how to compute modular exponentiation with large operators based on the right to left binary algorithm abstract when the lengths of the operators are at least 1024 binary or 300 decimal digits modular exponentiation can be time consuming and is often the dominant part of the computation in many computer algebra systems the prime approach on this computational problem is known as the square and multiply method which includes two versions i e the left to right binary algorithm and the right to left binary algorithm for the past years too many attentions have been paid to propose the fast modular exponentiation methods based on the left to right binary algorithm however extremely few attentions have been paid on developing the fast modular exponentiation methods based on the right to left binary algorithm in this paper we propose a t fold exponent method based on the right to left binary algorithm from the performance view our t fold exponent method is similar to the m ary method based on the left to right binary algorithm from the structure view our t fold exponent method offers a framework for the fast modular exponentiation methods based on the right to left binary algorithm more important it is the first but steady step to develop the fast modular exponentiation methods based on the right to left binary algorithm

ranks of elliptic curves this paper gives a general survey of ranks of elliptic curves over the field of rational numbers the rank is a measure of the size of the set of rational points the paper includes discussions of the birch and swinnertondyer conjecture the parity conjecture ranks in families of quadratic twists and ways to search for elliptic curves of large rank

feedback decoupling of structured systems the feedback decoupling of structured linear systems is considered using results gathered on decoupling and the graph characterization of the structure at infinity a necessary and sufficient decoupling condition for structured systems is developed this condition has a graphical interpretation in terms of shortest input output paths

analysis of the variable length nonzero window method for exponentiation abstract the variable length nonzero window method is a way of computing exponentiation and modular exponentiation this method has been analyzed but there is a substantial difference between the analysis and experimental results the difference results from some logical errors in the analysis we analyze the variable length nonzero window method by modeling it as a markov chain we consider all details and our analysis predicts the behavior of the variable length nonzero window method exactly

the double base number system and its application to elliptic curve cryptography we describe an algorithm for point multiplication on generic elliptic curves based on a representation of the scalar as a sum of mixed powers of 2 and 3 the sparseness of this so called double base number system combined with some efficient point tripling formulae lead to efficient point multiplication algorithms for curves defined over both prime and binary fields side channel resistance is provided thanks to side channel atomicity

a survey of public key cryptosystems we give an overview of the most important public key cryptosystems and discuss the difficult task of evaluating the merit of such systems

a knapsack public key cryptosystem based on elliptic curve discrete logarithm there is no question that knapsack systems still warrant continuous researches as a result of the np completeness nature the faster speed and a desire to have a wide variety of available cryptosystems this paper presents a new knapsack type public key cryptosystem based on a novel application of the elliptic curve logarithm problem the difficulty can be made sufficient to frustrate attacks against the survey major knapsack public key cryptosystems developed in the past more than 20 years and hope fully shed some light on the future of the knapsack cryptosystems the proposed new cryptosystem can be found some theoretic support on confidentiality

the attractor for a nonlinear reaction diffusion system in an unbounded domain in this paper the quasi linear second order parabolic systems of reaction diffusion type in an unbounded domain are considered our aim is to study the long time behavior of parabolic systems for which the nonlinearity depends explicitly on the gradient of the unknown functions to this end we give a systematic study of given parabolic systems and their attractors in weighted sobolev spaces dependence of the hausdorff dimension of attractors on the weight of the sobolev spaces is considered 2001 john wiley sons inc

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

the smallest grammar problem this paper addresses the smallest grammar problem what is the smallest context free grammar that generates exactly one given string spl sigma this is a natural question about a fundamental object connected to many fields such as data compression kolmogorov complexity pattern identification and addition chains due to the problem s inherent complexity our objective is to find an approximation algorithm which finds a small grammar for the input string we focus attention on the approximation ratio of the algorithm and implicitly the worst case behavior to establish provable performance guarantees and to address shortcomings in the classical measure of redundancy in the literature our first results are concern the hardness of approximating the smallest grammar problem most notably we show that every efficient algorithm for the smallest grammar problem has approximation ratio at least 8569 8568 unless p np we then bound approximation ratios for several of the best known grammar based compression algorithms including lz78 b isection sequential longest match greedy and re pair among these the best upper bound we show is o n sup 1 2 we finish by presenting two novel algorithms with exponentially better ratios of o log sup 3 n and o log n m sup where m sup is the size of the smallest grammar for that input the latter algorithm highlights a connection between grammar based compression and lz77

