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

block recombination approach for subquadratic space complexity binary field multiplication based on toeplitz matrix vector product in this paper we present a new method for parallel binary finite field multiplication which results in subquadratic space complexity the method is based on decomposing the building blocks of the fan hasan subquadratic toeplitz matrix vector multiplier we reduce the space complexity of their architecture by recombining the building blocks in comparison to other similar schemes available in the literature our proposal presents a better space complexity while having the same time complexity we also show that block recombination can be used for efficient implementation of the ghash function of galois counter mode gcm

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

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

orthogonal polynomials and measures on the unit circle the geronimus transformations in this paper we analyze a perturbation of a nontrivial positive measure supported on the unit circle this perturbation is the inverse of the christoffel transformation and is called the geronimus transformation we study the corresponding sequences of monic orthogonal polynomials as well as the connection between the associated hessenberg matrices finally we show an example of this kind of transformation

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

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

new block recombination for subquadratic space complexity polynomial multiplication based on overlap free approach in this paper we present new parallel polynomial multiplication formulas which result in subquadratic space complexity the schemes are based on a recently proposed block recombination of polynomial multiplication formula the proposed two way three way and four way split polynomial multiplication formulas achieve the smallest space complexities moreover by providing area time tradeoff method the proposed formulas enable one to choose a parallel formula for polynomial multiplication which is suited for a design environment

on the arithmetic operations over finite fields of characteristic three with low complexity in this paper the hermite polynomial representation is adapted as a new way to represent certain finite fields of characteristic three we give the multiplication method to multiply two elements of f 3 n in the hermite polynomial representation with subquadratic computational complexity by using a divide and conquer idea we show that in some cases there is a set of irreducible binomials in the hermite polynomial representation to obtain modular reduction with a lower addition complexity than the standard polynomial representation we also investigate the matrix vector product method for the multiplication of the field elements represented by hermite polynomials

finite perturbations of orthogonal polynomials in this paper we study orthogonal polynomials pn which arise from a given system of orthogonal polynomials pn by a finite perturbation of the recurrence coefficients i e the recurrence coefficients n n 1 of pn respectively n n 1 of pn are related to each other by n l n m and n 1 l n 1 m for n n where l and m are fixed nonnegative integers closed expressions for the polynomial pn in terms of pn and its associated polynomial pn 1 1 as well as for the orthogonality measure of pn in terms of the orthogonality measure of pn are given in fact the results are presented for the most general case when is an arbitrary not necessarily positive measure

low space complexity multiplication over binary fields with dickson polynomial representation we study dickson bases for binary field representation such a representation seems interesting when no optimal normal basis exists for the field we express the product of two field elements as toeplitz or hankel matrix vector products this provides a parallel multiplier which is subquadratic in space and logarithmic in time using the matrix vector formulation of the field multiplication we also present sequential multiplier structures with linear space complexity

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

on the favard theorem and its extensions in this paper we present a survey on the favard theorem and its extensions c 2001 elsevier science b v all rights reserved

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

look up table based large finite field multiplication in memory constrained cryptosystems many cryptographic systems use multiplication in the finite field gf 2 sup n for their underlying computations in the recent past a number of look up table based algorithms have been proposed for the software implementation of gf 2 sup n multiplication look up table based algorithms can provide speed advantages but they either require a large memory space or do not fully utilize the resources of the processor on which the software is executed in this work an algorithm for gf 2 sup n multiplication is proposed which can alleviate this problem in each iteration of the proposed algorithm a group of bits of one of the input operands are examined and two look up tables are accessed the groupsize determines the table sizes but does not affect the utilization of the processor resources it can be used for both software and hardware realizations and is particularly suitable for implementations in memory constrained environment such as smart cards and embedded cryptosystems

efficient subquadratic space complexity binary polynomial multipliers based on block recombination some applications like cryptography involve a large number of multiplications of binary polynomial in this paper we consider two three and four way methods for parallel implementation of binary polynomial multiplication we propose optimized three and four way split formulas which reduce the space and time complexity of the best known methods moreover we present a block recombination method which provides some further reduction in the space complexity of the considered two three and four way split multipliers

