toward a theory of pollard s rho method abstract pollard s rho method for integer factorization iterates a simple polynomial map and produces a nontrivial divisor of n when two such iterates agree modulo this divisor experience and heuristic arguments suggest that a prime divisor p should be detected in o p steps but this has never been proved indeed nothing seems to be have been rigorously proved about the probability of success that would improve the obvious lower bound of 1 p this paper shows that for fixed k this probability is at least 2 k p o p 3 2 as p this leads to an log 2 p p estimate of the success probability

equations for the modular curve and models of elliptic curves with torsion points we describe an algorithm for constructing plane models of the modular curve x 1 n and discuss the resulting equations when n 51

ecm using edwards curves this paper introduces eecm mpfq a fast implementation of the elliptic curve method of factoring integers eecm mpfq uses fewer modular multiplications than the well known gmp ecm software takes less time than gmp ecm and finds more primes than gmp ecm the main improvements above the modular arithmetic level are as follows 1 use edwards curves instead of montgomery curves 2 use extended edwards coordinates 3 use signed sliding window addition subtraction chains 4 batch primes to increase the window size 5 choose curves with small parameters and base points 6 choose curves with large torsion

solving structured linear systems with large displacement rank linear systems with structures such as toeplitz vandermonde or cauchy likeness can be solved in email protected a 2n operations where n is the matrix size a is its displacement rank and email protected denotes the omission of logarithmic factors we show that for such matrices this cost can be reduced to email protected a w 1n where w is a feasible exponent for matrix multiplication over the base field the best known estimate for w is w 2 38 resulting in costs of order email protected a 1 3 8n we present consequences for hermite pade approximation and bivariate interpolation

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

the multiple polynomial quadratic sieve a modification due to peter montgomery of pomerance s quadratic sieve for factoring large integers is discussed along with its implementation using it allows factorization with over an order of magnitude less sieving than the basic algorithm it enables one to factor numbers in the 60 digit range in about a day using a large minicomputer the algorithm has features which make it well adapted to parallel implementation

alternative formulations of a combined trip generation trip distribution modal split and trip assignment model abstract the traditional four step model has been widely used in travel demand forecasting by considering trip generation trip distribution modal split and traffic assignment sequentially in a fixed order however this sequential approach suffers from the inconsistency among the level of service and flow values in each step of the procedure in the last two decades this problem has been addressed by many researchers who have sought to develop combined or integrated models that can consider travelers choice on different stages simultaneously and give consistent results in this paper alternative formulations including mathematical programming mp formulation and variational inequality vi formulations are provided for a combined travel demand model that integrates trip generation trip distribution modal split and traffic assignment using the random utility theory framework thus the proposed alternative formulations not only allow a systematic and consistent treatment of travel choice over different dimensions but also have behavioral richness qualitative properties of the formulations are also given to ensure the existence and uniqueness of the solution particularly the model is analyzed for a special but useful case where the probabilistic travel choices are assumed to be a hierarchical logit model furthermore a self adaptive goldstein levitin polyak glp projection algorithm is adopted for solving this special case

video deblurring algorithm using accurate blur kernel estimation and residual deconvolution based on a blurred unblurred frame pair blurred frames may happen sparsely in a video sequence acquired by consumer devices such as digital camcorders and digital cameras in order to avoid visually annoying artifacts due to those blurred frames this paper presents a novel motion deblurring algorithm in which a blurred frame can be reconstructed utilizing the high resolution information of adjacent unblurred frames first a motion compensated predictor for the blurred frame is derived from its neighboring unblurred frame via specific motion estimation then an accurate blur kernel which is difficult to directly obtain from the blurred frame itself is computed using both the predictor and the blurred frame next a residual deconvolution is applied to both of those frames in order to reduce the ringing artifacts inherently caused by conventional deconvolution the blur kernel estimation and deconvolution processes are iteratively performed for the deblurred frame simulation results show that the proposed algorithm provides superior deblurring results over conventional deblurring algorithms while preserving details and reducing ringing artifacts

factorization of the eighth fermat number we describe a monte carlo factorization algorithm which was used to factorize the fermat number f8 2256 1 previously f8 was known to be composite but its factors were unknown comments only the abstract is given here the full paper appeared as 2 for a succinct proof of primality of the larger factor q8 of f8 see 3 at the time of this paper lenstra s elliptic curve method ecm had not been invented thus a modification of pollard s rho method 1 was used to factor f8 and several mersenne numbers 4 for the factorization of a larger fermat number by ecm see 5 the smaller factor of f8 is 1238926361552897 the epigram i am now entirely persuaded to employ the method a handy trick on gigantic composite numbers may appeal to readers who wish to memorize this factor