global existence for reaction diffusion systems modelling ignition the pair of parabolic equations with a 0 and b 0 models the temperature and concentration for an exothermic chemical reaction for which just one species controls the reaction rate f of particular interest is the case where which appears in the frank kamenetskii approximation to arrhenius type reactions we show here that for a large choice of the nonlinearity f u v in 1 2 including the model case 3 the corresponding initial value problem for 1 2 in the whole space with bounded initial data has a solution which exists for all times finite time blow up might occur though for other choices of function f u v and we discuss here a linear example which strongly hints at such behaviour

achieving ntru with montgomery multiplication we propose a new unified architecture that utilizes the montgomery multiplication algorithm to perform a modular multiplication for both integers and binary polynomials and ntru s polynomial multiplications the unified design is capable of supporting a majority of public key cryptosystems such as ntru rsa diffie hellman key exchange and elliptic curve schemes among others furthermore the architecture is highly efficient in terms of area and speed

extended elliptic curve montgomery ladder algorithm over binary fields with resistance to simple power analysis in this paper we propose a scalar multiplication algorithm on elliptic curves over gf 2 m the proposed algorithm is an extended version of the montgomery ladder algorithm with the quaternary representation of the scalar in addition in order to improve performance we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm the proposed composite formulas are 2p 1 2p 2 3p 1 p 2 and 4p 1 where p 1 and p 2 are points on an elliptic curve they can be computed using only the x coordinate of a point p x y in the affine coordinate system however the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks because different operations are performed depending on the bits of the scalar unlike the original montgomery ladder algorithm therefore we combine the concept of the side channel atomicity with the proposed composite operation formulas to prevent simple power analysis furthermore to optimize the computational cost we use the montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system as the result the proposed scalar multiplication algorithm saves at least 26 of running time with small storage compared to the previous algorithms such as window based methods and comb based methods

computing elliptic curve discrete logarithms with the negation map it is clear that the negation map can be used to speed up the computation of elliptic curve discrete logarithms with the pollard rho method however the random walks defined on elliptic curve points equivalence class p used by pollard rho will always get trapped in fruitless cycles we propose an efficient alternative approach to resolve fruitless cycles besides the theoretical analysis we also examine the performance of the new algorithm in experiments with elliptic curve groups the experiment results show that we can achieve the speedup by a factor extremely close to 2 using the new algorithm with the negation map which is the best performance reported in the literature

conversion algorithms and implementations for koblitz curve cryptography in this paper we discuss conversions between integers and tau adic expansions and we provide efficient algorithms and hardware architectures for these conversions the results have significance in elliptic curve cryptography using koblitz curves a family of elliptic curves offering faster computation than general elliptic curves however in order to enable these faster computations scalars need to be reduced and represented using a special base tau expansion hence efficient conversion algorithms and implementations are necessary existing conversion algorithms require several complicated operations such as multiprecision multiplications and computations with large rationals resulting in slow and large implementations in hardware and microcontrollers with limited instruction sets our algorithms are designed to utilize only simple operations such as additions and shifts which are easily implementable on practically all platforms we demonstrate the practicability of the new algorithms by implementing them on altera stratix ii fpgas the implementations considerably improve both computation speed and required area compared to the existing solutions

efficient generation of minimal length addition chains an addition chain for a positive integer n is a set 1 a0 a1 ar n of integers such that for each i ge 1 a i a j a k for some k le j i this paper is concerned with some of the computational aspects of generating minimal length addition chains for an integer n particular attention is paid to various pruning techniques that cut down the search time for such chains certain of these techniques are influenced by the multiplicative structure of n later sections of the paper present some results that have been uncovered by searching for minimal length addition chains

a nonlinear elliptic curve cryptosystem based on matrices we propose a new mathematical problem that is applicable to public key cryptography based on the discrete logarithm problem dlp it uses certain elements formed by two matrices with elements in a finite field and a matrix whose elements are points of an elliptic curve with this system we get a larger key space without increasing the underlying elliptic curve and consequently without the computational requirements inherent to the set up of elliptic curves at random also we expose the diffie hellman key agreement protocol with this system acting as the underlying mathematical problem

