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

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

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

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

the generalized triangular decomposition given a complex matrix h we consider the decomposition h qrp where r is upper triangular and q and p have orthonormal columns special instances of this decomposition include the singular value decomposition svd and the schur decomposition where r is an upper triangular matrix with the eigenvalues of h on the diagonal we show that any diagonal for r can be achieved that satisfies weyl s multiplicative majorization conditions k k k k i 1 r i i 1 1 k k k i 1 r i i 1 i where k is the rank of h i is the i th largest singular value of h and r i is the i th largest in magnitude diagonal element of r given a vector r which satisfies weyl s conditions we call the decomposition h qrp where r is upper triangular with prescribed diagonal r the generalized triangular decomposition gtd a direct nonrecursive algorithm is developed for computing the gtd this algorithm starts with the svd and applies a series of permutations and givens rotations to obtain the gtd the numerical stability of the gtd update step is established the gtd can be used to optimize the power utilization of a communication channel while taking into account quality of service requirements for subchannels another application of the gtd is to inverse eigenvalue problems where the goal is to construct matrices with prescribed eigenvalues and singular values

analysis of low bit rate image transform coding calculations based on high resolution quantizations prove that the distortion rate d r of an image transform coding is proportional to 2 sup 2r when r is large enough in wavelet and block cosine bases we show that if r 1 bit pixel then d r varies like r sup 1 2 spl gamma where spl gamma remains of the order of 1 for most natural images the improved performance of embedded codings in wavelet bases is analyzed at low bit rates we show that the compression performance of an orthonormal basis depends mostly on its ability to approximate images with a few nonzero vectors

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

some undecidability results concerning the property of preserving regularity a finite string rewriting system r preserves regularity if and only if it preserves regularity where is the alphabet containing exactly those letters that have occurrences in the rules of r this proves a conjecture of gyenizse and vagvolgyi 1997 in addition some undecidability results are presented that generalize results of gilleron and tison 1995 from term rewriting systems to string rewriting systems it follows that the property of being regularity preserving is undecidable for term rewriting systems thus answering another question of gyenizse and vagvolgyi 1997 finally it is shown that it is undecidable in general whether a finite lengthreducing and confluent string rewriting system yields a regular set of normal forms for each regular language

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

the role of gtd in optimizing perfect reconstruction filter banks filter bank optimization for specific input statistics has been of great interest in both theory and practice in many signal processing applications in this paper we propose gtd generalized triangular decomposition filter banks as a subband coder for optimizing the theoretical coding gain we focus on perfect reconstruction orthonormal gtd filter banks and biorthogonal gtd filter banks we show that in both cases there are two fundamental properties in the optimal solutions namely total decorrelation and spectrum equalization the optimal solutions can be obtained by performing the frequency dependent gtd on the cholesky factor of the input power spectrum density matrices we also show that in both theory and numerical simulations the optimal gtd subband coders have superior performance than optimal traditional subband coders in addition the uniform bit loading scheme can with no loss of optimality be used in the optimal biorthogonal gtd coders which solves the granularity problem in the conventional optimum bit loading formula we then extend the use of gtd filter banks to wireless communication systems where linear precoding and zero forcing decision feedback equalization is used in frequency selective channels we consider the quality of service qos problem of minimizing the transmitted power subject to the bit error rate and total bit rate constraints optimal systems with orthonormal precoder and unconstrained precoder are both derived and shown to be related to the frequency dependent gtd of the channel frequency response