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

the factorization of the ninth fermat number in this paper we exhibit the full prime factorization of the ninth fermat number f9 2 512 1 it is the product of three prime factors that have 7 49 and 99 decimal digits we found the two largest prime factors by means of the number field sieve which is a factoring algorithm that depends on arithmetic in an algebraic number field in the present case the number field used was q fifth root 2 the calculations were done on approximately 700 workstations scattered around the world and in one of the final stages a supercomputer was used the entire factorization took four months

understanding blind deconvolution algorithms blind deconvolution is the recovery of a sharp version of a blurred image when the blur kernel is unknown recent algorithms have afforded dramatic progress yet many aspects of the problem remain challenging and hard to understand the goal of this paper is to analyze and evaluate recent blind deconvolution algorithms both theoretically and experimentally we explain the previously reported failure of the naive map approach by demonstrating that it mostly favors no blur explanations we show that using reasonable image priors a naive simulations map estimation of both latent image and blur kernel is guaranteed to fail even with infinitely large images sampled from the prior on the other hand we show that since the kernel size is often smaller than the image size a map estimation of the kernel alone is well constrained and is guaranteed to succeed to recover the true blur the plethora of recent deconvolution techniques makes an experimental evaluation on ground truth data important as a first step toward this experimental evaluation we have collected blur data with ground truth and compared recent algorithms under equal settings additionally our data demonstrate that the shift invariant blur assumption made by most algorithms is often violated

temporal coherence based deblurring using non uniform motion optimization non uniform motion blur due to object movement or camera jitter is a common phenomenon in videos however the state of the art video deblurring methods used to deal with this problem can introduce artifacts and may sometimes fail to handle motion blur due to the movements of the object or the camera in this paper we propose a non uniform motion model to deblur video frames the proposed method is based on superpixel matching in the video sequence to reconstruct sharp frames from blurry ones to identify a suitable sharp superpixel to replace a blurry one we enrich the search space with a non uniform motion blur kernel and use a generalized patchmatch algorithm to handle rotation scale and blur differences in the matching step instead of using pixel based or regular patch based representation we adopt a superpixel based representation and use color and motion to gather similar pixels our non uniform motion blur kernels are estimated from the motion field of these superpixels and our spatially varying motion model considers spatial and temporal coherence to find sharp superpixels experimental results showed that the proposed method can reconstruct sharp video frames from blurred frames caused by complex object and camera movements and performs better than the state of the art methods

solving the combined modal split and traffic assignment problem with two types of transit impedance function abstract the gradient projection gp algorithm has been shown as a successful path based algorithm for solving various traffic assignment problems in this paper the gp algorithm is adapted for solving the combined modal split and traffic assignment cmsta problem which can be viewed as an elastic demand traffic equilibrium problem edtep with two modes using the excess demand formulation of edtep the cmsta problem is reformulated and solved by a modified gp algorithm numerical results based on a real bi modal network in the city of winnipeg canada are provided to demonstrate the efficiency and robustness of the modified path based gp algorithm for solving the cmsta problem in addition the cmsta problem is investigated with two types of impedance function for the transit mode and with different degrees of dispersion for the modal split function the computational results show the modified gp algorithm outperforms the classical evan s algorithm for both types of transit impedance function and it can be as efficient as the original gp algorithm for solving the traffic assignment problem with fixed demand

families of elliptic curves over quartic number fields with prescribed torsion subgroups in this paper we construct infinite families of elliptic curves with given torsion group structures over cubic number fields this result provides explicit examples of the theoretical result recently developed by the first two authors and a schweizer they determined all the group structures which occur infinitely often as the torsion of elliptic curves over cubic number fields in fact this paper presents an efficient way of constructing such families of elliptic curves with prescribed torsion group structures over cubic number fields

defining equations of modular curves we obtain defining equations of modular curves x0 n x1 n and x n by explicitly constructing modular functions using generalized dedekind eta functions as applications we describe a method of obtaining a basis for the space of cusp forms of weight 2 on a congruence subgroup we also use our model of x0 37 to find explicit modular parameterization of rational elliptic curves of conductor 37