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

evaluation of discrete logarithms in a group of p torsion points of an elliptic curve in characteristic p we show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs o ln p operations in this field

competition in ramped turing structures abstract stationary pattern selection and competition in the uniform brusselator in two 2d and three 3d dimensions are reviewed including reentrant hexagonal and striped zig zag phases influences of linear or chain like profiles of the pool chemicals on this selection are presented in the form of numerical experiments the relation with the recent experimental patterns obtained with the cima reaction is discussed

survey generic properties and control of linear structured systems a survey in this survey paper we consider linear structured systems in state space form where a linear system is structured when each entry of its matrices like a b c and d is either a fixed zero or a free parameter the location of the fixed zeros in these matrices constitutes the structure of the system indeed a lot of man made physical systems which admit a linear model are structured a structured system is representative of a class of linear systems in the usual sense it is of interest to investigate properties of structured systems which are true for almost any value of the free parameters therefore also called generic properties interestingly a lot of classical properties of linear systems can be studied in terms of genericity moreover these generic properties can in general be checked by means of directed graphs that can be associated to a structured system in a natural way we review here a number of results concerning generic properties of structured systems expressed in graph theoretic terms by properties we mean here system specific properties like controllability the finite and infinite zero structure and so on as well as solvability issues of certain classical control problems like disturbance rejection input output decoupling and so on in this paper we do not try to be exhaustive but instead by a selection of results we would like to motivate the reader to appreciate what we consider as a wonderful modelling and analysis tool we emphasize the fact that this modelling technique allows us to get a number of important results based on poor information on the system only moreover the graph theoretic conditions are intuitive and are easy to check by hand for small systems and by means of well known polynomially bounded combinatorial techniques for larger systems

linear and sublinear time algorithms for the basis of abelian groups it is well known that every finite abelian group g can be represented as a direct product of cyclic groups email protected g 1xg 2x xg t where each g i is a cyclic group of order p j for some prime p and integer j 1 if a i generates the cyclic group of g i i 1 2 t then the elements a 1 a 2 a t are called a basis of g we show a randomized algorithm such that given a set of generators m x 1 x k for an abelian group g and the prime factorization of order ord x i i 1 k it computes a basis of g in o m logn email protected i 1 tn ip i n i 2 time where n g has prime factorization p 1 n 1p 2 n 2 p t n t which is not a part of input this generalizes buchmann and schmidt s algorithm that takes o m g time in another model all elements in an abelian group are put into a list as a part of input we obtain an o n time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group

paper transfer matrix approach to the triangular block decoupling problem the triangular block decoupling problem for linear multivariable systems is studied via the transfer matrix approach this approach clearly separates the problem of admissible control laws from the one of desired decoupled system specifications necessary and sufficient triangular decoupling conditions are given for various control laws these conditions are expressed in a very simple way in terms of linear dependance among the transfer matrix rows it turns out that when the problem is solvable this can be done by static state feedback on a minimal realization of the system furthermore it is shown that whenever triangular block decoupling is possible it is also attainable with stability

a hardware algorithm for modular multiplication division a mixed radix 4 2 algorithm for modular multiplication division suitable for vlsi implementation is proposed the algorithm is based on montgomery method for modular multiplication and on the extended binary gcd algorithm for modular division both algorithms are modified and combined into the proposed algorithm so that almost all the hardware components are shared the new algorithm carries out both calculations using simple operations such as shifts additions and subtractions the radix 2 signed digit representation is used to avoid carry propagation in all additions and subtractions a modular multiplier divider based on the algorithm performs an n bit modular multiplication division in o n clock cycles where the length of the clock cycle is constant and independent of n the modular multiplier divider has a linear array structure with a bit slice feature and can be implemented with much smaller hardware than that necessary to implement both multiplier and divider separately

