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

inequalities for the tails of some elementary series we prove that if dn k x n k 2 n k 2 in k x in k x n 2 in x 2 with formula then we have for all x 0 dn k x k 0 and dn 1 k x 2 dn k x dn 2 k x n k 1 0

fast elliptic scalar multiplication using new double base chain and point halving the fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication based on the double base chain representation of scalar using powers of 2 and 3 we propose a new representation with powers of 1 2 and 3 instead thus the efficient point halving operation can be incorporated in the new double base chain to achieve fast scalar multiplication experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems

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

state space solutions to hadamard weighted ℋ and ℋ 2 control problems variations of the standard h and h 2 control problems are presented state space solutions are given for the problems of weighting the individual elements of a closed loop transfer function while i satisfying an h frobenius norm bound or ii minimizing the h 2 norm of the closed loop the optimal h frobenius case is analysed and it is shown that the optimal controller is often unique in contrast to the standard h control problem the solutions of an algebraic riccati equation are examined in the h 2 case to reveal the structure of the h 2 optimal controller

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

inequalities for residuals of power expansions for the exponential function and completely monotone functions letin x be the residual after thenth term of power series for a functionf in the casef x ex x 0 inequalities forinhave been studied by many authors but there are almost no results forx 0 in this paper we present some bounds forinwithf x e xand we give some results for completely monotone functions

low complexity bit parallel square root computation over gf 2 m for all trinomials in this contribution we introduce a low complexity bit parallel algorithm for computing square roots over binary extension fields our proposed method can be applied to any type of irreducible polynomials we derive explicit formulas for the space and time complexities associated with the square root operator when working with binary extension fields generated using irreducible trinomials we show that for those finite fields it is possible to compute the square root of an arbitrary field element with equal or better hardware efficiency than the one associated with the field squaring operation furthermore a practical application of the square root operator in the domain of field exponentiation computation is presented

fast architectures for the eta_t pairing over small characteristic supersingular elliptic curves this paper is devoted to the design of fast parallel accelerators for the cryptographic t pairing on supersingular elliptic curves over finite fields of characteristics two and three we propose here a novel hardware implementation of miller s algorithm based on a parallel pipelined karatsuba multiplier after a short description of the strategies that we considered to design our multiplier we point out the intrinsic parallelism of miller s loop and outline the architecture of coprocessors for the t pairing over f 2m and f 2m thanks to a careful choice of algorithms for the tower field arithmetic associated with the t pairing we manage to keep the pipelined multiplier at the heart of each coprocessor busy a final exponentiation is still required to obtain a unique value which is desirable in most cryptographic protocols we supplement our pairing accelerators with a coprocessor responsible for this task an improved exponentiation algorithm allows us to save hardware resources according to our place and route results on xilinx fpgas our designs improve both the computation time and the area time trade off compared to previously published coprocessors

all optimal hankel norm approximations of linear multivariable systems and their l error bounds the problem of approximating a multivariable transfer function g s of mcmillan degree n by s of mcmillan degree k is considered a complete characterization of all approximations that minimize the hankel norm is derived the solution involves a characterization of all rational functions s f s that minimize where s has mcmillan degree k and f s is anticavisal the solution to the latter problem is via results on balanced realizations all pass functions and the inertia of matrices all in terms of the solutions to lyapunov equations it is then shown that where k 1 g s is the k l st hankel singular value of g s and for one class of optimal hankel norm approximations the method is not computationally demanding and is applied to a 12 state model

functional inequalities for the incomplete gamma function abstract we present several inequalities for f a x a x a 0 a 0 x 0 where a x is the incomplete gamma function one of our theorems states that the inequalities f a s p x 1 x n f a x 1 f a x n f a s q x 1 x n p q 0 hold for all nonnegative real numbers x 1 x n n 2 if and only if p m i n a 1 and q m a x a 1 here s t x 1 x n denotes the power sum of order t this extends and complements a result published by ismail and laforgia in 2006

