optimal las vegas reduction from one way set reconciliation to error correction suppose we have two players a and c where player a has a string s 0 u 1 and player c has a string t 0 u 1 and none of the two players knows the other s string assume that s and t are both over an integer alphabet 0 1 where the first string contains n non zero entries we would wish to answer the following basic question assuming that s and t differ in at most k positions how many bits does player a need to send to player c so that he can recover s with certainty further how much time does player a need to spend to compute the sent bits and how much time does player c need to recover the string s this problem has a certain number of applications for example in databases where each of the two parties possesses a set of n key value pairs where keys are from the universe u and values are from and usually n u in this paper we show a time and message size optimal las vegas reduction from this problem to the problem of systematic error correction of k errors for strings of length n over an alphabet of size 2 log log u n the additional running time incurred by the reduction is linear expected randomized for player a and linear worst case deterministic for player c but the correction works with certainty when using the popular reed solomon codes the reduction gives a protocol that transmits o k log u log bits and runs in time o n polylog n log u log for all values of k the time is expected for player a encoding time and worst case for player c decoding time the message size is optimal whenever k u 1 1

a geometric approach to root finding in gt q sup m the problem of finding roots in f of polynomials in f x for f gf q sup m where q is a prime or prime power and m is a positive integer greater than 1 is considered the problem is analyzed by making use of the finite affine geometry ag m q a new method is proposed for finding roots of polynomials over finite extension fields it is more efficient than previous algorithms when the degree of the polynomial whose roots are to be found is less than dimension m of the extension field implementation of the algorithm can be enhanced in cases in which optimal normal bases for the coefficient field are available

recognising simplicity of black box groups by constructing involutions and their centralisers abstract we investigate the complexity of constructing involutions and their centralisers in groups of lie type over finite fields of odd order and discuss applications to the problem of deciding whether a matrix group or a black box group of known characteristic is simple we show that if the characteristic is odd then simplicity can be recognised in monte carlo polynomial time

sublinear root detection and new hardness results for sparse polynomials over finite fields we present a deterministic 2 o t q frac t 2 t 1 o 1 algorithm to decide whether a univariate polynomial f with t monomial terms and degree q has a root in the finite field math

on r th root extraction algorithm in mathbb f _q for q equiv lr s 1 mathrm mod r s 1 with 0 and small s we present an r th root extraction algorithm over a finite field mathbb f q our algorithm precomputes a primitive r s th root of unity xi where s is the largest positive integer satisfying r s q 1 and is applicable for the cases when s is small the proposed algorithm requires one exponentiation for the r th root computation and is favorably compared to the existing algorithms

mechanical output from individual muscles during explosive leg extensions the role of biarticular muscles the main result of this study is that biarticular leg muscles contribute significantly to the work done at joints due to transfer of power during explosive leg extensions in particular a net power transfer was shown from hip to knee joint during jumping and sprinting seven elite athletes performed explosive one legged jump and sprint push offs kinematics ground reaction forces and el tromyography emg of leg muscles were recorded the mechanical output of six individu muscle groups was estimated by using hill based muscle models the emg and kinematics served as input to these models for jumping as well as for sprinting the model estimated similar results for the relative work contribution done about a joint due to transfer of power by the biarticular muscles rectus femoris showed a power transfer from hip to knee joint while in contrast hamstrings showed a power transfer from knee to hip joint regardless of these opposite directions of power transfer a net transfer occurred from the hip to the knee joint the relative work contribution of hamstrings done in hip extension was 7 in jumping and 11 in sprinting for rectus femoris the relative work contribution done in knee extension was 21 in jumping and 31 in sprinting power transferring actions by gastrocnemius from knee to ankle contributed 25 in jumping and 28 in sprinting to the work done in piantar flexion these results support the hypothesis that the action of biarticular muscles contributes to a net transfer of power from proximal to distal joints during explosive leg extensions this action of the biarticular muscles causes an efficient conversion of body segment rotations into the desired translation of the body centre of gravity

generation of finite classical groups by pairs of elements with large fixed point spaces abstract we study good elements in finite 2 n dimensional classical groups g namely t is a good element if o t is divisible by a primitive prime divisor of q n 1 for the relevant field order q and t fixes pointwise an n space the group sl 2 n q contains such elements and they are present in su 2 n q sp 2 n q so 2 n e q only if n is odd even even respectively we prove that there is an absolute positive constant c such that two random conjugates of t generate g with probability at least c if g so 2 n e 2 and g sp 2 n q with q even in the exceptional case g sp 2 n q with q even two conjugates of t never generate g in this case we prove that two random conjugates of t generate a subgroup so 2 n e q with probability at least c the results underpin analysis of new constructive recognition algorithms for classical groups in even characteristic which succeed where methods utilising involution centralisers are not available

on instantiating pairing based protocols with elliptic curves of embedding degree one since the discovery of identity based encryption schemes in 2000 bilinear pairings have been used in the design of hundreds of cryptographic protocols the most commonly used pairings are constructed from elliptic curves over finite fields with small embedding degree these pairings can have different security performance and functionality characteristics and were therefore classified into types 1 2 3 and 4 in this paper we observe that this conventional classification is not applicable to pairings from elliptic curves with embedding degree one it is important to understand the security efficiency and functionality of these pairings in light of recent attacks on certain pairings constructed from elliptic curves with embedding degree greater than one we define three kinds of pairings from elliptic curves with embedding degree one discuss some subtleties with using them to implement pairing based protocols and provide an estimated cost of implementing them on modern processors

