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

exact energy and momentum conserving algorithms for general models in nonlinear elasticity implicit time integration schemes that inherit the conservation laws of total energy linear and angular momentum are considered for initial boundary value problems in finite deformation elastodynamics conserving schemes are constructed for general hyperelastic material models both compressible and incompressible and are formulated in a way that is independent of spatial discretization three numerical examples for ogden type material models implemented using a finite element discretization in space are given to illustrate the performance of the proposed schemes these examples show that relative to the standard implicit mid point rule the conserving schemes exhibit superior numerical stability properties without a compromise in accuracy

numerical solution of a nonlinear wave equation in polar coordinates we study a finite element scheme and a difference scheme for the radial solutions of a nonlinear klein gordon equation

high order schemes for conservative or dissipative systems we propose a new method for designing high order finite difference schemes that inherit conservation or dissipation properties from conservative or dissipative systems such as hamiltonian systems with without damping terms the proposed method has a feature that the computational costs of the resulting schemes do not increase in practice even when the order of accuracy is increased

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

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

fast arithmetic for public key algorithms in galois fields with composite exponents the article describes a novel class of arithmetic architectures for galois fields gf 2 sup k the main applications of the architecture are public key systems which are based on the discrete logarithm problem for elliptic curves the architectures use a representation of the field gf 2 sup k as gf 2 sup n sup m where k n spl middot m the approach explores bit parallel arithmetic in the subfield gf 2 sup n and serial processing for the extension field arithmetic this mixed parallel serial hybrid approach can lead to fast implementations as the core module a hybrid multiplier is introduced and several optimizations are discussed we provide two different approaches to squaring we develop exact expressions for the complexity of parallel squarers in composite fields which can have a surprisingly low complexity the hybrid architectures are capable of exploring the time space trade off paradigm in a flexible manner in particular the number of clock cycles for one field multiplication which is the atomic operation in most public key schemes can be reduced by a factor of n compared to other known realizations the acceleration is achieved at the cost of an increased computational complexity we describe a proof of concept implementation of an asic for multiplication and squaring in gf 2 sup n sup m m variable

finite dimensional comodules over the hopf algebra of rooted trees the aim of this paper is an algebraic study of the hopf algebra h r of rooted trees which was introduced in cite kreimer1 connes broadhurst kreimer2 we first construct comodules over h r from finite families of primitive elements furthermore we classify these comodules by restricting the possible families of primitive elements and taking the quotient by the action of certain groups in the next section we give a formula about primitive elements of the subalgebra of ladders and construct a projection operator on the space of primitive elements it allows us to obtain a basis of the primitive elements by an inductive process which answers one of the questions of cite kreimer in the last sections we classify the hopf algebra endomorphisms and the coalgebras endomorphisms using the graded hopf algebra gr h r associated to the filtration by deg p of cite kreimer we then prove that h r is isomorphic to gr h r and deduce a decomposition of the group of the hopf algebra automorphisms of h r as a semi direct product

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

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

classification of systems of dyson schwinger equations in the hopf algebra of decorated rooted trees abstract we consider systems of combinatorial dyson schwinger equations briefly sdse x 1 b 1 f 1 x 1 x n x n b n f n x 1 x n in the connes kreimer hopf algebra h i of rooted trees decorated by i 1 n where b i is the operator of grafting on a root decorated by i and f 1 f n are non constant formal series the unique solution x x 1 x n of this equation generates a graded subalgebra h s of h i we characterise here all the families of formal series f 1 f n such that h s is a hopf subalgebra more precisely we define three operations on sdse change of variables dilatation and extension and give two families of sdse cyclic and fundamental systems and prove that any sdse s such that h s is hopf is the concatenation of several fundamental or cyclic systems after the application of a change of variables a dilatation and iterated extensions

finite difference solutions of a non linear schrodinger equation abstract this paper describes a finite difference method to approximate a schrodinger equation with a power non linearity a special case of this equation is currently used to model the propagation of a laser beam in a plasma the main feature of the method we present is that it satisfies a discrete analogue of some important conservation laws of the equations we present numerical results which show in particular the propagation and the formation of solitions in the one dimensional case

conservative numerical methods for x f x abstract two distinctly different numerical methods are developed for solving the conservative initial value problem x f x x o dot x o both methods conserve exactly the same total energy as does the given differential equation computer examples are described and

on overlapping divergences using set theoretic considerations we show that the forest formula for overlapping divergences comes from the hopf algebra of rooted trees

