uniform q series asymptotics for staircase polygons we present a uniform asymptotic expansion for the area perimeter generating function of staircase polygons by calculating the asymptotic behaviour of the alternating q series 1 phi 1 0 y q x as q to 1 from a new integral representation this leads to a direct calculation of the scaling function for this model

fluid dynamic limits of kinetic equations i formal derivations the connection between kinetic theory and the macroscopic equations of fluid dynamics is described in particular our results concerning the incompressible navier stokes equations are based on a formal derivation in which limiting moments are carefully balanced rather than on a classical expansion such as those of hilbert or chapman enskog the moment formalism shows that the limit leading to the incompressible navier stokes equations like that leading to the compressible euler equations is a natural one in kinetic theory and is contrasted with the systematics leading to the compressible navier stokes equations some indications of the validity of these limits are given more specifically the connection between the diperna lions renormalized solution of the classical boltzmann equation and the leray solution of the navier stokes equations is discussed

the alternating greedy expansion and applications to computing digit expansions from left to right in cryptography the central topic of this paper is the alternating greedy expansion of integers which is defined to be a binary expansion with digits 0 1 with the property that the nonzero digits have alternating signs we collect known results about this alternating greedy expansion and complement it with other useful properties and algorithms in the second part we apply it to give an algorithm for computing a joint expansion of d integers of minimal joint hamming weight from left to right i e from the column with the most significant bits towards the column with the least significant bits furthermore we also compute an expansion equivalent to the so caled w naf from left to right using the alternating greedy expansion

plane partition vesicles we examine partitions and their natural three dimensional generalizations plane partitions as models of vesicles undergoing an inflation deflation transition the phase diagrams of these models include a critical point corresponding to an inflation deflation transition and exhibits multicritical scaling in the vicinity of a multicritical point located elsewhere on the critical curve we determine the locations of the multicritical points by analysing the generating functions using analytic and numerical means in addition we determine the numerical values of the multicritical scaling exponents associated with the multicritical scaling regimes in these models

convex polyominoes and heaps of segments builds a one to one correspondence between some polyominoes called parallelogram polyominoes and some heaps of segments then the author derives explicit expressions for the generating function of convex polyominoes according to their height width and area the author also enumerates two subsets of convex polyominoes namely the directed and convex polyominoes and the parallelogram polyominoes according to these three parameters generating functions of polyominoes having a fixed width are also studied

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

global validity of the boltzmann equation for two and three dimensional rare gas in vacuum erratum and improved result we point out an error in our earlier papers 1 and 2 and present a more direct and natural proof which although based on the same physical ideas of the previous ones saves and actually improves the validity results for the boltzmann equation given in 1 and 2

fluid dynamic limits of kinetic equations ii convergence proofs for the boltzmann equation using relative entropy estimates about an absolute maxwellian it is shown that any properly scaled sequence of diperna lions renormalized solutions of some classical boltzmann equations has fluctuations that converge to an infinitesimal maxwellian with fluid variables that satisfy the incompressibility and boussinesq relations moreover if the initial fluctuations entropically converge to an infinitesimal maxwellian then the limiting fluid variables satisfy a version of the leray energy inequality if the sequence satisfies a local momentum conservation assumption the momentum densities globaly converge to a solution of the stokes equation a similar discrete time version of this result holds for the navier stokes limit with an additional mild weak compactness assumption the continuous time navier stokes limit is also discussed 01993 john wiley sons inc

fast exponentiation using split exponents we propose a new method to speed up discrete logarithm dl based cryptosystems by considering a new variant of the dl problem where the exponents are formed as e1 e2 for some fixed a and two integers e1 ae2 with a low weight representation we call this class of exponents split exponents and we show that with certain choice of parameters the dl problem on split exponents is essentially as secure as the standard dl problem while the exponentiation operation using exponents of this class is significantly faster than best exponentiation algorithms given for standard exponents for example the speed of scalar multiplication on the standard koblitz curve k163 is estimated to be accelerated by up to 51 5 and 23 5 at the cost of memory for one precomputed point compared to the tnaf and window tnaf methods respectively as for security we show that the provable security of the dl problem using split exponents is only by a small constant e g 1 4 worse than the security of the standard dl problem split exponents can be adopted to speed up various dl based cryptosystems we exemplify this on the recent cca secure public key encryption of bellare kohno and shoup