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

blowup in reaction diffusion systems with dissipation of mass we prove possible blowup in finite time of the solutions to reaction diffusion systems which preserve nonnegativity and for which the total mass of the components is nonincreasing in time two natural properties in applications this is done by presenting explicit counterexamples constructed with the help of formal computation software several partial results of global existence had been obtained previously in the literature our counterexamples explain a posteriori why extra conditions were needed negative results of independent interest are also provided as a by product for linear parabolic equations in nondivergence form and with discontinuous coefficients and for nonlinear hamilton jacobi evolution equations

on the structure at infinity of a structured system abstract we develop a graph theoretic characterization of the generic structure at infinity of the transfer matrix of a structured system we show that the generic structure at infinity can be determined by means of algorithms from combinatorial optimization based on the max flow min cut theorem and on results concerning minimal cost flows as an application of the obtained characterization we propose a structural version of two well known disturbance decoupling problems and we derive graph theoretic necessary and sufficient conditions for the solvability of each of the two problems

arithmetic operations in finite fields of medium prime characteristic using the lagrange representation in this paper we propose a complete set of algorithms for the arithmetic operations in finite fields of prime medium characteristic the elements of the fields ifpk are represented using the newly defined lagrange representation where polynomials are expressed using their values at sufficiently many points our multiplication algorithm which uses a montgomery approach can be implemented in o k multiplications and o k2 log k additions in the base field ifp for the inversion we propose a variant of the extended euclidean gcd algorithm where the inputs are given in the lagrange representation the lagrange representation scheme and the arithmetic algorithms presented in the present work represent an interesting alternative for elliptic curve cryptography

construction of cm picard curves in this article we generalize the cm method for elliptic and hyperelliptic curves to picard curves we describe the algorithm in detail and discuss the results of our implementation

a behavioral approach to estimation and dead beat observer design with applications to state space models the observer design problem is investigated in the context of linear left shift invariant discrete behaviors whose trajectories have supports on z necessary and sufficient conditions for the existence of a dead beat observer of some relevant variables from some measured ones in the presence of some unmeasured and irrelevant variables are introduced and a complete parametrization of all dead beat observers is given equivalent conditions for the existence of causal dead beat observers are then derived finally several classical problems addressed for state space models like state estimation the design of unknown input observers or the design of fault detectors and identifiers possibly in the presence of disturbances are cast in this general framework and the aforementioned equivalent conditions and parametrizations are specialized to these cases

attractors of reaction diffusion systems in unbounded domains and their spatial complexity the nonlinear reaction diffusion system in an unbounded domain is studied it is proven that under some natural assumptions on the nonlinear term and on the diffusion matrix this system possesses a global attractor in the corresponding phase space since the dimension of the attractor happens to be infinite we study its kolmogorov s entropy upper and lower bounds of this entropy are obtained moreover we give a more detailed study of the attractor for the spatially homogeneous rde in n in this case a group of spatial shifts acts on the attractor in order to study the spatial complexity of the attractor we interpret this group as a dynamical system with multidimensional time if n 1 acting on a phase space it is proven that the dynamical system thus obtained is chaotic and has infinite topological entropy in order to clarify the nature of this chaos we suggest a new model dynamical system that generalizes the symbolic dynamics to the case of the infinite entropy and construct the homeomorphic and even lipschitz continuous embedding of this system into the spatial shifts on the attractor finally we consider also the temporal evolution of the spatially chaotic structures in the attractor and prove that the spatial chaos is preserved under this evolution 2003 wiley periodicals inc

id based threshold digital signature schemes on the elliptic curve discrete logarithm problem we present three identification group oriented t n threshold digital signature schemes based on the elliptic curve discrete logarithm problem with a view to optimizing these techniques for implementation within a network by employing these schemes any t out of n users in a group can represent this group to sign the group signature this paper has the following five notable advantages 1 this work demonstrates the feasibility of constructing a fast and extremely secure user identification system 2 any group signature can be mutually generated by at least t group member 3 the signature verification process is simplified because there is only one group public key required 4 the group signature can be verified by any outsider 5 these schemes offer higher speeds lower power consumption smaller bandwidth requirements and smaller certificate sizes than other existing public key schemes