hopf algebras of rooted forests cocyles and free rota baxter algebras the hopf algebra and the rota baxter algebra are the two algebraic structures underlying the algebraic approach of connes and kreimer to renormalization of perturbative quantum field theory in particular the hopf algebra of rooted trees serves as the baby model of feynman graphs in their approach and can be characterized by certain universal properties involving a hochschild 1 cocycle decorated rooted trees have also been applied to study feynman graphs we will continue the study of universal properties of various spaces of decorated rooted trees with such a 1 cocycle leading to the concept of a cocycle hopf algebra we further apply the universal properties to equip a free rota baxter algebra with the structure of a cocycle hopf algebra

structure of the loday ronco hopf algebra of trees abstract loday and ronco defined an interesting hopf algebra structure on the linear span of the set of planar binary trees they showed that the inclusion of the hopf algebra of non commutative symmetric functions in the malvenuto reutenauer hopf algebra of permutations factors through their hopf algebra of trees and these maps correspond to natural maps from the weak order on the symmetric group to the tamari order on planar binary trees to the boolean algebra we further study the structure of this hopf algebra of trees using a new basis for it we describe the product coproduct and antipode in terms of this basis and use these results to elucidate its hopf algebraic structure in the dual basis for the graded dual hopf algebra our formula for the coproduct gives an explicit isomorphism with a free associative algebra we also obtain a transparent proof of its isomorphism with the non commutative connes kreimer hopf algebra of foissy and show that this algebra is related to non commutative symmetric functions as the commutative connes kreimer hopf algebra is related to symmetric functions

combinatorics of rooted trees and hopf algebras we begin by considering the graded vector space with a basis consisting of rooted trees with grading given by the count of non root vertices we define two linear operators on this vector space the growth and pruning operators which respectively raise and lower grading their commutator is the operator that multiplies a rooted tree by its number of vertices and each operator naturally associates a multiplicity to each pair of rooted trees by using symmetry groups of trees we define an inner product with respect to which the growth and pruning operators are adjoint and obtain several results about the associated multiplicities now the symmetric algebra on the vector space of rooted trees after a degree shift can be endowed with a coproduct to make a hopf algebra this was defined by kreimer in connection with renormalization we extend the growth and pruning operators as well as the inner product mentioned above to kreimer s hopf algebra on the other hand the vector space of rooted trees itself can be given a noncommutative multiplication with an appropriate coproduct this leads to the hopf algebra of grossman and larson we show that the inner product on rooted trees leads to an isomorphism of the grossman larson hopf algebra with the graded dual of kreimer s hopf algebra correcting an earlier result of panaite

a stable finite element method for initial boundary value problems for first order hyperbolic systems a nonstandard finite element method for initial boundary value problems for first order hyperbolic systems in one space dimension with general boundary conditions is analyzed the method can be considered as a generalization of the box scheme we first establish a stability result for the method and then derive several error estimates u x t a x ux x t b x u x t f x t 1 1 o 0 t g kxu l t u x 0 u0 x for x t e i x j 0 1 x 0 where t 0 here aix and 5 x are n x n matrices we assume that aix has n real eigenvalues xx x a2 x a x and that there exist a constant t0 0 and an integer m 0 t0 for all xg furthermore we assume that for each x e there is a nonsingular matrix qix such that

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

constructing composite field representations for efficient conversion we describe a method of construction of a composite field representation from a given binary field representation we derive the conversion change of basis matrix the special case of when the degree of the ground field is relatively prime to the extension degree where the irreducible polynomial generating the composite field has its coefficients from the binary prime field rather than the ground field is also treated furthermore certain generalizations of the proposed construction method e g the use of nonprimitive elements and the construction of composite fields with special irreducible polynomials are also discussed finally we give storage efficient conversion algorithms between the binary and composite fields when the degree of the ground field is relatively prime to the extension degree

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 new solution procedure for application of energy conserving algorithms to general constitutive models in nonlinear elastodynamics abstract this paper develops a novel solution strategy for temporal discretization of initial value problems in nonlinear elastodynamics enabling exact energy and momentum conservation for a spatially discretized problem this new energy momentum algorithm fits into the general framework established by simo and tarnow z angew math phys 43 1992 757 however in contrast to the presentation given in that reference the current approach extends the critical stress update formula to encompass generic stored energy functions for the hyperelastic continuum in a manner which also allows for quadratically convergent newton raphson solution of the nonlinear algorithmic equations numerical examples using a neo hookean material model are given to illustrate the excellent performance of the proposed scheme not only is the new algorithm guaranteed to be second order accurate and unconditionally stable in the fully nonlinear regime as are other energy momentum strategies but also the approach advocated is demonstrated to be more robust and versatile than its predecessors