incompressible navier stokes and euler limits of the boltzmann equation we consider solutions of the boltzmann equation in a d dimensional torus d 2 3 for macroscopic times t n 1 t 0 when the space variations are on a macroscopic scale x n 1r n 2 x in the unit torus let u x t be for t t0 a smooth solution of the incompressible navier stokes equations ins for n 2 and of the incompressible euler equation ie for n 2 we prove that has solutions for t t0 which are close to o 2 in a suitable norm to the local maxwellian p 2 t d 2 exp v u x t 2 2t with constant density p and temperature t this is a particular case defined by the choice of initial values of the macroscopic variables of a class of such solutions in which the macroscopic variables satisfy more general hydrodynamical equations for n 3 these equations correspond to variable density ie while for n 2 they involve higher order derivatives of the density

some theorems on the rogers ramanujan continued fraction in ramanujan s lost notebook in his first two letters to g h hardy and in his notebooks ramanujan recorded many theorems about the rogers ramanujan continued fraction in his lost notebook he offered several further assertions the purpose of this paper is to provide proofs for many of the claims about the rogers ramanujan and generalized rogers ramanujan continued fractions found in the lost notebook these theorems involve among other things modular equations transformations zeros and class invariants

double point compression with applications to speeding up random point multiplication this paper presents two main results relating to elliptic curve cryptography first a double point compression scheme is proposed which allows a compact representation of elliptic curve points without the computational cost associated with ordinary single point compression a triple point compression scheme is also proposed which can result in more savings in memory and or bandwidth second a new approach to speeding up random point multiplication is given for the case where the base point is variable but available in a certificate in this approach some redundant information a few multiples of the base point is added to the certificate it is shown that a significant speed up can be obtained by optimizing the moller s algorithm for the case where only a portion of the lookup table is available it is also shown how to use redundant information to compute random point multiplication using parallel processors the proposed point compression schemes can be employed to reduce the required bandwidth when single point compression is computationally expensive

inflating square and rectangular lattice vesicles in this paper two simple models of lattice vesicles in two and in three dimensions are examined lattice vesicles have a rich mathematical nature and the combinatorial and scaling properties of square and rectangular lattice vesicle models are investigated and then generalized to three dimensions the phase diagram of the vesicles also includes a multicritical point and scaling of the generating functions of these models are studied to test the models for tricritical scaling behaviour in the models of square vesicles in the square lattice and cubical vesicles in the cubic lattice the tricritical scaling forms are explicitly confirmed however rectangular vesicles in the square lattice and rectangular box vesicles in the cubic lattice appear not to conform to the expected tricritical scaling forms

stacking models of vesicles and compact clusters we investigate three simple lattice models of two dimensional vesicles these models differ in their behavior from the universality class of partially convex polygons which has been recently established they do not have the tricritical scaling of those models and furthermore display a surprising feature their perimeter free energy is discontinuous with an isolated value at zero pressure we give the full asymptotic descriptions of the generating functions in area and perimeter variables from theq series solutions and obtain the scaling functions where applicable

scaling behaviour of two dimensional polygon models exactly solvable two dimensional polygon models counted by perimeter and area are described by q algebraic functional equations we provide techniques to extract the scaling behaviour of these models up to arbitrary order and apply them to some examples these are then used to analyze the unsolved model of self avoiding polygons where we numerically confirm predictions about its scaling function and its first two corrections to scaling

statistical properties of lorentz gas with periodic configuration of scatterers in our previous paper markov partitions for some classes of dispersed billiards were constructed using these partitions we estimate the decay of velocity auto correlation function and prove the central limit theorem of probability theory and donsker s invariance principle for lorentz gas with periodic configuration of scatterers