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

the montgomery modular inverse revisited we modify an algorithm given by kaliski to compute the montgomery inverse of an integer modulo a prime number we also give a new definition of the montgomery inverse and introduce efficient algorithms for computing the classical modular inverse the kaliski montgomery inverse and the new montgomery inverse the proposed algorithms are suitable for software implementations on general purpose microprocessors

observer based fault detection and isolation for structured systems fault detection and isolation fdi problems are here consid ered for linear systems with faults and disturbances we design a set of ob server based residuals in such a way that the transfer from the disturbances to the residuals is zero and the transfer from the faults to the residuals al lows fault isolation we are interested in obtaining a transfer function from faults to residuals with either a diagonal structure i e a dedicated struc tured residuals set or a triangular one we deal with this problem when the system under consideration is structured that is the entries of the system matrices are either fixed zeros or free parameters to a structured system one can associate in a natural way a directed graph we can then provide necessary and sufficient conditions under which the fdi problems have a solution for almost any value of the free parameters these conditions are simply expressed in terms of input output paths in the associated graph index terms fault detection and isolation graph theory model based approach structured residuals structured systems

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

attractor and spatial chaos for the brusselator in rn abstract brusselator is an important dynamical system which can be described by a reaction diffusion system in this paper it is proved that this reaction diffusion system possesses a global attractor a in the corresponding phase space the upper and lower bounds of kolmogorov e entropy of a are obtained moreover we give a more detailed study of spatial chaos of the attractor a for the brusselator in r n we interpret a group of spatial shifts as a dynamical system which acts on the attractor a by using the technique of unstable manifolds it is proved that this dynamical system is chaotic in order to clarify the nature of this chaos we construct the lipschitz continuous homeomorphic embedding of a typical model dynamical system whose chaotic behavior is evident into the spatial shifts on the attractor a this typical dynamical system generalizes the symbolic system it was first introduced by zelik

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

analysis of linear structured systems using a primal dual algorithm in this paper we give a polynomial algorithm to compute the infinite structure of a structured system a directed graph is associated with such structured systems the infinite zero orders can be computed on the associated graph via the determination of the minimal length of vertex disjoint input output paths this search corresponds to a minimum cost flow determination on an appropriate directed graph the proposed algorithm is based on the primal dual algorithm linked to linear programming this polynomial algorithm is one of the most efficient for this type of problem moreover it allows an iterative determination of the generic infinite structure which is a key tool for solving numerous control problems

multicore curve based cryptoprocessor with reconfigurable modular arithmetic logic units over gf 2 n this paper presents a reconfigurable curve based cryptoprocessor that accelerates scalar multiplication of elliptic curve cryptography ecc and hyperelliptic curve cryptography hecc of genus 2 over gf 2n by allocating a copies of processing cores that embed reconfigurable modular arithmetic logic units malus over gf 2n the scalar multiplication of ecc hecc can be accelerated by exploiting instruction level parallelism ilp the supported field size can be arbitrary up to a n 1 1 the superscaling feature is facilitated by defining a single instruction that can be used for all field operations and point divisor operations in addition the cryptoprocessor is fully programmable and it can handle various curve parameters and arbitrary irreducible polynomials the cost performance and security trade offs are thoroughly discussed for different hardware configurations and software programs the synthesis results with a 0 13 mum cmos technology show that the proposed reconfigurable cryptoprocessor runs at 292 mhz whereas the field sizes can be supported up to 587 bits the compact and fastest configuration of our design is also synthesized with a fixed field size and irreducible polynomial the results show that the scalar multiplication of ecc over gf 2163 and hecc over gf 283 can be performed in 29 and 63 mus respectively

a graph theoretic approach to fault detection and isolation for structured bilinear systems in this paper using a graph theoretic approach we address some issues related to the fault detection and isolation for structured bilinear systems considering a structured bilinear system submitted to faults and disturbances we give necessary and sufficient conditions to the solvability of the so called bilinear fundamental problem of residual generation we also treat the cases where the system is submitted to multiple failures occurring simultaneously or only one at a time one of the main advantages of the proposed analysis tool is that all the given conditions are easy to check because they deal with finding paths in a digraph this makes our approach well suited to studying large scale systems