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

recognition of finite exceptional groups of lie type let q be a prime power and let g be an absolutely irreducible subgroup of gld f where f is a finite field of the same characteristic as fq the field of q elements assume that g g q a quasisimple group of exceptional lie type over fq which is neither a suzuki nor a ree group we present a las vegas algorithm that constructs an isomorphism from g to the standard copy of g q if g 6 3d4 q with q even then the algorithm runs in polynomial time subject to the existence of a discrete log oracle

presentations on standard generators for classical groups abstract for each family of finite classical groups and their associated simple quotients we provide an explicit presentation on a specific generating set of size at most 8 since there exist efficient algorithms to construct this generating set in any copy of the group our presentations can be used to verify claimed isomorphisms between representations of the classical group the presentations are available in magma

a fast algorithm for reversion of power series we give an algorithm for reversion of formal power series based on an efficient way to evaluate the lagrange inversion formula our algorithm requires o n 1 2 m n mm n 1 2 operations where m n and mm n are the costs of polynomial and matrix multiplication respectively this matches an algorithm of brent and kung but we achieve a constant factor speedup whose magnitude depends on the polynomial and matrix multiplication algo rithms used benchmarks confirm that the algorithm performs well in practice

knee and ankle joint torque angle relationships of multi joint leg extension abstract the force length relation f l r is an important property of skeletal muscle to characterise its function whereas for in vivo human muscles torque angle relationships t a r represent the maximum muscular capacity as a function of joint angle however since in vivo force torque length data is only available for rotational single joint movements the purpose of the present study was to identify torque angle relationships for multi joint leg extension therefore inverse dynamics served for calculation of ankle and knee joint torques of 18 male subjects when performing maximum voluntary isometric contractions in a seated leg press measurements in increments of 10 knee angle from 30 to 100 knee flexion resulted in eight discrete angle configurations of hip knee and ankle joints for the knee joint we found an ascending descending t a r with a maximum torque of 289 5 43 3 n m which closely matches literature data from rotational knee extension in comparison to literature we observed a shift of optimum knee angle towards knee extension in contrast the t a r of the ankle joint vastly differed from relationships obtained for isolated plantar flexion for the ankle t a r derived from multi joint leg extension subjects operated over different sections of the force length curve but the ankle t a r derived from isolated joint efforts was over the ascending limb for all subjects moreover mean maximum torque of 234 7 56 6 n m exceeded maximal strength of isolated plantar flexion 185 7 27 8 n m from these findings we conclude that muscle function between isolated and more physiological multi joint tasks differs this should be considered for ergonomic and sports optimisation as well as for modelling and simulation of human movement

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

an emg driven musculoskeletal model to estimate muscle forces and knee joint moments in vivo this paper examined if an electromyography emg driven musculoskeletal model of the human knee could be used to predict knee moments calculated using inverse dynamics across a varied range of dynamic contractile conditions muscle tendon lengths and moment arms of 13 muscles crossing the knee joint were determined from joint kinematics using a three dimensional anatomical model of the lower limb muscle activation was determined using a second order discrete non linear model using rectified and lowpass filtered emg as input a modified hill type muscle model was used to calculate individual muscle forces using activation and muscle tendon lengths as inputs the model was calibrated to six individuals by altering a set of physiologically based parameters using mathematical optimisation to match the net flexion extension fe muscle moment with those measured by inverse dynamics the model was calibrated for each subject using 5 different tasks including passive and active fe in an isokinetic dynamometer running and cutting manoeuvres recorded using three dimensional motion analysis once calibrated the model was used to predict the fe moments estimated via inverse dynamics from over 200 isokinetic dynamometer running and sidestepping tasks the inverse dynamics joint moments were predicted with an average r 2 of 0 91 and mean residual error of b12 nm a re calibration of only the emg to activation parameters revealed fe moments prediction across weeks of similar accuracy changing the muscle model to one that is more physiologically correct produced better predictions the modelling method presented represents a good way to estimate in vivo muscle forces during movement tasks r 2003 elsevier science ltd all rights reserved