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

a scaling theory of the collapse transition in geometric cluster models of polymers and vesicles much effort has been expended in the past decade to calculate numerically the exponents at the collapse transition point in walk polygon and animal models the crossover exponent phi has been of special interest and sometimes is assumed to obey the relation 2 alpha 1 phi with the alpha the canonical thermodynamic exponent that characterizes the divergence of the specific heat the reasons for the validity of this relation are not widely known the authors present a scaling theory of collapse transitions in such models the free energy and canonical partition functions have finite length scaling forms whilst the grand partition function has a tricritical scaling form the link between the grand and canonical ensembles leads to the above scaling relation they then comment on the validity of current estimates of the crossover exponent for interacting self avoiding walks in two dimensions and propose a test involving the scaling relation which may be used to check these values

critical exponents from nonlinear functional equations for partially directed cluster models we present a method for the derivation of the generating function and computation of critical exponents for several cluster models staircase bar graph and directed column convex polygons as well as partially directed self avoiding walks starting with nonlinear functional equations for the generating function by linearizing these equations we first give a derivation of the generating functions the nonlinear equations are further used to compute the thermodynamic critical exponents via a formal perturbation ansatz alternatively taking the continuum limit leads to nonlinear differential equations from which one can extract the scaling function we find that all the above models are in the same universality class with exponents y 1 2 7 1 3 and b 2 3 all models have as their scaling function the logarithmic derivative of the airy function

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 new minimal average weight representation for left to right point multiplication methods this paper introduces a new radix 2 representation with the same average weight as the width w nonadjacent form w naf in both w naf and the proposed representations each nonzero digit is an odd integer with absolute value less than m however for w naf m is of the form 2 sup w 1 while for the proposed representation it can be any positive integer therefore using the proposed integer representation we can use the available memory efficiently which is attractive for devices with limited memory another advantage of the proposed representation over w naf is that it can be obtained by scanning the bits from left to right this property is also useful for memory constrained devices because it can reduce both the time and space complexity of fast point multiplication techniques

billiards in a general domain with random reflections we study stochastic billiards on general tables a particle moves according to its constant velocity inside some domain fancyscript d subset mathbb r d until it hits the boundary and bounces randomly inside according to some reflection law we assume that the boundary of the domain is locally lipschitz and almost everywhere continuously differentiable the angle of the outgoing velocity with the inner normal vector has a specified absolutely continuous density we construct the discrete time and the continuous time processes recording the sequence of hitting points on the boundary and the pair location velocity we mainly focus on the case of bounded domains then we prove exponential ergodicity of these two markov processes we study their invariant distribution and their normal gaussian fluctuations of particular interest is the case of the cosine reflection law the stationary distributions for the two processes are uniform in this case the discrete time chain is reversible though the continuous time process is quasi reversible also in this case we give a natural construction of a chord picked at random in fancyscript d and we study the angle of intersection of the process with a d 1 dimensional manifold contained in fancyscript d

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

fast multicomputation with asynchronous strategy we propose a new strategy to improve the performance of multicomputations such as computing xa yb in an additive group or a xby in a multiplicative group this new strategy is called asynchronous strategy in each iteration it tries to compute partial results by pairing nearby zero and nonzero bits the new strategy can be applied to any binary code of x and y to further reduce the computational cost however our experiments show that sparse signed digit code can achieve minimum computational cost the algorithms using the asynchronous strategy need only o h extra registers when they try to do computation with bits which are h positions apart

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

anomalous diffusion limit induced on a kinetic equation using a compactness argument based on the velocity averaging lemma of golse et al it is shown that the limiting behavior of a kinetic linearized bgk gas model confined between two plates with maxwell boundary conditions when the distance between the plates goes to zero under a suitable anomalous scaling is diffusive we do not require the use of central limit theorems as in the method of borgers et al

scaling function and universal amplitude combinations for self avoiding polygons we analyse new data for self avoiding polygons saps on the square and triangular lattices enumerated by both perimeter and area providing evidence that the scaling function is the logarithm of an airy function the results imply universal amplitude combinations for all area moments and suggest that rooted saps may satisfy a q algebraic functional equation