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

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

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

a 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 orthogonal polynomials with perturbed recurrence relations abstract orthogonal polynomials may be fully characterized by the following recurrence relation pn x x n 1 pn 1 x n 1pn 2 x with p0 x 1 p1 x x 0 and n 0 here we study how the structure and the spectrum of these polynomials get modified by a local perturbation in the and parameters of a co recursive k k co dilated k k and co modified k k k k nature for an arbitrary but fixed kth element 1 k specifically stieltjes functions differential equations and distributions of zeros as well as representations of the new perturbed polynomials in terms of the old unperturbed ones are given this type of problems is strongly related to the boundary value problems of finite difference equations and to the quantum mechanical study of physical many body systems atoms molecules nuclei and solid state systems

vlsi architectures for soft decision decoding of reed solomon codes soft decision decoding of reed solomon codes delivers significant coding gains over classical minimum distance decoding in this paper we present architectures for polynomial interpolation and factorization the two main steps of the soft decoding algorithm we introduce an algorithmic transformation for reducing the iterations required in generating the interpolation polynomial and present efficient architectures by sharing computations we also describe algorithmic transformations for further reducing the interpolation and factorization latency an area efficient folded pipelined version of the interpolation architecture is also described finally we present an example of a reed solomon soft decoder utilizing the presented architectures having a 250 mbps throughput

spectral transformations for hermitian toeplitz matrices in this paper we deal with some perturbations of probability measures supported on the unit circle as well as in a more general framework with hermitian linear functionals we focus our attention in the hessenberg matrix associated with the multiplication operator in terms of an orthogonal basis in the linear space of polynomials with complex coefficients the lu and qr factorizations of such a matrix are introduced then the connection between the above mentioned perturbations and such factorizations is presented

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

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

rational spectral transformations and orthogonal polynomials abstract we show that generic rational transformations of the stieltjes function with polynomial coefficients st can be presented as a finite superposition of four fundamental elementary transforms christoffel transform ct geronimus transform gt and forward and backward associated transformations a t a t it is shown that the laguerre hahn polynomials lhp on arbitrary nonuniform lattice are covariant with respect to st i e st of a lhp yields another lhp whereas the semi classical polynomials are covariant with respect to a subclass of linear st some applications of these results to the theory of the semi classical polynomials are considered

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

on perturbed szegő recurrences abstract the purpose of the present contribution is to investigate the effects of finite modifications of verblunsky coefficients on szeg recurrences more precisely we study the structural relations and the corresponding c functions of the orthogonal polynomials with respect to these modifications from the initial ones

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

compensatability and optimal compensation under randomly varying distributed delays this paper establishes necessary and sufficient conditions for existence uniqueness and global optimality of the linear quadratic coupled delay compensator lqcdc which is designed to circumvent the detrimental effects of randomly varying delays from sensor to controller and from controller to actuator as well as the time skew caused by missynchronization of sensor and controller sampling instants these conditions are derived on the basis of the concepts of stabilizability detectability and compensatability in the mean square sense in the absence of random delays from sensor to controller and from controller to actuator it has been shown that lqcdc problems reduce to the classical linear quadratic gaussian lqg

characterization of orthogonal polynomials with respect to a functional abstract in this paper we study questions of existence uniqueness and characterization of polynomials orthogonal with respect to a linear not necessarily definite functional l defined on the set of laurent polynomials first we characterize with the help of the function f l z l y z y z polynomials orthogonal with respect to l using this characterization which has wide applications we are able to settle the question of existence and uniqueness of orthogonal polynomials the uniquely determined orthogonal polynomials will be called basic orthogonal polynomials it is to be pointed out that in contrast to the real case there are natural numbers n such that there exist no polynomials of degree n which are orthogonal with respect to l if l is indefinite and if we have orthogonality jumps greater than 1 furthermore the functional to which the basic orthogonal polynomials of the second kind are orthogonal is determined finally we get explicit expressions for all basic orthogonal polynomials with respect to a weight function the support of which consists of several arcs of the unit circle changes sign from arc to arc and has square root singularities at the boundary points of the arcs these polynomials can be considered as the basic polynomials in describing and generating orthogonal polynomials with periodic reflection coefficients

better circuits for binary polynomial multiplication we develop a new and simple way to describe karatsuba like algorithms for multiplication of polynomials over mathbb f 2 f 2 we restrict the search of small circuits to a class of circuits we call symmetric bilinear these are circuits in which and gates only compute functions of the form sum i in s a i cdot sum i in s b i quad quad s subseteq lbrace 0 ldots n 1 rbrace i s a i i s b i s 0 n 1 these techniques yield improved recurrences for m kn m k n the number of gates used in a circuit that multiplies two kn k n term polynomials for k 4 5 6 k 4 5 6 and 7 we built and verified the circuits for n n term binary polynomial multiplication for values of n n of practical interest circuits for n n up to 100 are posted at http cs www cs yale edu homes peralta circuitstuff binpolmult tar gz