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

a survey on normal bases over finite fields normal bases are important in efficient implementation of finite field arithmetic for many applications in coding theory and cryptography this paper gives a survey on various topics of normal bases including basic properties of normal elements primitive normal bases dual and self dual normal bases fast arithmetic under normal bases and constructions for low complexity normal bases

the hopf algebra approach to feynman diagram calculations the hopf algebra structure underlying feynman diagrams which governs the process of renormalization in perturbative quantum field theory is reviewed recent progress is briefly summarized with an emphasis on further directions of research

conservation properties of a time fe method part iv higher order energy and momentum conserving schemes in the present paper a systematic development of higher order accurate time stepping schemes which exactly conserve total energy as well as momentum maps of underlying finite dimensional hamiltonian systems with symmetry is shown the result of this development is the enhanced galerkin eg finite element method in time the conservation of the eg method is generally related to its collocation property total energy conservation in particular is obtained by a new projection technique the eg method is moreover based on objective time discretization of the used strain measure this paper is concerned with particle dynamics and semi discrete non linear elastodynamics the related numerical examples show good performance in presence of stiffness as well as for calculating large strain motions copyright 2005 john wiley sons ltd

on the enforcing energy conservation of time finite elements for discrete elasto dynamics problems in the context of the time finite element method algorithmic stresses which enable the conservation of energy are designed for temporal integrators derived from the midpoint and trapezoidal schemes this is achieved through an appropriate modification of the standard midpoint and trapezoidal quadrature rules used for the numerical integration of time integrals either scalar scaling or vectorial adjustments can be employed for the modification and well designed simple tests allow to investigate the quality of these different strategies of energy conserving enforcements numerical examples with semi discrete elasto dynamics problems are presented to show the superior stability of energy conserving schemes copyright 2006 john wiley sons ltd

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

low complexity word level sequential normal basis multipliers for efficient hardware implementation of finite field arithmetic units the use of a normal basis is advantageous in this paper two classes of architectures for multipliers over the finite field gf 2 sup m are proposed these multipliers are of sequential type i e after receiving the coordinates of the two input field elements they go through k 1 spl les k spl les m iterations i e clock cycles to finally yield all the coordinates of the product in parallel the value of k depends on the word size w spl lceil m k spl rceil for w 1 these multipliers are highly area efficient and require fewer number of logic gates even when compared with the most area efficient multipliers available in the open literature this makes the proposed multipliers suitable for applications where the value of m is large but space is of concern e g resource constrained cryptographic systems additionally if the field dimension m is composite i e m kn then the extension of one class of the architectures yields a highly efficient multiplier over composite fields

ramification of rough paths the stack of iterated integrals of a path is embedded in a larger algebraic structure where iterated integrals are indexed by decorated rooted trees and where an extended chen s multiplicative property involves the durr connes kreimer coproduct on rooted trees this turns out to be the natural setting for a non geometric theory of rough paths

continuous finite element methods which preserve energy properties for nonlinear problems abstract continuous finite elements are used to develop numerical methods which preserve energy properties for nonlinear ordinary and partial differential equations schemes are constructed that give high orders of accuracy time discretization algorithms for the cahn hilliard klein gordon korteweg de vries and schrodinger equations are presented comparisons are made with existing schemes

wg a family of stream ciphers with designed randomness properties in this paper we present a family of stream ciphers which generate a keystream with ideal two level autocorrelation the design also guarantees other randomness properties i e balance long period ideal tuple distribution and high and exact linear complexity we discuss how these properties are achieved by the proposed design and show how to select various parameters to obtain an efficient stream cipher for the desired security level we also show that the proposed generators are secure against time memory data tradeoff attacks algebraic attacks and correlation attacks finally we present wg 128 as a concrete example of a wg stream cipher with a key size of 128bits

mulitgraded dyson schwinger systems we study systems of combinatorial dyson schwinger equations with an arbitrary number n of coupling constants the considered hopf algebra of feynman graphs is mathbb n n graded and we wonder if the graded subalgebra generated by the solution is hopf or not we first introduce a family of pre lie algebras which we classify dually providing systems generating a hopf subalgebra we also describe the associated groups as extensions of groups of formal diffeomorphisms on several variables we then consider systems coming from feynman graphs of a quantum field theory we show that if the number n of independent coupling constants is the number of interactions of the considered qft then the generated subalgebra is hopf for qed varphi 3 and qcd we also prove that this is the minimal value of n all these examples are generalizations of the first family of dyson schwinger systems in the one coupling constant case called fundamental we also give a generalization of the second family called cyclic

