structures of the sets um um note that such kind of investigations enables to study birkhoff s problem for quadratic g doubly stochastic operators

birkhoff s theorem and multidimensional numerical range abstract we show that under certain conditions birkhoff s theorem on doubly stochastic matrices remains valid for countable families of discrete probability spaces which have nonempty intersections using this result we study the relation between the spectrum of a self adjoint operator a and its multidimensional numerical range it turns out that the multidimensional numerical range is a convex set whose extreme points are sequences of eigenvalues of the operator a every collection of eigenvalues which can be obtained by the rayleigh ritz formula generates an extreme point of the multidimensional numerical range however it may also have other extreme points

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 perturbation method for spinorial yamabe type equations on s m and its application for m ge 2 we prove the existence of non trivial solutions for a certain kind of nonlinear dirac equations with critical sobolev nonlinearities on s m via a perturbative variational method for the special case m 2 this establishes the existence of a conformal immersion s 2 rightarrow mathbb r 3 with prescribed mean curvature h which is close to a positive constant under an index counting condition on h

a generalization of addition chains and fast inversions in binary fields in this paper we study a generalization of addition chains where k previous values are summed together on each step instead of only two values as in traditional addition chains such chains are called k chains and we show that they have applications in finding efficient parallelizations in problems that are known to be difficult to parallelize in particular 3 chains improve computations of inversions in finite fields using hybrid double multipliers recently it was shown that this operation can be efficiently computed using a ternary algorithm but we show that 3 chains provide a significantly more efficient solution

solutions for conformally invariant fractional laplacian equations with multi bumps centered in lattices abstract in this paper we consider the following nonlinear elliptic equation involving the fractional laplacian with critical exponent s u k x u n 2 s n 2 s u 0 in r n where s 0 1 and n 2 2 s k 0 is periodic in x 1 x k with 1 k n 2 s 2 under some natural conditions on k near a critical point we prove the existence of multi bump solutions where the centers of bumps can be placed in some lattices in r k including infinite lattices on the other hand to obtain positive solution with infinite bumps such that the bumps locate in lattices in r k the restriction on 1 k n 2 s 2 is in some sense optimal since we can show that for k n 2 s 2 no such solutions exist

computing special powers in finite fields we study exponentiation in nonprime finite fields with very special exponents such as they occur for example in inversion primitivity tests and polynomial factorization our algorithmic approach improves the corresponding exponentiation problem from about quadratic to about linear time

non degeneracy of multi bubbling solutions for the prescribed scalar curvature equations and applications abstract we consider the following prescribed scalar curvature equations in r n 0 1 u k y u 2 1 u 0 in r n u d 1 2 r n where k r is a positive function 2 2 n n 2 we first prove a non degeneracy result for the positive multi bubbling solutions constructed in 26 by using the local pohozaev identities then we use this non degeneracy result to glue together bubbles with different concentration rate to obtain new solutions

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

a fast algorithm for multiplicative inversion in gf 2 sup m using normal basis a fast algorithm for multiplicative inversion in gf 2 sup m using normal basis is proposed it is an improvement on those proposed by itoh and tsujii and by chang et al which are based on fermat s theorem and require o logm multiplications the number of multiplications is reduced by decomposing m 1 into several factors and a small remainder

efficient generation of minimal length addition chains an addition chain for a positive integer n is a set 1 a0 a1 ar n of integers such that for each i ge 1 a i a j a k for some k le j i this paper is concerned with some of the computational aspects of generating minimal length addition chains for an integer n particular attention is paid to various pruning techniques that cut down the search time for such chains certain of these techniques are influenced by the multiplicative structure of n later sections of the paper present some results that have been uncovered by searching for minimal length addition chains

multi bump solutions and multi tower solutions for equations on rn let 0 be a small parameter in this paper we study existence of multiple multi bump positive solutions for the semilinear schrodinger equation u u 1 a x u p 2u u h1 rn where n 1 2 2 if n 1 or n 2 a c rn a x 0 for x rn and lim x a x 0 we also study existence of multiple multi tower positive solutions for the prescribed scalar curvature equation u 1 k x un 2n 2 u d1 2 rn where n 3 k c 0 k r 0 for r 0 limr 0k r 0 and limr k r 0