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

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

efficient hardware implementation of fp arithmetic for pairing friendly curves this paper describes a new method to speed up ifp arithmetic in hardware for pairing friendly curves such as the well known barreto naehrig bn curves we explore the characteristics of the modulus defined by these curves and choose curve parameters such that ifp multiplication becomes more efficient the proposed algorithm uses montgomery reduction in a polynomial ring combined with a coefficient reduction phase using a pseudo mersenne number as an application we show that the performance of pairings on bn curves in hardware can be significantly improved resulting in a factor 2 5 speedup compared with state of the art hardware implementations

optimal pairings in this paper we introduce the concept of an optimal pairing which by definition can be computed using only log 2 r k basic miller iterations with r the order of the groups involved and k the embedding degree we describe an algorithm to construct optimal ate pairings on all parametrized families of pairing friendly elliptic curves finally we conjecture that any nondegenerate pairing on an elliptic curve without efficiently computable endomorphisms different from powers of frobenius requires at least log 2 r k basic miller iterations

approach to pairing inversions without solving miller inversion in the present paper we show that the pairing inversion problem of atei pairing can be solved under the assumption that we have a generic algorithm for solving exponentiation inversion problem with such an algorithm the inversion problem of atei pairing can be solved without solving the miller inversion thus the pairing inversion problem of atei pairing is reduced to the exponentiation inversion problem

robust decoupling of discrete systems using spl lscr sup 1 optimization we address a control design problem related to decoupling namely robust decoupling the idea of robust decoupling is defined and shown to be equivalent to robust performance of a hadamard weighted spl lscr sup 1 performance measure a necessary and sufficient condition to determine if a compensator satisfies robust performance of a hadamard weighted spl lscr sup 1 performance measure in the presence of structured uncertainty is given this condition can be used to determine if a compensator robustly decouples a plant finally an iterative scheme for synthesizing a robust decoupling controller is discussed

efficient and generalized pairing computation on abelian varieties in this paper we propose a new method for constructing a bilinear pairing over hyper elliptic curves which we call the r ate pairing this pairing is a generalization of the ate and atei pairing and can be computed more efficiently using the r ate pairing the loop length in miller s algorithm can be as small as log r1 phi k some pairing friendly elliptic curves which have not reached this lower bound therefore we obtain savings of between 29 and 69 in overall costs compared to the atei pairing on supersingular hyperelliptic curves of genus 2 we show that this approach makes the loop length in miller s algorithm shorter than that of the ate pairing

log concavity for bernstein type operators using stochastic orders abstract this paper aims to study the preservation of log concavity for bernstein type operators in particular attention is focused on positive linear operators defined on the positive semi axis admitting a probabilistic representation in terms of a process with independent increments this class includes the classical gamma szasz and szasz durrmeyer operators with respect to the first and second operators the results of this paper correct two erroneous counterexamples in 10 as a main tool in our results we use stochastic order techniques our results include as a particular case the log concavity of certain functions related to the incomplete gamma function

on the final exponentiation in tate pairing computations the tate pairing computation consists of two parts miller step and final exponentiation step in this paper we investigate the structure of the final exponentiation step consider an order r subgroup of an elliptic curve defined over fq with embedding degree k the final exponentiation in the tate pairing is an exponentiation of an element in fqk by qk 1 r the hardest part of this computation is to raise to the power k q r where k denotes the kth cyclotomic polynomial write it as 0 1q k 1q k 1 in the q ary representation the final exponentiation cost mostly depends on the size of the maximum of i in many parameterized pairing friendly curves the value is about 1 1 k log2q where log2q log2r while random curves will have log2q we investigate how this small is obtained for parameterized pairing friendly elliptic curves and show that 1 1 k log2q is the lower bound for all known construction methods of parameterized pairing friendly curves in the second part of our paper we propose a method to obtain a modified tate pairing with small for any pairing friendly elliptic curves including those not belonging to parameterized families more precisely our method finds an integer m using the lattice basis reduction such that m 1 1 k log2q using this modified tate pairing we can reduce the number of squarings in the final exponentiation by a factor of 1 1 k from the usual tate pairing we apply our method to several known pairing friendly curves to verify the expected speedup