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

efficient elliptic curve scalar multiplication algorithms resistant to power analysis this paper presents four algorithms for securing elliptic curve scalar multiplication against power analysis the highest weight binary form hbf of scalars and randomization are applied to resist power analysis by using a special method to recode the scalars the proposed algorithms do not suffer from simple power analysis spa with the randomization of the secret scalar or base point three of the four algorithms are secure against differential power analysis dpa refined power analysis rpa and zero value point attacks zpa the countermeasures are also immune to the doubling attack fast shamir s method is used in order to improve the efficiency of parallel scalar multiplication compared with previous countermeasures the new countermeasures achieve higher security and do not impact overall performance

optimal left to right binary signed digit recoding this paper describes new methods for producing optimal binary signed digit representations this can be useful in the fast computation of exponentiations contrary to existing algorithms the digits are scanned from left to right i e from the most significant position to the least significant position this may lead to better performances in both hardware and software

complexity and fast algorithms for multiexponentiations in this paper we propose new algorithms for multiple modular exponentiation operations the major aim of these algorithms is to speed up the performance of some cryptographic protocols based on multiexponentiation our new algorithms are based on binary like complex arithmetic introduced by k pekmestzi 1989 and generalized in this paper

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

low latency digit serial systolic double basis multiplier over mbi gf 2 m using subquadratic toeplitz matrix vector product approach recently in cryptography and security the multipliers with subquadratic space complexity for trinomials and some specific pentanomials have been proposed for such kind of multipliers alternatively we use double basis multiplication which combines the polynomial basis and the modified polynomial basis to develop a new efficient digit serial systolic multiplier the proposed multiplier depends on trinomials and almost equally space pentanomials aesps and utilizes the subquadratic toeplitz matrix vector product scheme to derive a low latency digit serial systolic architecture if the selected digit size is d bits the proposed digit serial multiplier for both polynomials i e trinomials and aesps requires the latency of 2 left lceil sqrt m over d right rceil while traditional ones take at least o left left lceil m over d right rceil right clock cycles analytical and application specific integrated circuit asic synthesis results indicate that both the area and the time times area complexities of our proposed architecture are significantly lower than the existing digit serial systolic multipliers

efficient multiplier architectures for galois fields gf 2 sup 4n this contribution introduces a new class of multipliers for finite fields gf 2 sup n sup 4 the architecture is based on a modified version of the karatsuba ofman algorithm koa by determining optimized field polynomials of degree four the last stage of the koa and the module reduction can be combined this saves computation and area in vlsi implementations the new algorithm leads to architectures which show a considerably improved gate complexity compared to traditional approaches and reduced delay if compared with koa based architectures with separate module reduction the new multipliers lead to highly modular architectures and are thus well suited for vlsi implementations three types of field polynomials are introduced and conditions for their existence are established for the small fields where n 2 3 8 which are of primary technical interest optimized field polynomials were determined by an exhaustive search for each field order exact space and time complexities are provided

sensor location for diagnosis in linear systems a structural analysis we consider here the fault detection and isolation fdi problem for linear systems we are interested in designing a set of observer based residuals in such a way that the transfer from the faults to the residuals is diagonal and the transfer from the disturbances to the residuals is zero we deal with this problem when the system under consideration is structured that is the entries of the system matrices are either fixed zeros or free parameters this problem can be solved in terms of the graph that can be associated in a natural way with a structured system when the fdi solvability conditions are not satisfied we assume that internal variables can be measured at a cost and look into the question of wether the problem is solvable with these new measurements we give solvability conditions for a solution with a minimal number of additional sensors and among such solutions provide a minimal cost solution for the sensor location problem under consideration we pay particular attention to the internal analysis of the system and we propose a structural decomposition of the system associated graph based on some particular separators this analysis leads to the definition of a reduced system we prove that some potential additional sensors are inefficient for solving our fdi problem and that the fdi problem can be solved using only measurements on the reduced system

