moment estimate and existence for solutions of stochastic functional differential equations abstract in this paper we give the existence uniqueness theorems and the moment estimates of solutions for a large class of sfdes these estimates improve and extend some related results including exponential stability decay stability and asymptotic behavior their corollaries improve and extend the classical halanay inequality and some of its generalizations moreover the stochastic version of the wintner theorem in continuous function space is established by the compare principle which improves and extends the main results of xu et al 2008 2013 when the methods presented are applied to the sfdes with impulses and sfdes in hilbert spaces we extend the related results of govindan and ahmed 2013 liu et al 2007 2010 vinodkumar 2010 and xu et al 2012 two examples are provided to illustrate the effectiveness of our results

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

integration with respect to fractal functions and stochastic calculus i the classical lebesgue stieltjes integral b a fdg of real or complex valued functions on a finite interval a b is extended to a large class of integrands f and integrators g of unbounded variation the key is to use composition formulas and integration by part rules for fractional integrals and weyl derivatives in the special case of holder continuous functions f and g of summed order greater than 1 convergence of the corresponding riemann stieltjes sums is proved the results are applied to stochastic integrals where g is replaced by the wiener process and f by adapted as well as anticipating random functions in the anticipating case we work within slobodeckij spaces and introduce a stochastic integral for which the classical ito formula remains valid moreover this approach enables us to derive calculation rules for pathwise defined stochastic integrals with respect to fractional brownian motion

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

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

a random fixed point theorem and the random graph transformation in the first part of this article we formulate a random fixed point theorem for random dynamical systems where a contraction condition is formulated as an expectation of a particular expression in the following we are going to formulate a random graph transformation this transformation defines a random dynamical system under certain conditions this random dynamical system has a random fixed point which can be found by the mentioned random fixed point theorem in one application we show that random dynamical systems given by particular differential equations have a random unstable invariant manifold

conjugate dynamics on center manifolds for stochastic partial differential equations abstract in this paper we prove that for a random differential equation with the driving noise constructed from a q wiener process and the wiener shift there exists a local center unstable stable center unstable center stable manifold and a local stable foliation an unstable foliation on the center unstable manifold and a stable foliation on the center stable manifold the smoothness of which depend on the vector fields of the equation also we show that any two arbitrarily local center manifolds constructed as above are conjugate

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

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

evolution equations driven by a fractional brownian motion abstract in this paper we study nonlinear stochastic evolution equations in a hilbert space driven by a cylindrical fractional brownian motion with hurst parameter h 1 2 and nuclear covariance operator we establish the existence and uniqueness of a mild solution under some regularity and boundedness conditions on the coefficients and for some values of the parameter h this result is applied to stochastic parabolic equation perturbed by a fractional white noise in this case if the coefficients are lipschitz continuous and bounded the existence and uniqueness of a solution holds if h d 4 the proofs of our results combine techniques of fractional calculus with semigroup estimates

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

the stable manifold theorem for stochastic differential equations type equations are treated starting with the existence of a stochastic flow for a sde we introduce the notion of a hyperbolic stationary trajectory we prove the existence of invariant random stable and unstable manifolds in the neighborhood of the hyperbolic stationary solution for stratonovich sdes the stable and unstable manifolds are dynamically characterized using forward and backward solutions of the anticipating sde the proof of the stable manifold theorem is based on ruelle oseledec multiplicative ergodic theory

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

unstable invariant manifolds for stochastic pdes driven by a fractional brownian motion abstract in this paper we consider a class of stochastic partial differential equations spdes driven by a fractional brownian motion fbm with the hurst parameter bigger than 1 2 the existence of local random unstable manifolds is shown if the linear parts of these spdes are hyperbolic for this purpose we introduce a modified lyapunov perron transform which contains stochastic integrals by the singularities inside these integrals we obtain a special lyapunov perron s approach by treating a segment of the solution over time interval 0 1 as a starting point and setting up an infinite series equation involving these segments as time evolves using this approach we establish the existence of local random unstable manifolds in a tempered neighborhood of an equilibrium

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

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

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

invariant manifolds for stochastic partial differential equations invariant manifolds provide the geometric structures for describing and understanding dynamics of nonlinear systems the theory of invariant manifolds for both finite and infinite dimensional autonomous deterministic systems and for stochastic ordinary differential equations is relatively mature in this paper we present a unified theory of invariant manifolds for infinite dimensional random dynamical systems generated by stochastic partial differential equations we first introduce a random graph transform and a fixed point theorem for nonautonomous systems then we show the existence of generalized fixed points which give the desired invariant manifolds

iterative wiener filters for image restoration the iterative wiener filter which successively uses the wiener filtered signal as an improved prototype to update the covariance estimates is investigated the convergence properties of this iterative filter are analyzed it has been shown that this iterative process converges to a signal which does not correspond to the minimum mean squared error solution based on the analysis an alternate iterative filter is proposed to correct for the convergence error the theoretical performance of the filter has been shown to give minimum mean squared error in practical implementation when there is unavoidable error in the covariance computation the filter may still result in undesirable restoration its performance has been investigated and a number of experiments in a practical setting were conducted to demonstrate its effectiveness