are the maximum shortening velocity and the shape parameter in a hill type model of whole muscle related to activation abstract mathematical models of the inter relationship of muscle force velocity and activation are useful in forward dynamic simulations of human movement tasks the objective of this work was to determine whether the parameters maximum shortening velocity v max and shape parameter k of a hill type muscle model interrelating muscle force velocity and activation are themselves dependent on the activation to fulfill this objective surface emg signals from four muscles as well as the kinematics and kinetics of the arm were recorded from 14 subjects who performed rapid release elbow extension tasks at 25 50 75 and 100 activation mvc the experimental elbow flexion angle was tracked by a forward dynamic simulation of the task in which v max and k of the triceps brachii were varied at each activation level to minimize the difference between the simulated and experimental elbow flexion angle because a preliminary analysis demonstrated no dependency of k on activation additional simulations were performed with constant k values of 0 15 0 20 and 0 25 the optimized values of v max normalized to the average value within a subject were then regressed onto the activation normalized v max depended significantly on the activation p k furthermore the estimated v max values were not sensitive to the selected k value the results support the use of hill type models in which v max depends on activation in forward dynamic simulations modeling muscles with mixed fiber type composition recruited in the range of 25 100 activation the use of more accurate models will lend greater confidence to the results of forward dynamic simulations

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

finding involutions in finite lie type groups of odd characteristic abstract let g be a finite group of lie type in odd characteristic defined over a field with q elements we prove that there is an absolute and explicit constant c such that if g is a classical matrix group of dimension n 2 then at least c log n of its elements are such that some power is an involution with fixed point subspace of dimension in the interval n 3 2 n 3 if g is exceptional or g is classical of small dimension then for each conjugacy class c of involutions we find a very good lower bound for the proportion of elements of g for which some power lies in c

constructive recognition of sl3 q abstract we present a constructive recognition algorithm for groups of lie type sl 3 q this is a necessary component for constructive recognition algorithms of quasisimple groups of lie type

new parallel approaches for scalar multiplication in elliptic curve over fields of small characteristic we present two new strategies for parallel implementation of scalar multiplication over elliptic curves we first introduce a montgomery halving algorithm which is a variation of the original montgomery ladder for point multiplication this montgomery halving can be run in parallel with the original montgomery ladder in order to concurrently compute part of the scalar multiplication we also present two point thirding formulas in some subfamilies of curves e mathbb f 3 m we use these thirding formulas to implement scalar multiplication through third double and add and third triple and add parallel approaches we also provide some implementation results of the presented parallel strategies which show a speed up of 5 14 percent on an intel core i7 processor and a speed up of 8 19 percent on a qualcomm snapdragon processor compared to non parallelized approaches

constructive recognition of classical groups in even characteristic abstract let g x gl d f be a classical group in its natural representation defined over a finite field f of even characteristic we present las vegas algorithms to construct standard generators for g which permit us to write an element of g as a straight line program in x and to construct an involution as a straight line program in x if f 4 then the algorithms run in time polynomial in the size of the input subject to the existence of a discrete logarithm oracle for f

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

a power flow analysis based on continuum dynamics based on the governing equations of continuum mechanics a power flow analysis is presented in developing the mathematical model the concept of energy flow density vector is introduced which uniquely defines the energy transmission between one part of a material body system and another this approach allows the energy flow line the energy flow potential and the equipotential surface to be defined from this model the local equation of energy flow balance the equation of energy exchange between two or many subsystems and the time average equations are derived to describe the characteristics of energy flow and energy exchange within the continuum to demonstrate the applicability of the proposed mathematical model the energy flow relation between two simple oscillators is discussed and the concept generalized to sequential and non sequential multiple systems such multiple systems are examined and for non sequential systems which are analogous to statically indeterminate structural systems an approach is developed for the solution of their power flow and energy exchange it is further shown that the governing equation of energy flow is a first order partial differential equation which does not directly correspond to the equation describing the flow of thermal energy in a heat conduction problem

comparing different approaches for determining joint torque parameters from isovelocity dynamometer measurements strength or maximum joint torque is a fundamental factor governing human movement and is regularly assessed for clinical and rehabilitative purposes as well as for research into human performance this study aimed to identify the most appropriate protocol for fitting a maximum voluntary torque function to experimental joint torque data three participants performed maximum isometric and concentric eccentric knee extension trials on an isovelocity dynamometer and a separate experimental protocol was used to estimate maximum knee extension angular velocity a nine parameter maximum voluntary torque function which included angle angular velocity and neural inhibition effects was fitted to the experimental torque data and three aspects of this fitting protocol were investigated using an independent experimental estimate of maximum knee extension angular velocity gave lower variability in the high concentric velocity region of the maximum torque function compared to using dynamometer measurements alone a weighted root mean square difference rmsd score function that forced the majority 73 92 of experimental data beneath the maximum torque function was found to best account for the one sided noise in experimental torques resulting from sub maximal effort by the participants the suggested protocol an appropriately weighted rmsd score function and an independent estimate of maximum knee extension angular velocity gave a weighted rmsd of between 11 and 13 nm 4 5 of maximum isometric torque it is recommended that this protocol be used in generating maximum voluntary joint torque functions in all torque based modelling of dynamic human movement