inequalities for the chi square distribution function abstract letpa x 1 x a0 e ttx 1 dtbe the chi square distribution function and letmt u v be the weighted power mean of ordert we prove the following ifx y 0 x y and 0 1 are fixed real numbers then the inequalities formula are valid for all real numbersa 0 if and only ifr 0 ands in particular we obtain that for alla 0 the functionx pa x is log concave on 0 this proves a conjecture of m merkle

frequency weighted optimal hankel norm approximation of stable transfer functions abstract this paper shows how to introduce arbitrary frequency weighting into the optimal hankel norm approximation problem for scalar finite dimensional linear time invariant systems the procedure has applications to the closed form design of reduced order controllers

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

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

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

multiple objective optimal control of linear systems the quadratic norm case control of a linear system to meet requirements embodied in multiple performance criteria is recognized as the typical design problem the authors study the design of control when every criterion of interest takes the form of an h sub 2 norm defined over certain exogenous inputs and controlled outputs of the system the objective is to find all stabilizing controllers that are noninferior with respect to the defined objectives that is to find the set of pareto optimal controllers the novel feature is the use of the youla parametrization to represent the admissible controllers this representation enables the authors to show that each criterion and hence their weighted sum is convex over the class of admissible controllers since the set of admissible controllers is itself convex the pareto optimal controllers can be determined by the standard scalarization technique it is concluded that in general the controller order is higher than the order of the plant

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

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

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

identity based encryption from the weil pairing we propose a fully functional identity based encryption ibe scheme the scheme has chosen ciphertext security in the random oracle model assuming a variant of the computational diffie hellman problem our system is based on bilinear maps between groups the weil pairing on elliptic curves is an example of such a map we give precise definitions for secure ibe schemes and give several applications for such systems

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

on the use of hadamard weightings in the partial decoupling of h based designs abstract in this paper a solution to the hadamard weighted h hwh 1 block problem is proposed the solution is based on a method by safonov and le 1988 the robust stability of designs based on the hwh method is also discussed

field inversion and point halving revisited we present a careful analysis of elliptic curve point multiplication methods that use the point halving technique of knudsen and schroeppel and compare these methods to traditional algorithms that use point doubling the performance advantage of halving methods is clearest in the case of point multiplication kp where p is not known in advance and smaller field inversion to multiplication ratios generally favor halving although halving essentially operates on affine coordinate representations we adapt an algorithm of knuth to allow efficient use of projective coordinates with halving based windowing methods for point multiplication

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

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 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

a structured approximation problem with applications to frequency weighted model reduction the authors consider the problem of finding necessary and sufficient conditions for the existence of a q with no more than k poles in the closed left half plane such that the norm of a certain matrix

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

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

faster computation of self pairings self pairings have found interesting applications in cryptographic schemes in this paper we present a novel method for constructing a self pairing on supersingular elliptic curves with even embedding degrees which we call the ateil pairing this new pairing improves the efficiency of the self pairing computation on supersingular curves over finite fields with large characteristic based on the t pairing we propose a generalization of the ateil pairing which we call the ateili pairing the optimal ateili pairing which has the shortest miller loop is faster than previously known self pairings on supersingular elliptic curves over finite fields with small characteristic we also present a new self pairing based on the weil pairing which is faster than the self pairing based on the tate pairing on ordinary elliptic curves with embedding degree one