factoring large numbers with a quadratic sieve the quadratic sieve algorithm was used to factor a 47 digit number into primes a comparison with wagstaff s results using the continued fraction early abort algorithm suggests that qs should be faster than cfea when the number being factored exceeds 60 digits plus or minus ten or more digits depending on details of the hardware and software

new cullen primes numbers of the forms c n n 2 n 1 and w n n 2 n 1 are both called cullen numbers new primes c n are presented for n 4713 5795 6611 18496 for w n several new primes are listed the largest one having n 18885 furthermore all efforts made to factorize numbers c n and w n are described and the result the complete factorization for all n 300 is given in a supplement

factoring integers with elliptic curves this paper is devoted to the description and analysis of a new algorithm to factor positive integers it depends on the use of elliptic curves the new method is obtained from pollard s p 1 method proc cambridge philos soc 76 1974 521 528 by replacing the multiplicative group by the group of points on a random elliptic curve it is conjectured that the algorithm determines a non trivial divisor of a composite number n in expected time at most k p log n 2 where p is the least prime dividing n and k is a function for which log k x 2 o 1 log x log log x for x x o in the worst case when n is the product of two primes of the same order of magnitude this is

finding suitable curves for the elliptic curve method of factorization using the parametrizations of kubert we show how to produce in nite families of elliptic curves which have prescribed nontrivial torsion over q and rank at least one these curves can be used to speed up the ecm factorization algorithm of lenstra we also brie y discuss curves with complex multiplication in this context

simultaneous estimation of the origin destination matrices and the parameters of a nested logit model in a combined network equilibrium model abstract this paper deals with the inverse problem of a type of traffic equilibrium models with combined modes this problem consists of obtaining a parametrization of the equilibrium model from a set of observations of the outputs for the model the inputs for the model are an origin destination o d trip matrix for the various alternatives that have been considered and a set of parameters for a nested logit model used as a demand model this model has been formulated by using a new approach called the calibration and demand adjustment model cdam based on bilevel programming which simultaneously estimates an o d matrix and the parameters for the nested logit model in this paper the cdam properties are studied and it is proved the cdam solution exists this paper develops a heuristic column paths hyperpaths generation algorithm for the cdam this algorithm iterates between the network equilibrium problem and that which is used to obtain a set of paths when equilibrium is attained and the cdam is restricted to the set of previously generated columns the so called restricted cdam the restricted cdam is formulated as a single level model under certain hypotheses and it furnishes a solution upon which to base a new estimation of the o d matrix and the vector of parameters the computational tests on the algorithm have been carried out using data from a multi modal network in madrid

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

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

gonality of the modular curve x1 n in this paper we compute the gonality over q of the modular curve x1 n for all n 40 and give upper bounds for each n 250 this allows us to determine all n for which x1 n has infinitely points of degree 8 we conjecture that the modular units of q x1 n are freely generated by f 2 f n 2 1 where f k is obtained from the equation for x1 k

new frameworks for montgomery s modular multiplication method we present frameworks for fast modular multiplication based on a modification of montgomery s original method for fixed large integers our algorithms may be significantly faster than conventional methods our techniques may also be extended to modular polynomial arithmetic

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

factorization of the tenth fermat number we describe the complete factorization of the tenth fermat number f10 by the elliptic curve method ecm f10 is a product of four prime factors with 8 10 40 and 252 decimal digits the 40 digit factor was found after about 140 mflop years of computation we also discuss the complete factorization of other fermat numbers by ecm and summarize the factorizations

prime factorization using square root approximation many cryptosystems are based on the factorization of large integers the complexity of this type of factorization is still an advantage for data security developers and a challenge for both mathematicians and cryptanalysts the security of rsa relies on the difficulty of factoring large integers the factorization was studied earlier by old civilizations like the greek but their methods were extended after the emergence of computers the paradox of rsa is that in order to make rsa more efficient we use a modulus n pq which is as small as possible on the other hand it is sufficient to factor n in order to decrypt the encrypted messages in this paper we propose a new factorization method based on the square root approximation this method allows in reducing the search for candidate prime factors of a given integer by approximating each prime factor before considering it as a candidate

linear recurrences with polynomial coefficients and application to integer factorization and cartier manin operator we study the complexity of computing one or several terms not necessarily consecutive in a recurrence with polynomial coefficients as applications we improve the best currently known upper bounds for factoring integers deterministically and for computing the cartier manin operator of hyperelliptic curves