hopf algebras cyclic cohomology and the transverse index theorem in this paper we solve a longstanding internal problem of noncommutative geometry namely the computation of the index of transversally elliptic operators on foliations we show that the computation of the local index formula for transversally hypoelliptic operators can be settled thanks to a very specific hopf algebra associated to each integer codimension this hopf algebra reduces transverse geometry to a universal geometry of affine nature the structure of this hopf algebra its relation with the lie algebra of formal vector fields as well as the computation of its cyclic cohomology are done in the present paper in which we also show that under a suitable unimodularity condition the cosimplicial space underlying the hochschild cohomology of a hopf algebra carries a highly nontrivial cyclic structure

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

faa di bruno subalgebras of the hopf algebra of planar trees from combinatorial dyson schwinger equations we consider the combinatorial dyson schwinger equation x b p x in the non commutative connes kreimer hopf algebra of planar rooted trees hnck where b is the operator of grafting on a root and p a formal series the unique solution x of this equation generates a graded subalgebra an p of hnck we describe all the formal series p such that an p is a hopf subalgebra we obtain in this way a 2 parameters family of hopf subalgebras of hnck organized into three isomorphism classes a first one restricted to a polynomial ring in one variable a second one restricted to the hopf subalgebra of ladders isomorphic to the hopf algebra of quasi symmetric functions a last infinite one which gives a non commutative version of the faa di bruno hopf algebra by taking the quotient the last class gives an infinite set of embeddings of the faa di bruno algebra into the connes kreimer hopf algebra of rooted trees moreover we give an embedding of the free faa di bruno hopf algebra on d variables into a hopf algebra of decorated rooted trees together with a non commutative version of this embedding

galerkin based mechanical integrators for finite elastodynamics formulated in principal stretches pitfalls and remedies in the present paper we deal with integrators relying on finite elements in time for general hyperelastic models which guarantee the conservation of the momentum maps as well as of the total energy within the calculation accuracy in the isotropic case eigenvalue based constitutive laws are on the one hand very significant to model the behaviour of various materials on the other hand it will be shown that the efficient algorithmic treatment of principal stretches based on a perturbation approach requires especially within the context of conserving schemes advanced techniques to circumvent numerical problems accordingly we propose an adequate solution strategy to avoid potential pitfalls which are related to numerical artefacts of the applied non standard quadrature rule finally the excellent numerical and mechanical performance of the resulting galerkin based time stepping schemes will be demonstrated by means of representative numerical examples

efficient montgomery based semi systolic multiplier for even type gnb of gf 2 m efficient finite field multiplication is crucial for implementing public key cryptosystem to achieve this multipliers using gaussian normal basis have been widely explored in previous works in this paper based on proposed gaussian normal basis montgomery gnbm representation a semi systolic even type gnbm multiplier is developed analysis shows that the proposed architecture saves about 57 percent space complexity and 50 percent time complexity when compared with the only existing semi systolic even type gnb multiplier moreover due to properties of regularity and modularity the proposed multiplier is very suitable for vlsi implementation

an efficient algorithm for computing the baker campbell hausdorff series and some of its applications we provide a new algorithm for generating the baker campbell hausdorff bch series z log exey in an arbitrary generalized hall basis of the free lie algebra l x y generated by x and y it is based on the close relationship of l x y with a lie algebraic structure of labeled rooted trees with this algorithm the computation of the bch series up to degree of 20 111 013 independent elements in l x y takes less than 15min on a personal computer and requires 1 5gbytes of memory we also address the issue of the convergence of the series providing an optimal convergence domain when x and y are real or complex matrices

feynman graphs rooted trees and ringel hall algebras we construct symmetric monoidal categories mathcal lrf mathcal lfg of rooted forests and feynman graphs these categories closely resemble finitary abelian categories and in particular the notion of ringel hall algebra applies the ringel hall hopf algebras of mathcal lrf mathcal lfg bf h mathcal lrf bf h mathcal lfg are dual to the corresponding connes kreimer hopf algebras on rooted trees and feynman diagrams we thus obtain an interpretation of the connes kreimer lie algebras on rooted trees and feynman graphs as ringel hall lie algebras

etude de l algebre de lie double des arbres enracines decores the double lie algebra ld of rooted trees decorated by a set d is introduced generalising the construction of connes and kreimer it is shown that it is a simple lie algebra its derivations and its automorphisms are described as well as some central extensions finally the category of lowest weight modules is introduced and studied