equal diagonal qr decomposition and its application to precoder design for successive cancellation detection in multiple input multiple output mimo multiuser detection theory the qr decomposition of the channel matrix h can be used to form the back cancellation detector in this paper we propose an optimal qr decomposition which we call the equal diagonal qr decomposition or briefly the qrs decomposition we apply the decomposition to precoded successive cancellation detection where we assume that both the transmitter and the receiver have perfect channel knowledge we show that for any channel matrix h there exists a unitary precoder matrix s such that hs qr where the nonzero diagonal entries of the upper triangular matrix r in the qr decomposition of hs are all equal to each other the precoder and the resulting successive cancellation detector have the following properties a the minimum euclidean distance between two signal points at the channel output is equal to the minimum euclidean distance between two constellation points at the precoder input up to a multiplicative factor that equals the diagonal entry in the r factor b the superchannel hs naturally exhibits an optimally ordered column permutation i e the optimal detection order for the vertical bell labs layered space time v blast detector is the natural order c the precoder s minimizes the block error probability of the qr successive cancellation detector d a lower and an upper bound for the free distance at the channel output is expressible in terms of the diagonal entries of the r factor in the qr decomposition of a channel matrix e the precoder s maximizes the lower bound of the channel s free distance subject to a power constraint f for the optimal precoder s the performance of the qr detector is asymptotically at large signal to noise ratios snrs equivalent to that of the maximum likelihood detector mld that uses the same precoder further we consider two multiplexing schemes time division multiple access tdma and orthogonal frequency division multiplexing ofdm we d

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

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

generalized triangular decomposition in transform coding a general family of optimal transform coders tcs is introduced here based on the generalized triangular decomposition gtd developed by jiang this family includes the karhunen loeve transform klt and the generalized version of the prediction based lower triangular transform plt introduced by phoong and lin as special cases the coding gain of the entire family with optimal bit allocation is equal to that of the klt and the plt even though the original plt introduced by phoong is not applicable for vectors that are not blocked versions of scalar wide sense stationary processes the gtd based family includes members that are natural extensions of the plt and therefore also enjoy the so called minlab structure of the plt which has the unit noise gain property other special cases of the gtd tc are the geometric mean decomposition gmd and the bidiagonal decomposition bid transform coders the gmd tc in particular has the property that the optimum bit allocation is a uniform allocation this is because all its transform domain coefficients have the same variance implying thereby that the dynamic ranges of the coefficients to be quantized are identical

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

minlab minimum noise structure for ladder based biorthogonal filter banks we introduce a minimum noise structure for ladder based biorthogonal minlab filter bank the minimum noise structure ensures that the quantization error has a unity noise gain even though the filter bank is biorthogonal the coder has a very low design and implementation cost the perfect reconstruction property is structurally preserved optimal bit allocation and coding gain formulas are derived we show that the coding gain of the optimal minlab coder is always greater than or equal to unity for both ar 1 process and ma 1 process the minlab coder with two taps has a higher coding gain than the optimal orthonormal coder with an infinite number of taps in addition to its superior decorrelation ability it has many other desired features that make it a potentially valuable and attractive alternative to the orthonormal coder especially for a high fidelity compression

a theoretical high rate analysis of causal versus unitary online transform coding backward adaptive or online transform coding tc of gaussian sources is investigated the karhunen loe spl grave ve transform klt unitary approach and the causal transform ct causal approach are compared in this context when the covariance matrix r sub x of the source is used in the tc scheme klt and ct present similar coding gains at high rates the aim of this study is to model analytically the behavior of these two coding structures when the ideal tc scheme gets perturbed that is when only a perturbed value r sub x spl delta r is known at the encoder in the online tc schemes considered here this estimate is used to compute both the transform and the bit assignment spl delta r is caused by two noise sources estimation noise finite set of available data at the encoder and quantization noise quantized data at the decoder furthermore not only does the transformation itself get perturbed but also the bit assignment does as well in this framework theoretical expressions for the coding gains in both the unitary and the causal cases are derived under the high rate assumption

efficient bit parallel gf 2 m multiplier for a large class of irreducible pentanomials this work studies efficient bit parallel multiplication in gf 2m for irreducible pentanomials based on the so called shifted polynomial bases spbs we derive a closed expression of the reduced spb product for a class of polynomials xm xk s xk s 1 hellip xk 1 1 with ks k1 les m 1 2 then we apply the above formulation to the case of pentanomials the resulting multiplier outperforms or is as efficient as the best proposals in the technical literature but it is suitable for a much larger class of pentanomials than those studied so far unlike previous works this property enables the choice of pentanomials optimizing different field operations for example inversion yet preserving an optimal implementation of field multiplication as discussed and quantitatively proved in the last part of the paper