a w nnaf method for the efficient computation of scalar multiplication in elliptic curve cryptography based on the non adjacent form naf expression here we present a new method to express a large positive integer k with the base of any positive integer w other than 2 we call this method the w nnaf method as the expression is near to the naf one and with base w this expression leads to the minimal hamming weight of all the base w expressions based on the proposed expression and some existing methods an algorithm is developed to compute kp efficiently

computing l l isogenies in polynomial time on jacobians of genus 2 curves in this paper we compute l isogenies between abelian varieties over a field of characteristic different from 2 in polynomial time in l when l is an odd prime which is coprime to the characteristic we use level n symmetric theta structure where n 2 or n 4 in a second part of this paper we explain how to convert between mumford coordinates of jacobians of genus 2 hyperelliptic curves to theta coordinates of level 2 or 4 combined with the preceding algorithm this gives a method to compute l l isogenies in polynomial time on jacobians of genus 2 curves

efficient algorithms for speeding up the computations of elliptic curve cryptosystems as we know the performance of the elliptic curve cryptosystem ecc deeply depends on the computation of scalar multiplication thus how to speed up the computation of the elliptic curve scalar multiplication is a significant issue in 1994 lim and lee proposed a more flexible precomputation method used in wireless networks environments for speeding up the computation of exponentiation this method can be also used for speeding up the scalar multiplication of elliptic curves we call it llecc method however the less storage is equipped with the computing devices the less efficient it is for this reason we propose a more efficient algorithm than llecc s in this paper first we modify llecc method to reduce the storage of precomputed values and then propose an efficient algorithm based on the nonadjacent form naf representation and multidoubling method furthermore the proposed algorithm can be also used for speeding up the multi point multiplication of elliptic curves according to the simulation results the proposed algorithm can reduce 11 and 21 in the aspect of the computational complexity and storage cost respectively in an elliptic curve of size 160 bit over finite fields with characteristic greater than 3 as compared with llecc s finally we implement the elliptic curve digital signature algorithm like ecdsa like system in the personal digital assistant pda using the proposed algorithms to improve the scalar multiplication

low weight polynomial form integers for efficient modular multiplication in 1999 solinas introduced families of moduli called the generalized mersenne numbers gmns which are expressed in low weight polynomial form p f t where t is limited to a power of 2 gmns are very useful in elliptic curve cryptosystems over prime fields since modular reduction by a gmn requires only integer additions and subtractions however since there are not many gmns and each gmn requires a dedicated implementation gmns are hardly useful for other cryptosystems here we modify gmn by removing restriction on the choice of t and restricting the coefficients of f t to 0 and plusmn1 we call such families of moduli low weight polynomial form integers lwpfis we show an efficient modular multiplication method using lwpfi moduli lwpfis allow general implementation and there exist many lwpfi moduli one may consider lwpfis as a trade off between general integers and gmns

disturbance rejection for structured systems the disturbance rejection problem for structured systems is solved a graph is associated with such a structured system disturbance rejection is shown to be solvable by state feedback whenever the control inputs reach the outputs more quickly than the disturbances in the associated graph the infinite structure of a structured system is characterized in terms of input output paths in the associated graph

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 generic polar harmonic transforms generic polar harmonic transforms have recently been proposed to extract rotation invariant features from images and their usefulness has been demonstrated in a number of pattern recognition problems however direct computation of these transforms from their definition is inefficient and is usually slower than some efficient computation strategies that have been proposed for other methods this paper presents a number of novel computation strategies to compute these transforms rapidly the proposed methods are based on the inherent recurrence relations among complex exponential and trigonometric functions used in the definition of the radial and angular kernels of these transforms the employment of these relations leads to recursive and addition chain based strategies for fast computation of harmonic function based kernels experimental results show that the proposed method is about 10 times faster than direct computation and 5 times faster than fast computation of zernike moments using the q recursive strategy thus among all existing rotation invariant feature extraction methods polar harmonic transforms are the fastest