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

fast binary multiplication by performing dot counting and complement recoding abstract in this paper we present a new computation method by combining the dots counting method and the complement recoding method to efficiently evaluate modular multiplication in binary multiplications the lut look up table technique can be adopted in our proposed method to efficiently reduce the number of multiplications and the proposed method can be easily implemented for the hardware numerous examples are provided to show the efficient and easy operations of the binary multiplication method in average case the worst case for the hamming weight of the product x y is uv 4 where u is the bit length of multiplicand x v is the bit length of multiplier y if the proposed method is applied we could effectively reduce the hamming weight of the product x y to 0 25 u

on the asymptotics of the finite perimeter partition function of two dimensional lattice vesicles we derive the dominant asymptotic form and the order of the correction terms of the finite perimeter partition function of self avoiding polygons on the square lattice which are weighted according to their area a as qa in the inflated regime q 1 the approach q 1 of the asymptotic form is examined

algebraic languages and polyominoes enumeration abstract in this paper the use of algebraic languages theory in solving an open problem in combinatorics is shown by constructing a bijection between convex polyominoes and words of an algebraic language and by solving the corresponding algebraic system we prove that the number of convex polyominoes with perimeter 2n 8 is 2n 11 4 n 4 2n 1 2n n

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

diffusion approximation and hyperbolic automorphisms of the torus abstract in this article a diffusion equation is obtained as a limit of a reversible kinetic equation scaled appropriately this limiting diffusion is produced by the collisions of the particles with the boundary indeed these particles follow a reversible reflection law having convenient mixing properties this model based on arnold s cat map can be handled with fourier series instead of the symbolic dynamics associated to a markov partition as a consequence optimal convergence results can be obtained by elementary means and illustrate the apparition of irreversibility in macroscopic limits

minimality and other properties of the width w nonadjacent form let w 2 be an integer and let d w be the set of integers that includes zero and the odd integers with absolute value less than 2 w 1 every integer n can be represented as a finite sum of the form n a i 2 i with a i e d w such that of any w consecutive i s at most one is nonzero such representations are called width w nonadjacent forms w nafs when w 2 these representations use the digits 0 1 and coincide with the well known nonadjacent forms width w nonadjacent forms are useful in efficiently implementing elliptic curve arithmetic for cryptographic applications we provide some new results on the w naf we show that w nafs have a minimal number of nonzero digits and we also give a new characterization of the w naf in terms of a right to left lexicographical ordering we also generalize a result on w nafs and show that any base 2 representation of an integer with digits in d w that has a minimal number of nonzero digits is at most one digit longer than its binary representation

analyzing the galbraith lin scott point multiplication method for elliptic curves over binary fields galbraith lin and scott recently constructed efficiently computable endomorphisms for a large family of elliptic curves defined over ifq2 and showed in the case where q is a prime that the gallant lambert vanstone point multiplication method for these curves is significantly faster than point multiplication for general elliptic curves over prime fields in this paper we investigate the potential benefits of using galbraith lin scott elliptic curves in the case where q is a power of 2 the analysis differs from the q prime case because of several factors including the availability of the point halving strategy for elliptic curves over binary fields our analysis and implementations show that galbraith lin scott point multiplication method offers significant acceleration for curves over binary fields in both doubling and halving based approaches experimentally the acceleration surpasses that reported for prime fields for the platform in common a somewhat counterintuitive result given the relative costs of point addition and doubling in each case

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

rectangular vesicles in three dimensions the phase diagram of a model of rectangular vesicles in three dimensions is examined the scaling of the generating function is determined in the area perimeter volume area and volume perimeter ensembles the results are interpreted within the framework of tricritical scaling and the crossover exponents associated with the transitions are determined we identify three phases in the phase diagram a needle phase a disc phase and a cubical phase these phases are separated by three curves of transitions that meet in a multicritical point