conservation properties of a time fe method part ii time stepping schemes for non linear elastodynamics in the present paper one step implicit integration algorithms for non linear elastodynamics are developed the discretization process rests on galerkin methods in space and time in particular the continuous galerkin method applied to the hamiltonian formulation of semidiscrete non linear elastodynamics lies at the heart of the time stepping schemes algorithmic conservation of energy and angular momentum are shown to be closely related to quadrature formulas that are required for the calculation of time integrals we newly introduce the assumed strain method in time which enables the design of energy momentum conserving schemes and which can be interpreted as temporal counterpart of the well established assumed strain method for finite elements in space the numerical examples deal with quasi rigid motion as well as large strain motion copyright 2001 john wiley sons ltd

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

hopf algebras renormalization and noncommutative geometry we explore the relation between the hopf algebra associated to the renormalization of qft and the hopf algebra associated to the ncg computations of tranverse index theory for foliations

hopf algebraic structure of families of trees in this paper we describe hopf algebras which are associated with certain families of trees these hopf algebras originally arose in a natural fashion one of the authors 5 was investigating data structures based on trees which could be used to efficiently compute certain differential operators given data structures such as trees which can be multiplied and which act as higherorder derivations on an algebra one expects to find a hopf algebra of some sort we were pleased to find that not only was there a hopf algebra associated with these data structures but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees previous work applying hopf algebras to combinatorial objects such as 10 13 or 14 has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets rather than on trees themselves

a bijection between rooted trees and fermionic fock states grafting and growth operators in fock space and fermionic operators for rooted trees we show that fermionic fock states in the occupation number representation can be indexed uniquely by rooted trees our main ingredients in this construction are the matula numbers the fundamental theorem of arithmetic and a relabeling of fermionic quantum states by natural numbers as a byproduct of the correspondence mentioned above we realize the grafting and the growth operators comprising important constructions in the context of hopf algebras in the fermionic fock space also we show how to construct fermionic creation and annihilation operators in the context of rooted trees new representations of the solutions of combinatorial dyson schwinger equations and of the antipode in the connes kreimer hopf algebra of rooted trees related to the occupation number picture are presented

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

new hardware implementationsof wg bf 29 11 and wg bf 16 streamciphers using polynomial basis the wg stream ciphers are based on the wg welch gong transformation and possess proved randomness properties in this paper we propose nine new hardware designs for the two classes of wg 29 11 and wg 16 for each class we design and implement three versions of standard pipelined and serial for the first time we use the polynomial basis pb representation to design and implement the wg 29 11 and wg 16 we consider traditional pb multiplier for the wg 29 11 and the traditional and karatsuba multipliers for the wg 16 for efficient field operations we propose an irreducible trinomial for the wg 29 11 for the wg 16 a new formulation of its permutation which requires only 8 multipliers is introduced in these designs the multipliers in the transforms arefurther reduced by utilizing a novel computation for the trace of the multiplication of two field elements we have implemented theproposed designs in asic using cmos 65 nm technology the results show that the proposed standard wg 29 11 consumes less area and slightly enhances the normalized throughput compared to the existing counterparts for the wg 16 throughput of theproposed pipelined instance outperforms the previous designs moreover the speed of the proposed wg 16 designs meet the peak bit rates for the 4 g specifications

general dyson schwinger equations and systems we classify combinatorial dyson schwinger equations giving a hopf subalgebra of the hopf algebra of feynman graphs of the considered quantum field theory we first treat single equations with an arbitrary eventually infinite number of insertion operators we distinguish two cases in the first one the hopf subalgebra generated by the solution is isomorphic to the faa di bruno hopf algebra or to the hopf algebra of symmetric functions in the second case we obtain the dual of the enveloping algebra of a particular associative algebra seen as a lie algebra we also treat systems with an arbitrary finite number of equations with an arbitrary number of insertion operators with at least one of degree 1 in each equation

lie algebras associated to systems of dyson schwinger equations we consider systems of combinatorial dyson schwinger equations in the connes kreimer hopf algebra hi of rooted trees decorated by a set i let h s be the subalgebra of hi generated by the homogeneous components of the unique solution of this system if it is a hopf subalgebra we describe it as the dual of the enveloping algebra of a lie algebra g s of one of the following types 1 g s is an associative algebra of paths associated to a certain oriented graph 2 or g s is an iterated extension of the faa di bruno lie algebra 3 or g s is an iterated extension of an infinite dimensional abelian lie algebra we also describe the character groups of h s