joint unitary triangularization for mimo networks this paper considers communication networks where individual links can be described as those based on multiple input multiple output channels unlike orthogonal modulation methods such as the singular value decomposition we allow interference between subchannels which can be removed by the receivers via successive cancellation the degrees of freedom earned by this relaxation are used for obtaining a basis and corresponding decomposition which are simultaneously good for more than one link specifically we derive necessary and sufficient conditions for shaping the ratio vector of subchannel gains of two broadcast channel receivers we then apply this decomposition to two scenarios first in digital multicasting we present a practical capacity achieving scheme which uses only scalar codes and linear processing then we consider the joint source channel problem of transmitting a gaussian source over a two user multiple input multiple output channel where we show the existence of nontrivial cases where the optimal distortion pair which for high signal to noise ratios snrs equals the optimal point to point distortions of the individual users may be achieved by employing a hybrid digital analog scheme over the induced equivalent channel these scenarios demonstrate the advantage of choosing a modulation basis based upon multiple links in the network thus we coin the approach network modulation

biorthogonal and nonuniform lapped transforms for transform coding with reduced blocking and ringing artifacts new lapped transforms are introduced the lapped biorthononal transform lbt and hierarchical lapped biorthogonal transform hlbt are appropriate for image coding and the modulated hlbt biorthogonal transform mmlbt and nonuniform modulated lapped biorthogonal transform nmlbt are appropriate for audio coding the hlbt has a significantly lower computational complexity than the lapped orthogonal transform lot essentially no blocking artifacts and fewer ringing artifacts than the commonly used discrete cosine transform dct the lbt and hlbt have transform coding gains that are typically between 0 5 and 1 2 db higher than that of the dct image coding examples using jpeg and embedded zerotree coders demonstrate the better performance of the let and hlbt the nmlbt has fewer ringing artifacts and better reproduction of transient sounds than the mlt as shown in audio coding examples fast algorithms for both the hlbt and the nmlbt are presented

theory of optimal orthonormal subband coders the theory of the orthogonal transform coder and methods for its optimal design have been known for a long time we derive a set of necessary and sufficient conditions for the coding gain optimality of an orthonormal subband coder for given input statistics we also show how these conditions can be satisfied by the construction of a sequence of optimal compaction filters one at a time several theoretical properties of optimal compaction filters and optimal subband coders are then derived especially pertaining to behavior as the number of subbands increases significant theoretical differences between optimum subband coders transform coders and predictive coders are summarized finally conditions are presented under which optimal orthonormal subband coders yield as much coding gain as biorthogonal ones for a fixed number of subbands

parallel multipliers based on special irreducible pentanomials the state of the art galois field gf 2 sup m multipliers offer advantageous space and time complexities when the field is generated by so special irreducible polynomial to date the best complexity results have been obtained when the irreducible polynomial is either a trinomial or an equally spaced polynomial esp unfortunately there exist only a few irreducible esps in the range of interest for most of the applications e g error correcting codes computer algebra and elliptic curve cryptography furthermore it is not always possible to find an irreducible trinomial of degree m in this range for those cases where neither an irreducible trinomial nor an irreducible esp exists the use of irreducible pentanomials has been suggested irreducible pentanomials are abundant and there are several eligible candidates for a given m we promote the use of two special types of irreducible pentanomials we propose new mastrovito and dual basis multiplier architectures based on these special irreducible pentanomials and give rigorous analyses of their space and time complexity