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

on the number of elliptic pseudoprimes for an elliptic curve e with complex multiplication by an order in k q v d a point p of infinite order on e and any prime p with d i p 1 we have that p 1 p o mod p where 0 is the point at infinity and calculations are done using the addition law for e any composite number which satisfies these conditions is called an elliptic pseudoprime in this paper it is shown that assuming the generalized riemann hypothesis elliptic pseudoprimes are less numerous than primes in particular on the grh the number of elliptic pseudoprimes less than x is o x log log x log2 x for certain curves it is shown that infinitely many elliptic pseudoprimes exist

tables of fibonacci and lucas factorizations we list the known prime factors of the fibonacci numbers fn for n 999 and lucas numbers ln for n 500 we discuss the various methods used to obtain these factorizations and primality tests and give some history of the subject

explicit determination of nontrivial torsion structures of elliptic curves over quadratic number fields we determine equations of the modular curves x1 n for n 11 13 14 15 16 17 and 18 except for n 17 these are the only existing elliptic or hyperelliptic x1 n applying these x1 n we calculate tables of elliptic curves e over quadratic fields k with torsion groups of one of the following isomorphism types etor k z mz m 11 13 14 15 16 and 18

removing motion blur with space time processing although spatial deblurring is relatively well understood by assuming that the blur kernel is shift invariant motion blur is not so when we attempt to deconvolve on a frame by frame basis this is because in general videos include complex multilayer transitions indeed we face an exceedingly difficult problem in motion deblurring of a single frame when the scene contains motion occlusions instead of deblurring video frames individually a fully 3 d deblurring method is proposed in this paper to reduce motion blur from a single motion blurred video to produce a high resolution video in both space and time unlike other existing approaches the proposed deblurring kernel is free from knowledge of the local motions most importantly due to its inherent locally adaptive nature the 3 d deblurring is capable of automatically deblurring the portions of the sequence which are motion blurred without segmentation and without adversely affecting the rest of the spatiotemporal domain where such blur is not present our method is a two step approach first we upscale the input video in space and time without explicit estimates of local motions and then perform 3 d deblurring to obtain the restored sequence

exploring trust region method for the solution of logit based stochastic user equilibrium problem abstract in this research paper we explored using the trust region method to solve the logit based sue problem we proposed a modified trust region newton mtrn algorithm for this problem when solving the trust region sue subproblem we showed that applying the well known steihaug toint method is inappropriate since it may make the convergence rate of the major iteration very slow in the early stage of the computation to overcome this drawback a modified steihaug toint method was proposed we proved the convergence of our mtrn algorithm and showed its convergence rate is superlinear for the implication of our algorithm we proposed an important principle on how to select the basic route for each od pair we indicated that it is a crucial principle to accelerate the convergence rate of the minor iteration i e trust region subproblem solving iteration in this study other implication issues for the sue problem are also considered including the computation of the trial step and the strategy to ensure strict feasibility iteration point we compared the mtrn algorithm with the gradient projection gp algorithm on the sioux falls network some results of numerical analysis are also reported

a comparison of linear reformulations for multinomial logit choice probabilities in facility location models abstract in the last decade several papers appeared on facility location problems that incorporate customer demand by the multinomial logit model three linear reformulations of the original non linear model have been proposed so far in this paper we discuss these models in terms of solvability we present empirical findings based on synthetic data

traffic assignment model with fuzzy level of travel demand an efficient algorithm based on quasi logit formulas abstract the place of fuzzy concepts in traffic assignment ta models has been studied in recent literature keeping fuzzy level of travel demand in mind we propose a new ta model in which the travel costs of links are depended on their congestion from the results of such fuzzy ta model network planners are able to estimate the number of travelers on network links by using zero one variables the proposed model is transformed into a crisp mixed integer problem with respect to path flow variables in order to produce the logit flows from this problem damberg et al algorithm is modified then the level of certainty is maximized and perceived travel delays are minimized for a fixed certainty degree the obtained solution which is named the fuzzy equilibrium flow satisfies a quasi logit formula similar to ordinary expression of the logit route choice model eventually we examine the quality of different path enumeration techniques in the proposed model