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

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

a scalable dual field elliptic curve cryptographic processor we propose an elliptic curve ec cryptographic processor architecture that can support galois fields gf p and gf 2 sup n for arbitrary prime numbers and irreducible polynomials by introducing a dual field multiplier a montgomery multiplier with an optimized data bus and an on the fly redundant binary converter boost the throughput of the ec scalar multiplication all popular cryptographic functions such as dsa ec dsa rsa crt and prime generation are also supported all commands are organized in a hierarchical structure according to their complexity our processor has high scalability and flexibility between speed hardware area and operand size in the hardware evaluation using a 0 13 spl mu m cmos standard cell library the high speed design using 117 5 kgates with a 64 bit multiplier achieved operation times of 1 21 ms and 0 19 ms for a 160 bit ec scalar multiplication in gf p and gf 2 sup n respectively a compact version with an 8 bit multiplier requires only 28 3 k gates and executes the operations in 7 47 ms and 2 79 ms not only 160 bit operations but any bit length can be supported by any hardware configuration so long as the memory capacity is sufficient

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

comments on a new architecture for a parallel finite field multiplier with low complexity based on composite field in this comment we show that the gate complexity of karatsuba polynomial multipliers presented in the above paper can be reduced by exploring the common subexpressions

optimal computer control via communication channels with irregular transmission times we study a linear discrete time partially observed system perturbed by white noises the observations are transmitted to the controller via communication channels with random transmission times various measurement signals may incur independent delays be lost or corrupted under certain assumptions a recursive minimum variance state estimator is proposed a finite horizon linear quadratic optimal control problem is solved we also find a solution for the problem of minimizing the average cost per unit time over the infinite planning horizon

a karatsuba based algorithm for polynomial multiplication in chebyshev form in this paper we present a new method for multiplying polynomials in chebyshev form our approach has two steps first the well known karatsuba s algorithm is applied to polynomials constructed by using chebyshev coefficients then from the obtained result extra arithmetic operations are used to write the final result in chebyshev form the proposed algorithm has a quadratic computational complexity we also compare our method to other approaches

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

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

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

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

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

optimal design of networked control systems computer control via asynchronous communication channels we study a linear discrete time partially observed system perturbed by white noises the observations are transmitted to the controller and its outputs are sent to the actuators via communication channels that provide random transmission times and may lose messages various signals may incur independent delays and arrive at the destination point out of order under certain assumptions a recursive minimum variance state estimator is proposed a finite horizon linear quadratic optimal control problem is also solved

an architecture for computing zech s logarithms in gf 2 sup m in this paper a new method for calculation of zech s logarithm in gf 2 sup m is presented for a given element the logarithm is calculated by bit operations performed on its binary representation no look up tables are used the proposed method makes feasible the implementation of an universal type device for finite field arithmetic

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

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

primitive polynomials and m sequences over gf q sup m procedures for obtaining primitive polynomials and m sequences over gf q sup m in terms of primitive polynomials and m sequences over gf q are presented using a degree mn primitive polynomial g x in gf g x an m sequence over gf q sup m can be expressed as a vector m sequence whose component m sequences are shifted versions of the m sequence generated by g x the degree n primitive polynomials in gf q sup m x with root alpha q sup i that are factors of g x with root alpha when g x is viewed in gf q sup m x are then developed from the m sequence over gf q sup m expressions for the shifts and corresponding primitive polynomial for the m sequence generated by the uth decimation of the m sequence generated by the polynomials are also given expressions for the shifts and corresponding primitive polynomial factors of g x for different bases expressing gf q sup m over gf q are presented

perturbations on the subdiagonals of toeplitz matrices let l be an hermitian linear functional defined on the linear space of laurent polynomials it is very well known that the gram matrix of the associated bilinear functional in the linear space of polynomials is a toeplitz matrix in this contribution we analyze some linear spectral transforms of l such that the corresponding toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial toeplitz matrix we focus our attention in the analysis of the quasi definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one

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