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

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

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

prediction based lower triangular transform a new nonunitary transform called the prediction based lower triangular transform plt is introduced for signal compression the new transform has the same decorrelation property as the kahurnen loeve transform klt but its implementational cost is less than one half of klt compared with the klt the design cost of an m spl times m plt is much lower and is only of the order of o m sup 2 moreover the plt can be factorized into simple building blocks using two different factorizations we introduce two minimum noise structures that have roughly the same complexity as the direct implementation of plt these minimum noise structures have the following properties 1 its noise gain is unity even though the transform is nonunitary 2 perfect reconstruction is structurally guaranteed 3 it can be used for both lossy lossless compression we show that the coding gain of plt implemented using the minimum noise structure is the same as that of klt furthermore universal transform coders using plt are derived for ar 1 process the m spl times m plt has a closed form and needs only m 1 multiplications and additions

suboptimality of the karhunen loeve transform for transform coding we examine the performance of the karhunen loeve transform klt for transform coding applications the klt has long been viewed as the best available block transform for a system that orthogonally transforms a vector source scalar quantizes the components of the transformed vector using optimal bit allocation and then inverse transforms the vector this paper treats fixed rate and variable rate transform codes of non gaussian sources the fixed rate approach uses an optimal fixed rate scalar quantizer to describe the transform coefficients the variable rate approach uses a uniform scalar quantizer followed by an optimal entropy code and each quantized component is encoded separately earlier work shows that for the variable rate case there exist sources on which the klt is not unique and the optimal quantization and coding stage matched to a worst klt yields performance as much as 1 5 db worse than the optimal quantization and coding stage matched to a best klt in this paper we strengthen that result to show that in both the fixed rate and the variable rate coding frameworks there exist sources for which the performance penalty for using a worst klt can be made arbitrarily large further we demonstrate in both frameworks that there exist sources for which even a best klt gives suboptimal performance finally we show that even for vector sources where the klt yields independent coefficients the klt can be suboptimal for fixed rate coding

mimo transceivers with decision feedback and bit loading theory and optimization this paper considers mimo transceivers with linear precoders and decision feedback equalizers dfes with bit allocation at the transmitter zero forcing zf is assumed considered first is the minimization of transmitted power for a given total bit rate and a specified set of error probabilities for the symbol streams the precoder and dfe matrices are optimized jointly with bit allocation it is shown that the generalized triangular decomposition gtd introduced by jiang li and hager offers an optimal family of solutions the optimal linear transceiver which has a linear equalizer rather than a dfe with optimal bit allocation is a member of this family this shows formally that under optimal bit allocation linear and dfe transceivers achieve the same minimum power the dfe transceiver using the geometric mean decomposition gmd is another member of this optimal family and is such that optimal bit allocation yields identical bits for all symbol streams no bit allocation is necessary when the specified error probabilities are identical for all streams the qr based system used in vblast is yet another member of the optimal family and is particularly well suited when limited feedback is allowed from receiver to transmitter two other optimization problems are then considered a minimization of power for specified set of bit rates and error probabilities the qos problem and b maximization of bit rate for fixed set of error probabilities and power it is shown in both cases that the gtd yields an optimal family of solutions

a novel architecture for galois fields gf 2 m multipliers based on mastrovito scheme in this paper a new gf 2m multiplier for standard basis representation is developed the proposed multiplier implements the mastrovito multiplication scheme and can be designed for every field gf 2m a minimum area implementation of the first block of mastrovito multiplier and a high speed delay driven tree architecture for the second block of the mastrovito multiplier are employed in the new circuit multiplier complexity and delay are analytically evaluated for many polynomial classes timing and area occupation performances of the proposed multiplier are also calculated for many fields used in reed solomon codes applications and compared with those of previously proposed solutions the comparison shows that the proposed multiplier outperforms previous architectures for every considered gf 2m field the effectiveness of the proposed solution in a real application is verified by implementing in a 0 25 mum cmos technology the key equation solving block of a 255 239 reed solomon decoder the use of the proposed multiplier in this application results in a substantial speed improvement without any penalty in the silicon area occupation

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

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

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