an algorithm for the ranking of shortest paths an efficient computational implementation of a path deletion k shortest paths algorithm and a new algorithm for the same problem are presented in a path deletion k shortest paths algorithm a sequence g1 g2 gk of networks is is defined such that g1 is given network and its k th shortes path is trivially determined from the shortest path in gk in essence as soon as the shortest path in gk is determined it is excluded from gk in such a way that no new paths are formed and no more paths are deleted so far each gk two procedures are executed a shortest path algorithm and a path deletion algorithm in the presented computational implementation all the information resulting from the determination of the k th shortest path is carried throughout gk 1 gk 2 gk the new algorithm not only keeps this characteristics but also avoids the last k 1 executions of a shortest path algorithm which results in a suprising and very substantial reduction in the execution time in fact for randomly generated networks with 104 nodes and 105 arcs once the shortest the new algorithm computes the next 100 shortest paths in times of the order of 10 1 seconds to illustrate the efficiency of this algorithm comparative computational experiments are reported

new regular radix 8 scheme for elliptic curve scalar multiplication without pre computation the recent advances in mobile technologies have increased the demand for high performance parallel computing schemes in this paper we present a new algorithm for evaluating elliptic curve scalar multiplication that can be used on any abelian group we show that the properties of the proposed algorithm enhance parallelism at both the point arithmetic and the field arithmetic levels then we employ this algorithm in proposing a new hardware design for the implementation of an elliptic curve scalar multiplication on a prime extended twisted edwards curve incorporating eight parallel operations we further show that in comparison to the other simple side channel attack protected schemes over prime fields the proposed design of the extended twisted edwards curve is the fastest scalar multiplication scheme reported in the literature

torus based compression by factor 4 and 6 we extend the torus based compression technique for cyclotomic subgroups and show how the elements of certain subgroups in characteristic two and three fields can be compressed by a factor of 4 and 6 respectively our compression and decompression functions can be computed at a negligible cost in particular our techniques lead to very efficient exponentiation algorithms that work with the compressed representations of elements and can be easily incorporated into pairing based protocols that require exponentiations or products of pairings

new integer factorizations new factorizations of fibonacci numbers lucas numbers and numbers of the form 2 i are presented together with the strategy a combination of known factorization methods used to obtain them

an fft extension to the p 1 factoring algorithm j m pollard in 1974 presented the p integer factoring al gorithm his paper couched the algorithm in theoretical terms based upon use of fast fourier transform techniques but he was unable to say whether the method could be made practical we discuss the mathematical basis of the al gorithm and show how it can work in practice the practical implementation depends for its success upon the use of residue number systems we also present an open problem as to how the method could be made to work for the elliptic curve factoring algorithm

modular multiplication without trial division let n 1 we present a method for multiplying two integers called n residues modulo n while avoiding division by n n residues are represented in a nonstandard way so this method is useful only if several computations are done modulo one n the addition and subtraction algorithms are unchanged 1 description some algorithms 1 2 4 5 require extensive modular arith metic we propose a representation of residue classes so as to speed modular multiplication without affecting the modular addition and subtraction algorithms other recent algorithms for modular arithmetic appear in 3 6 fix n 1 define an a residue to be a residue class modulo n select a radix r coprime to n possibly the machine word size or a power thereof such that r n and such that computations modulo r are inexpensive to process let r l and n be integers satisfying 0 n then return t n else return t to validate redc observe mn tn n tmod r so t is an integer also tr tmod n so t tr x mod n thirdly 0 t mn rn rn so 0 t 2n if r and n are large then t mn may exceed the largest double precision value one can circumvent this by adjusting m so r m 0 given two numbers x and y between 0 and n 1 inclusive let z redc xy then z xy r x mod n so xr l yr x zrx mod n also 0 z n so z is the product of x and y in this representation other algorithms for operating on n residues in this representation can be derived from the algorithms normally used the addition algorithm is unchanged since xr x yr x zr x mod n if and only if x y z mod n also unchanged are

counting points on hyperelliptic curves in average polynomial time let g 1 and let q2 z x be a monic squarefree polynomial of degree 2g 1 for an odd prime p not dividing the discriminant of q let zp t denote the zeta function of the hyperelliptic curve of genus g over the nite eld fp obtained by reducing the coecients of the equation y 2 q x modulo p we present an explicit deterministic algorithm that given as input q and a positive integer n computes zp t simultaneously for all such primes p n whose average complexity per prime is polynomial in g logn and the number of bits required to represent q

torsion of elliptic curves over cyclic cubic fields we determine all the possible torsion groups of elliptic curves over cyclic cubic fields over non cyclic totally real cubic fields and over complex cubic fields

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method