the relation between the resultant moments at a joint and the moments measured by an isokinetic dynamometer isokinetic dynamometers have frequently been used to determine in vivo properties of human skeletal muscles in these investigations the moments recorded by the dynamometer were often implicitly assumed to be the same as the joint moments produced by muscles this assumption may have contributed to the conflicting results about in vivo muscle properties of human skeletal muscles that are reported in the literature the purpose of this paper is to present a method for calculating the resultant joint moments from the moments recorded using a cybex ii dynamometer and to determine the differences between these two moments for some typical exercise conditions the differences between these these moments are shown to be due to a gravitational effects b inertial effects and c non rigidity of the cybex arm shank foot system the results obtained suggest that the moments recorded using the dynamometer should not be used to derive muscle properties without taking into account the relationship between the resultant joint moments and the moments measured using an isokinetic dynamometer

fast constructive recognition of black box orthogonal groups abstract we present an algorithm that constructively recognises when a given black box group is a nontrivial homomorphic image of the orthogonal group e d q for known e d and q the algorithm runs in polynomial time assuming oracles for handling sl 2 q subgroups and discrete logarithms in f q

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

on the cell probe complexity of polynomial evaluation abstract we consider the cell probe complexity of the polynomial evaluation problem with preprocessing of coefficients for polynomials of degree at most n over a finite field k we show that the trivial cell probe algorithm for the problem is optimal if k is sufficiently large compared to n as an application we give a new proof of the fact that p incr time o log n log log n

presentations of finite simple groups a quantitative approach there is a constant such that all nonabelian finite simple groups of rank over with the possible exception of the ree groups have presentations with at most generators and relations and total length at most as a corollary we deduce a conjecture of holt there is a constant such that for every finite simple group every prime and every irreducible module

on products of involutions in finite groups of lie type in even characteristic abstract let g be a finite simple group of lie type in characteristic 2 and t g an involution we provide a lower bound for the proportion of elements g g such that t t g has odd order this has applications to the theory of recognition algorithms for such groups

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

elliptic curves and primality proving this report describes the theory and implementation of the elliptic curve primality proving algorithm ecpp algorithm this includes the relationships between representing primes by quadratic forms and the explicit construction of class fields of imaginary quadratic fields the theory of elliptic curves with complex multiplication over the field of complex numbers as well as over finite fields we then use this theory to design a very powerful primality proving algorithm half of the paper is devoted to the description of its implementation in particular we give the currently best algorithms to speed up each part of the program the resulting program is very fast we can prove the primality of 100 digit numbers in less than five minutes on a sun 3 60 workstation and we can treat all numbers with less than 1000 digits in a reasonable amount of time using a distributed implementation

the cell probe complexity of succinct data structures we consider time space tradeoffs for static data structure problems in the cell probe model with word size 1 the bit probe model in this model the goal is to represent n bit data with s n r bits such that queries of a certain type about the data can be answered by reading at most t bits of the representation ideally we would like to keep both s and t small but there are tradeoffs between the values of s and t that limit the possibilities of keeping both parameters small in this paper we consider the case of succinct representations where s n r for some email protected n for a boolean version of the problem of polynomial evaluation with preprocessing of coefficients we show a lower bound on the redundancy query time tradeoff of the form r 1 t email protected n logn in particular for very small redundancies r we get an almost optimal lower bound stating that the query algorithm has to inspect almost the entire data structure up to a logarithmic factor we show similar lower bounds for problems satisfying a certain combinatorial properties of a coding theoretic flavor and obtain r 1 t email protected n for certain problems previously no w m lower bounds were known on t in the general model for explicit boolean problems even for very small redundancies by restricting our attention to systematic or index structures f satisfying f x email protected email protected x for some map f where denotes concatenation we show similar lower bounds on the redundancy query time tradeoff for the natural data structuring problems of prefix sum and substring search

improved computation of square roots in specific finite fields in this paper we study exponentiation in the specific finite fields f with very special exponents such as those that occur in algorithms for computing square roots here q is a prime power q pk where k 1 and k is odd our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent to the best of our knowledge it is the first major improvement to the tonelli shanks algorithm for example the number of multiplications can be reduced to at least 60 percent on the average when p 1 mod 16 several numerical examples are given that show the speedup of the proposed methods

square root computation over even extension fields this paper presents a comprehensive study of the computation of square roots over finite extension fields we propose two novel algorithms for computing square roots over even field extensions of the form bbf q 2 with q p n p an odd prime and n geq 1 both algorithms have an associate computational cost roughly equivalent to one exponentiation in bbf q 2 the first algorithm is devoted to the case when q equiv 1 rm mod 4 whereas the second one handles the case when q equiv 3 rm mod 4 numerical comparisons show that the two algorithms presented in this paper are competitive and in some cases more efficient than the square root methods previously known