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

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

image reconstruction and restoration overview of common estimation structures and problems developments in the theory of image reconstruction and restoration over the past 20 or 30 years are outlined particular attention is paid to common estimation structures and to practical problems not properly solved yet the problem of image reconstruction and restoration is first formulated some of the current regularization approaches used to solve the problem are then described the concepts of a priori information and compound criterion are introduced a bayesian interpretation of the regularization techniques is given which clarifies the role of the tuning parameters and indicates how they could be estimated the practical aspects of computing the solution first when the hyperparameters are known and second when they must be estimated are then considered conclusions are drawn and points that still need to be investigated are outlined

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

invariant manifolds for stochastic wave equations abstract in this paper we consider a class of stochastic wave equations with nonlinear multiplicative noise we first show that these stochastic wave equations generate random dynamical systems or stochastic flows by transforming the stochastic wave equations to random wave equations through a stationary random homeomorphism then we establish the existence of random invariant manifolds for the random wave equations due to the temperedness of the nonlinearity we obtain only local invariant manifolds no matter how large the spectral gap is unlike the deterministic cases based on these random dynamical systems we prove the existence of random invariant manifolds in a tempered neighborhood of an equilibrium finally we show that the images of these invariant manifolds under the inverse stationary transformation give invariant manifolds for the stochastic wave equations

restoration of differently blurred versions of an image with measurement errors in the psf s restoration of an object from t observations is considered each image is distorted by a different deterministic blur and additive noise the point spread function psf for each observation is unknown however a noisy measurement of it is available taking the errors in measurements of the psfs into consideration the maximum likelihood and wiener filters are derived it is shown that these filters give better results when the regression filter and the conventional wiener filter i e the one which ignores the presence of the noise in the psfs the consistency and the ill conditioning characteristics of the filters are discussed regularized forms for these filers are obtained

time space fractional stochastic ginzburg landau equation driven by fractional brownian motion abstract the current paper is devoted to the time space fractional ginzburg landau equation driven by fractional brownian motion the spatial temporal regularity of the nonlocal stochastic convolution is firstly established and then the existence and uniqueness of the global mild solution are obtained by the banach fixed point theorem and mittag leffler functions finally the numerical simulations for the time fractional stochastic ginzburg landau equation are provided to verify the mathematical analysis results

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

elliptic curve scalar multiplication algorithm using complementary recoding elliptic curve based cryptosystem is an efficient public key cryptosystem which is more suitable for limited environments the very expensive operation in elliptic curve based cryptographic protocol is the scalar multiplication in this paper scalar multiplication using complementary recoding method is proposed the intermediate operations carried out in conversion are less than the other existing scalar multiplication algorithm the efficiency of our proposed method is tested and found that our method is an efficient than the other methods

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

a recursive algorithm for maximum likelihood based identification of blur from multiple observations a maximum likelihood based method is proposed for blur identification from multiple observations of a scene when the relations among the blurring functions are known the estimate of blur obtained using the proposed method is very good since direct computation of the likelihood function becomes difficult as the number of images increases we propose an algorithm to compute the likelihood function recursively

on eigenstructure based direct multichannel blind image restoration existing eigenstructure based direct multichannel blind image restoration techniques include nullspace based and direct deconvolver estimation techniques the nullspace based approach can be formulated as an optimization problem we show that this formulation implies a new subspace based approach that uses matrix operations this new approach has the same advantages as the nullspace based one but requires less computational complexity under some mild conditions its complexity is equal to that of the fft furthermore the relation among the nullspace based approach the direct deconvolver estimation and the new subspace based approach is studied

exponential stability of stochastic evolution equations driven by small fractional brownian motion with hurst parameter in 1 2 1 abstract this paper addresses the exponential stability of the trivial solution of some types of evolution equations driven by holder continuous functions with holder index greater than 1 2 the results can be applied to the case of equations whose noisy inputs are given by a fractional brownian motion b h with covariance operator q provided that h 1 2 1 and tr q is sufficiently small

paper identification of scalar errors in variables models with dynamics the paper considers the task of identifying a causal linear dynamic system excited by stationary gaussian zero mean noise of unknown spectrum and given measurements of the system input and output contaminated by independent additive stationary noise signals of unknown spectra while the solution is normally not unique finite dimensional parameterizations of the solution set are given in some cases involving just one parameter even though the various spectra may not be rational

fast simultaneous scalar multiplication scalar multiplication is a very expensive operation in elliptic curve based cryptographic protocol in this paper we propose a modified simultaneous elliptic curve scalar multiplication algorithm using complementary recoding the signed binary representation of an integer is obtained by simple arithmetic operations such as complement and bitwise subtraction this is an efficient method to obtain the signed binary representation of scalar when compared to other standard methods such as naf mof and jsf

stochastic evolution equations with fractional brownian motion in this paper linear stochastic evolution equations driven by infinite dimensional fractional brownian motion are studied a necessary and sufficient condition for the existence and uniqueness of the solution is established and the spatial regularity of the solution is analyzed separate proofs are required for the cases of hurst parameter above and below 1 2 the particular case of the laplacian on the circle is discussed in detail

the wong zakai approximations of invariant manifolds and foliations for stochastic evolution equations abstract in this paper we study the wong zakai approximations given by a stationary process via the wiener shift and their associated dynamics of a class of stochastic evolution equations with a multiplicative white noise we prove that the solutions of wong zakai approximations almost surely converge to the solutions of the stratonovich stochastic evolution equation we also show that the invariant manifolds and stable foliations of the wong zakai approximations converge to the invariant manifolds and stable foliations of the stratonovich stochastic evolution equation respectively

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

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