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

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

software multiplication using gaussian normal bases fast algorithms for multiplication in finite fields are required for several cryptographic applications in particular for implementing elliptic curve operations over binary fields f2m in this paper we present new software algorithms for efficient multiplication over f2m that use a gaussian normal basis representation two approaches are presented direct normal basis multiplication and a method that exploits a mapping to a ring where fast polynomial based techniques can be employed our analysis including experimental results on an intel pentium family processor shows that the new algorithms are faster and can use memory more efficiently than previous methods despite significant improvements we conclude that the penalty in multiplication is still sufficiently large to discourage the use of normal bases in software implementations of elliptic curve systems

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

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

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

state space solutions to standard h sub 2 and h sub infinity control problems simple state space formulas are derived for all controllers solving the following standard h sub infinity problem for a given number gamma 0 find all controllers such that the h sub infinity norm of the closed loop transfer function is strictly less than gamma it is known that a controller exists if and only if the unique stabilizing solutions to two algebraic riccati equations are positive definite and the spectral radius of their product is less than gamma sup 2 under these conditions a parameterization of all controllers solving the problem is given as a linear fractional transformation lft on a contractive stable free parameter the state dimension of the coefficient matrix for the lft constructed using the two riccati solutions equals that of the plant and has a separation structure reminiscent of classical lqg i e h sub 2 theory this paper is intended to be of tutorial value so a standard h sub 2 solution is developed in parallel

aspects of pairing inversion in this paper we discuss some applications of the pairing inversion problem and outline some potential approaches for solving it our analysis of these approaches gives further evidence that pairing inversion is a difficult problem

the eta pairing revisited in this paper we simplify and extend the eta pairing originally discovered in the setting of supersingular curves by barreto to ordinary curves furthermore we show that by swapping the arguments of the eta pairing one obtains a very efficient algorithm resulting in a speed up of a factor of around six over the usual tate pairing in the case of curves that have large security parameters complex multiplication by an order of qopf radic 3 and when the trace of frobenius is chosen to be suitably small other more minor savings are obtained for more general curves

on a new method for h_ 2 based decomposition this note proposes a simple linear matrix inequality lmi system for the design of static precompensators to reduce the interactions of a multivariable system the approach is based on minimizing the h2 norm of a modified system and the lmis are derived from the well known technique of pseudodiagonalisation the approach is applied to two complex real life benchmark problems with high levels of interaction it is shown that its performance is significantly better than previously proposed lmi optimization techniques for designing static precompensators

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

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

a characterization of all solutions to the four block general distance problem all solutions to the four block general distance problem which arises in h infty optimal control are characterized the procedure is to embed the original problem in an all pass matrix which is constructed it is then shown that part of this all pass matrix acts as a generator of all solutions special attention is given to the characterization of all optimal solutions by invoking a new descriptor characterization of all pass transfer functions as an application necessary and sufficient conditions are found for the existence of an h infty optimal controller following that a descriptor representation of all solutions is derived

improved miller s algorithm for computing pairings on edwards curves since edwards curves were introduced to elliptic curve cryptography by bernstein and lange in 2007 they have received a lot of attention due to their very fast group law operation pairing computation on such curves is slightly slower than on weierstrass curves however in some pairing based cryptosystems they might require a number of scalar multiplications which is time consuming operation and this can be advantageous to use edwards in this scenario in this paper we present a variant of miller s algorithm for pairing computation on edwards curves our approach is generic it is able to compute both weil and tate pairings on pairing friendly edwards curves of any embedding degree our analysis shows that the new algorithm is faster than the previous algorithms for odd embedding degree and as fast as for even embedding degree hence the new algorithm is suitable for computing optimal pairings and in situations where the denominators elimination technique is not possible

an improvement of the elliptic net algorithm in this paper we propose a modified elliptic net algorithm to compute pairings by reducing the number of the intermediate variables which should be updated in the iteration loop of the elliptic net algorithm we speed up the computation of pairings experimental results show that the proposed method is more efficient than the original elliptic net algorithm on barreto naehrig bn curves which are the best choice for implementing pairings at 128 bit security level

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