high performance architecture of elliptic curve scalar multiplication a high performance architecture of elliptic curve scalar multiplication based on the montgomery ladder method over finite field gf 2m is proposed a pseudo pipelined word serial finite field multiplier with word size w suitable for the scalar multiplication is also developed implemented in hardware this system performs a scalar multiplication in approximately 6lceilm wrceil m 1 clock cycles and the gate delay in the critical path is equal to tand lceillog2 w k rceiltxor where tand and txor are delays due to two input and and xor gates respectively and 1 les k lt w is used to shorten the critical path

a comparison of different finite fields for use in elliptic curve cryptosystems we examine the relative efficiency of four methods for finite field representation in the context of elliptic curve cryptography ecc we conclude that a set of fields called the optimized extension fields oefs give greater performance even when used with affine coordinates when compared against the type of fields recommended in the emerging ecc standards although this performance advantage is only marginal and hence there is probably no need to change the current standards to allow oef fields in standards compliant implementations

constructing composition factors for a linear group in polynomial time abstract we present a las vegas polynomial time algorithm that takes as input a subgroup of gl d f q and subject to the existence of certain oracles determines its composition factors provided that none of those factors is isomorphic to one of b 2 2 2 2 k 1 f 4 2 2 2 k 1 d 4 3 2 k or g 2 2 3 2 k 1 for any k

reconfigurable computing approach for tate pairing cryptosystems over binary fields tate pairing based cryptosystems because of their ability to be used in multiparty identity based key management schemes have recently emerged as an alternative to traditional public key cryptosystems due to the inherent parallelism of the existing pairing algorithms high performance can be achieved via hardware realizations three schemes for tate pairing computations have been proposed in the literature cubic elliptic binary elliptic and binary hyperelliptic in this paper we propose a new fpga based architecture of the tate pairing based computation over binary fields even though our field sizes are larger than in the architectures based on cubic elliptic curves or binary hyperelliptic curves with the same security strength nevertheless fewer multiplications in the underlying field need to be performed as a result the computational latency for a pairing computation has been reduced and our implementation runs 2 20 times faster than the equivalent implementations of other pairing based schemes at the same level of security strength furthermore we ported our pairing designs for eight field sizes ranging from 239 to 557 bits to the reconfigurable computer sgi altix 4700 supported by silicon graphics inc and performance and cost are demonstrated

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

predicting maximum eccentric strength from surface emg measurements the origin of the well documented discrepancy between maximum voluntary and in vitro tetanic eccentric strength has yet to be fully understood this study aimed to determine whether surface emg measurements can be used to reproduce the in vitro tetanic force velocity relationship from maximum voluntary contractions five subjects performed maximal knee extensions over a range of eccentric and concentric velocities on an isovelocity dynamometer whilst emg from the quadriceps were recorded maximum voluntary mvc force length velocity data were estimated from the dynamometer measurements and a muscle model normalised amplitude length velocity data were obtained from the emg signals dividing the mvc forces by the normalised amplitudes generated emg corrected force length velocity data the goodness of fit of the in vitro tetanic force velocity function to the mvc and emg corrected forces was assessed based on a number of comparative scores the in vitro tetanic force velocity function provided a significantly better fit to the emg corrected forces compared to the mvc forces p or 0 05 furthermore the emg corrected forces generated realistic in vitro tetanic force velocity profiles a 58 19 increase in maximum eccentric strength is theoretically achievable through eliminating neural factors in conclusion emg amplitude can be used to estimate in vitro tetanic forces from maximal in vivo force measurements supporting neural factors as the major contributor to the difference between in vitro and in vivo maximal force

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

note on taking square roots modulo n in this article it is shown how gauss 1981 famous cyclotomic sum formula can be used for extracting square roots modulo n

a refinement of h c williams th root algorithm let p and q be primes such that p 1 mod q let a be an integer such that a p l q 1 mod p in 1972 h c williams gave an algorithm which determines a solution of the congruence xq a mod p in o q3 logp steps once an integer b has been found such that bq a p1 q 0 0 1 mod p a step is an arithmetic operation mod p or an arithmetic operation on q bit integers we present a refinement of this algorithm which determines a solution in o q4 o q2 logp steps once b has been determined thus the new algorithm is better when q is small compared with p

black box groups isomorphic to pgl 2 2e abstract a deterministic polynomial time algorithm constructs an isomorphism between pgl 2 2 e and a black box group to which it is isomorphic