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

exponentiation using division chains exponentiation may be performed faster than the traditional square and multiply method by iteratively reducing the exponent module numbers which as exponents themselves require few multiplications this mainly includes those with few nonzero bits for a suitable choice of such divisors the resulting mixed basis representation of the exponent reduces the expected number of nonsquaring multiplications by over half at the cost of a single extra register preprocessing effort depends entirely on the exponent and can be kept down to the work saved in a single exponentiation moreover no precomputed look up tables are required so the method is especially applicable where space is at a premium in particular it outperforms the instance of the m ary method which uses the same space however for 512 bit exponents it beats every instance of the m ary method achieving well under 635 multiplications on average both hardware and software implementations of the rsa crypto system can benefit from this algorithm

computing sequences with addition chains given a sequence n 1 cdots n m of positive integers what is the smallest number of additions needed to compute all m integers starting with 1 this generalization of the addition chain m 1 problem will be called the addition sequence problem we show that the sequence 2 0 2 1 cdots 2 n 1 2 n 1 can be computed with n 2 13 sqrt n log n additions and that n sqrt n 2 is a lower bound this lower bound result is applied to show that the addition sequence problem is np complete

on pattern structures of the n soliton solution of the discrete kp equation over a finite field the existence and properties of coherent pattern in the multisoliton solutions of the dkp equation over a finite field are investigated to that end starting with an algebro geometric construction over a finite field we derive a travelling wave formula for n soliton solutions in a finite field however despite it having a form similar to its analogue in the complex field case the finite field solutions produce patterns essentially different from those of classical interacting solitons

fast exponentiation using data compression we present the first exponentiation algorithm that uses the entropy of the source of the exponent to improve on existing exponentiation algorithms when the entropy is smaller than 1 w s l s 1 where w s is the hamming weight of the exponent and l s is its length for entropy 1 it is comparable to the best known general purpose exponentiation algorithms

an efficient undeniable group oriented signature scheme three main methods with favorable security and efficiency can be considered in the design of cryptosystems these methods are integer factorization systems of which rsa is the best known example discrete logarithm systems such as dsa and elliptic curve cryptosystems ecc math comput 48 1987 203 advances in cryptology crypto 85 springer verlag 1986 p 417 the elliptic curve cryptosystem has been applied in this study because of its low computational cost and short key size which reduces the overheads in a limited hardware environment thus the elliptic curve constitutes a suitable environment for the cryptosystems this study improves the undeniable signature scheme proceedings of the eleventh national conference on information security 2001 p 331 by reducing its time complexity such an improvement makes the proposed scheme more efficient and secure

brief generic solvability of the failure detection and identification problem necessary and sufficient conditions for the generic solvability of the extended fundamental problem in residual generation efprg massoumnia et al 1989 ieee trans automat control 34 316 321 in the case of simultaneous actuator and sensor failures are presented in the process of deriving these conditions the generic forms of supremal controllability and infimal unobservability subspaces of lti systems having diagonal state space matrices are obtained

security analysis of the public key algorithm based on chebyshev polynomials over the integer ring zn recently kocarev and tasev 20 proposed to use chebyshev polynomials over real numbers to design a public key algorithm by employing the semigroup property bergamo et al 4 pointed out that the public key algorithm based on chebyshev polynomials working on real numbers is not secure and devised an attack which permits to recover the corresponding plaintext from a given ciphertext later kocarev et al 19 generalized the chebyshev polynomials from real number fields to finite fields and finite rings to make the public key algorithm more secure and practical however we analyzed the period distribution of the sequences generated by the chebyshev polynomials over finite fields 21 when the modulus n is prime we found this algorithm was also not secure and proposed an attack on this algorithm over finite fields we then proposed some schemes to improve the security in this paper we further analyze in detail the period distribution of the sequences generated by chebyshev polynomials over the integer ring z n when n is composite it turns out that the period distribution is poor if n is not chosen properly and there are many small periods which are not secure in the sense of cryptology based on these findings we devise an attack on the public key algorithm based on chebyshev polynomials over the integer ring z n we also propose some suggestions to avoid this attack