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

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

a class of unidirectional bit serial systolic architectures for multiplicative inversion and division over gf 2 sup m a class of universal unidirectional bit serial systolic architectures for multiplicative inversion and division over galois field gf 2 sup m is presented the field elements are represented with polynomial standard basis these systolic architectures have no carry propagation structures and are suitable for hardware implementations where the dimension of the field is large and may vary this is the typical case for cryptographic applications these architectures are independent of any defining irreducible polynomial of a given degree as well the time complexity is constant and area complexity is linear w r t field dimension and these measures are equivalent to or exceed similar proposed designs

fast constructive recognition of black box symplectic groups abstract we present an algorithm which for any given black box group g isomorphic to sp d q constructs explicit inverse isomorphisms between g and the standard copy of sp d q we also report on the performance of an implementation of this algorithm in the group theory system gap

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

a power flow mode theory based on a system s damping distribution and power flow design approaches a power flow mode theory is developed to describe the natural power flow behaviour of a dynamic system based on its inherent damping distribution the system9s characteristic damping matrix is constructed and it is shown that the eigenvalues and eigenvectors of this matrix identify natural power flow characteristics these eigenvectors or power flow mode vectors are chosen as a set of base vectors spanning the power flow space and completely describe the power flow in the system the generalized coordinate of the velocity vector decomposed in this space defines the power flow response vector a time averaged power flow expression and theorems relating to its estimation are presented based on this theory power flow design approaches are proposed to identify energy flow patterns satisfying vibration control requirements the mode control factor defines the measure of the correlation between a power flow mode and a natural vibration mode of the system power flow design theorems are presented providing guidelines to construct damping distributions maximizing power dissipation or to suppress retain a particular vibration mode and or a motion the developed damping based power flow mode theory is compared with a mobility based power flow model it is shown that the proposed power flow model provides insight into the power flow dissipation mechanisms in dynamic systems examples are presented to demonstrate the applicability of the power flow mode theory and the power flow design approach these examples demonstrate the generality of the theory including non symmetric damping matrices and illustrate power flow design applications through modifications of the system9s damping distribution using passive and or active control components

do relevant shear forces appear in isokinetic shoulder testing to be implemented in biomechanical models abstract isokinetic dynamometers measure joint torques about a single fixed rotational axis previous studies yet suggested that muscles produce both tangential and radial forces during a movement so that the contact forces exerted to perform this movement are multidirectional then isokinetic dynamometers might neglect the torque components about the two other euclidean space axes our objective was to experimentally quantify the shear forces impact on the overall shoulder torque by comparing the dynamometer torque to the torque computed from the contact forces at the hand and elbow ten healthy women performed isokinetic maximal internal external concentric eccentric shoulder rotation movements the hand and elbow contact forces were measured using two six axis force sensors the main finding is that the contact forces at the hand were not purely tangential to the direction of the movement effectiveness indexes from 0 26 0 25 to 0 54 0 20 such that the resulting shoulder torque computed from the two force sensors was three dimensional therefore the flexion and abduction components of the shoulder torque measured by the isokinetic dynamometer were significantly underestimated up to 94 9 these findings suggest that musculoskeletal models parameters should not be estimated without accounting for the torques about the three space axes

taking roots over high extensions of finite fields we present a new algorithm for computing m th roots over the finite field fq where q pn with p a prime and m any positive integer in the particular case m 2 the cost of the new algorithm is an expected o m n log p c n log n operations in fp where m n and c n are bounds for the cost of polynomial multiplication and modular polynomial composition known results give m n o n log n log log n and c n o n1 67 so our algorithm is subquadratic in n

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

reducing ground reaction forces in gymnastics landings may increase internal loading the aim of this study was to use a subject specific seven link wobbling mass model of a gymnast and a multi layer model of a landing mat to determine landing strategies that minimise ground reaction forces grf and internal forces subject specific strength parameters were determined that defined the maximum voluntary torque angle angular velocity relationship at each joint these relationships were used to produce subject specific lumped linear muscle models for each joint muscle activation histories were optimised using a simplex algorithm to minimise grf or bone bending moments for forward and backward rotating vault landings optimising the landing strategy to minimise each of the grf reduced the peak vertical and horizontal grf by 9 for the backward rotating vault and by 8 and 48 for the forward rotating vault compared to a matching simulation however most internal loading measures bone bending moments joint reaction forces and muscle forces increased compared to the matching simulation optimising the landing strategy to minimise the peak bone bending moments resulted in reduced internal loading measures and in most cases reduced grf bone bending moments were reduced by 27 during the forward rotating vault and by 2 during the backward rotating vault landings when compared to the matching simulations it is possible for a gymnast to modify their landing strategy in order to minimise internal forces and lower grf however using a reduction in grf due to a change in landing strategy as a basis for a reduction in injury potential in vaulting movements may not be appropriate since internal loading can increase

a space averaged quadratic method for predicting one dimensional power transfers a method which is based on space averaged quadratic variables and does not assume the restrictive assumptions of the statistical energy analysis sea such as high overlap or damping conditions is described the results are spatially more detailed than an sea response and of a lower computational cost than the one of the usual displacement formulation some of the quadratic variables used like energy densities or structural intensity retain a strong energy meaning even when time and space averaged their governing equation is derived for a homogeneous and isotropic medium with hysteretic damping the main originality of this work lies in considering complex structural intensity and establishing appropriate energetic boundary conditions for both active and reactive space averaged structural intensities by using the usual boundary conditions for the displacement field and the stress tensor the numerical examples prove that this space averaged quadratic method is well suited to describe global energy transfers along one dimensional dissipative structures in a frequency range for which the overlap is too low to obtain a quadratic response from the sea and solving the wave equation by using the finite element method fem would require more elements besides describing junctions with impedances makes the resonant behaviour of the system still accessible on frequency responses

power flow analysis of complex coupled systems by progressive approaches a generalized mobility impedance power flow mathematical model is developed to analyze the dynamical behaviour of a complex coupled system consisting of any number of substructures with various configurations and multiple interaction interfaces the coupled system is subject to multiple excitations and selected boundary conditions generalized mobility impedance matrix formulations for three dimensional rigid and elastic structures of general configuration are first derived allowing the construction of equivalent mobility emm and equivalent impedance eim matrices to describe the dynamical behaviour of a substructure or a subsystem assembled from several inter connected substructures within the overall system based on these two proposed matrices two progressive approaches are developed to predict the force vectors and velocity response vectors as well as the power flows into and transmission between substructures in the complex coupled system the developed mathematical model avoids the generalized inverse process associated with rectangular matrices when dealing with multi input multi output mimo systems in which the dimensions of input and output are different it is also very flexible and conveniently extended if additional substructures are further connected to the original dynamic system without involving much additional computational effort the proposed methods are shown to reduce the complexity of the power flow analysis applied to complex dynamic coupled systems and they are applicable to a very large class of dynamical systems in engineering to illustrate and demonstrate their usage the dynamics of a flexible raft vibration isolation system is investigated this comprises two machines flexible raft and flexible foundation with connecting isolator attachments

hardware and software normal basis arithmetic for pairing based cryptography in characteristic three although identity based cryptography offers a number of functional advantages over conventional public key methods the computational costs are significantly greater the dominant part of this cost is the tate pairing which in characteristic three is best computed using the algorithm of duursma and lee however in hardware and constrained environments this algorithm is unattractive since it requires online computation of cube roots or enough storage space to precompute required results we examine the use of normal basis arithmetic in characteristic three in an attempt to get the best of both worlds an efficient method for computing the tate pairing that requires no precomputation and that may also be implemented in hardware to accelerate devices such as smart cards

constructive recognition of mathrm psl 2 q existing black box and other algorithms for explicitly recognising groups of lie type over gf q have asymptotic running times which are polynomial in q whereas the input size involves only log q this has represented a serious obstruction to the efficient recognition of such groups recently brooksbank and kantor devised new explicit recognition algorithms for classical groups these run in time that is polynomial in the size of the input given an oracle that recognises psl 2 q explicitly the present paper in conjunction with an earlier paper by the first two authors provides such an oracle the earlier paper produced an algorithm for explicitly recognising sl 2 q in its natural representation in polynomial time given a discrete logarithm oracle for gf q the algorithm presented here takes as input a generating set for a subgroup g of gl d f that is isomorphic modulo scalars to psl 2 q where f is a finite field of the same characteristic as gf q it returns the natural representation of g modulo scalars since a faithful projective representation of psl 2 q in cross characteristic or a faithful permutation representation of this group is necessarily of size that is polynomial in q rather than in log q elementary algorithms will recognise psl 2 q explicitly in polynomial time in these cases given a discrete logarithm oracle for gf q our algorithm thus provides the required polynomial time oracle for recognising psl 2 q explicitly in the remaining case namely for representations in the natural characteristic this leads to a partial solution of a question posed by babai and shalev if g is a matrix group in characteristic p determine in polynomial time whether or not op g is trivial

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

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

algorithms and arithmetic operators for computing the ηt pairing in characteristic three since their introduction in constructive cryptographic applications pairings over hyper elliptic curves are at the heart of an ever increasing number of protocols with software implementations being rather slow the study of hardware architectures became an active research area in this paper we discuss several algorithms to compute the etat pairing in characteristic three and suggest further improvements these algorithms involve addition multiplication cubing inversion and sometimes cube root extraction over f3m we propose a hardware accelerator based on a unified arithmetic operator able to perform the operations required by a given algorithm we describe the implementation of a compact coprocessor for the field f397 given by f3 x x97 x12 2 which compares favorably with other solutions described in the open literature

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

fast polynomial factorization and modular composition we obtain randomized algorithms for factoring degree n univariate polynomials over mathbb f q requiring o n 1 5 o 1 rm log 1 o 1 q n 1 o 1 rm log 2 o 1 q bit operations when rm log q n this is asymptotically faster than the best previous algorithms j von zur gathen and v shoup comput complexity 2 1992 pp 187 224 e kaltofen and v shoup math comp 67 1998 pp 1179 1197 for rm log q ge n it matches the asymptotic running time of the best known algorithms the improvements come from new algorithms for modular composition of degree n univariate polynomials which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields the best previous algorithms for modular composition use o n omega 1 2 field operations where omega is the exponent of matrix multiplication r p brent and h t kung j assoc comput mach 25 1978 pp 581 595 with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication x huang and v y pan j complexity 14 1998 pp 257 299 we show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent alpha implies an algorithm for the other with exponent alpha o 1 and vice versa we then give two new algorithms that solve the problem near optimally an algebraic algorithm for fields of characteristic at most n o 1 and a nonalgebraic algorithm that works in arbitrary characteristic the latter algorithm works by lifting to characteristic 0 applying a small number of rounds of multimodular reduction and finishing with a small number of multidimensional ffts the final evaluations are reconstructed using the chinese remainder theorem as a bonus this algorithm produces a very efficient data structure supporting polynomial evaluation queries which is of independent interest our algorithms use techniques that are commonly employed in practice in contrast to all previous subquadratic algorithms for these problems which relied on fast matrix multiplication

estimating proportions of elements in finite groups of lie type abstract we show that certain closure properties on subsets q of a finite group g of lie type enable the ratio q g to be determined by finding the proportions of elements of q in the maximal tori of g and the proportions of certain related subsets of the weyl group we prove fundamental results about these subsets q including those necessary for moving between these groups and their automorphism groups normal subgroups and central quotients as a sample application of these new techniques we derive upper and lower bounds for the proportion of elements with order divisible by certain natural sets of primes for the finite classical groups thereby improving existing results

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

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

optimal tower fields we introduce a new tower field representation optimal tower fields otfs that facilitates efficient finite field operations the recursive direct inversion method we present has significantly lower complexity than the known best method for inversion in optimal extension fields oefs i e itoh tsujii s inversion technique the complexity of our inversion algorithm is shown to be o m sup 2 significantly better than that of the itoh tsujii algorithm i e o m sup 2 log sub 2 m this complexity is further improved to o m sup log sub 2 sup 3 by utilizing the karatsuba ofman algorithm in addition we show that otfs may be converted to oef representation via a simple permutation of the coefficients and hence otf operations may be utilized to achieve the oef arithmetic operations whenever a corresponding otf representation exists while the original otf multiplication and squaring operations require slightly more additions than their oef counterparts due to the free conversion both otf operations may be achieved with the complexity of oef operations

modelling suppressed muscle activation by means of an exponential sigmoid function validation and bounds the aim of this study was to establish how well a three parameter sigmoid exponential function difact follows experimentally obtained voluntary neural activation angular velocity profiles and how robust it is to perturbed levels of maximal activation six male volunteers age 26 3 2 73 years were tested before and after an 8 session 3 week training protocol torque angular velocity t and experimental voluntary neural drive angular velocity va datasets obtained via the interpolated twitch technique were determined from pre and post training testing sessions non linear regression fits of the product of difact and a hill type tetanic torque function and of the difact function only were performed on the pre and post training t and va datasets for three different values of the difact upper bound max 100 95 90 the determination coefficients r2 and the rms of the fits were compared using a two way mixed anova and results showed that there was no significant difference p 0 05 due to changing max values indicating the difact remains robust to changes in maximal activation mean r2 values of 0 95 and 0 96 for pre and post training sessions show that the maximal voluntary torque function successfully reproduces the t raw dataset

maximum voluntary joint torque as a function of joint angle and angular velocity model development and application to the lower limb measurements of human strength can be important during analyses of physical activities such measurements have often taken the form of the maximum voluntary torque at a single joint angle and angular velocity however the available strength varies substantially with joint position and velocity when examining dynamic activities strength measurements should account for these variations a model is presented of maximum voluntary joint torque as a function of joint angle and angular velocity the model is based on well known physiological relationships between muscle force and length and between muscle force and velocity and was tested by fitting it to maximum voluntary joint torque data from six different exertions in the lower limb isometric concentric and eccentric maximum voluntary contractions were collected during hip extension hip flexion knee extension knee flexion ankle plantar flexion and dorsiflexion model parameters are reported for each of these exertion directions by gender and age group this model provides an efficient method by which strength variations with joint angle and angular velocity may be incorporated into comparisons between joint torques calculated by inverse dynamics and the maximum available joint torques

effective black box constructive recognition of classical groups abstract we describe a black box las vegas algorithm to construct standard generators for classical groups defined over finite fields we assume that the field has size at least 4 and that oracles to solve certain problems are available subject to these assumptions the algorithm runs in polynomial time a practical implementation of our algorithm is distributed with the computer algebra system magma

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