internal diffusion limited aggregation parallel algorithms and complexity the computational complexity of internal diffusion limited aggregation dla is examined from both a theoretical and a practical point of view we show that for two or more dimensions the problem of predicting the cluster from a given set of paths is complete for the complexity class cc the subset of p characterized by circuits composed of comparator gates cc completeness is believed to imply that in the worst case growing a cluster of size n requires polynomial time in n even on a parallel computer a parallel relaxation algorithm is presented that uses the fact that clusters are nearly spherical to guess the cluster from a given set of paths and then corrects defects in the guessed cluster through a nonlocal annihilation process the parallel running time of the relaxation algorithm for two dimensional internal dla is studied by simulating it on a serial computer the numerical results are compatible with a running time that is either polylogarithmic in n or a small power of n thus the computational resources needed to grow large clusters are significantly less on average than the worst case analysis would suggest for a parallel machine with k processors we show that random clusters in d dimensions can be generated in mathcal o n k logk n 2 d steps this is a significant speedup over explicit sequential simulation which takes mathcal o n 1 2 d time on average finally we show that in one dimension internal dla can be predicted in mathcal o logn parallel time and so is in the complexity class nc

knottedness is in np modulo grh abstract given a tame knot k presented in the form of a knot diagram we show that the problem of determining whether k is knotted is in the complexity class np assuming the generalized riemann hypothesis grh in other words there exists a polynomial length certificate that can be verified in polynomial time to prove that k is non trivial grh is not needed to believe the certificate but only to find a short certificate this result complements the result of hass lagarias and pippenger that unknottedness is in np our proof is a corollary of major results of others in algebraic geometry and geometric topology

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

constructing hyperelliptic curves of genus 2 suitable for cryptography in this article we show how to generalize the cm method for elliptic curves to genus two we describe the algorithm in detail and discuss the results of our implementation

on the singular values of weber modular functions the minimal polynomials of the singular values of the classical weber modular functions give far simpler defining polynomials for the class fields of imaginary quadratic fields than the minimal polynomials of singular moduli of level 1 we describe computations of these polynomials and give conjectural formulas describing the prime decomposition of their resultants and discriminants extending the formulas of gross zagier for the level 1 case

checking the odd goldbach conjecture up to 10 20 vinogradov s theorem states that any sufficiently large odd integer is the sum of three prime numbers this theorem allows us to suppose the conjecture that this is true for all odd integers in this paper we describe the implementation of an algorithm which allowed us to check this conjecture up to 10 20

dynamic programming alignment of sequences representing cyclic patterns string alignment by dynamic programming is generalized to include cyclic shift and corresponding optimal alignment cost for strings representing cyclic patterns a guided search algorithm uses bounds on alignment costs to find all optimal cyclic shifts the bounds are derived from submatrices of an initial dynamic programming matrix algorithmic complexity is analyzed for major stages in the search the applicability of the method is illustrated with satellite dna sequences and circularly permuted protein sequences

the primality of r1031 on decrit une technique permettant de demontrer que le nombre r1031 10 1031 1 9 compose uniquement de 1 est premier

phase only filtering for the masses of dna data a new approach to sequence alignment alignment of dna segments containing repetitive nucleotide base patterns is an important task in several genomics applications including dna sequencing dna fingerprinting pathogen detection and gene finding one of the most efficient procedures used for this task is the cross correlation method the main computations of the procedure are the discrete fourier transform dft and a pointwise multiplication of two complex fourier transform sequences in this paper the standard magnitude and phase cross correlation technique is compared with the lesser known but closely related phase only cross correlation method it is shown that for a periodic dna sequence the standard approach leads to significant sidelobes in the cross correlation the magnitude of which increases with sequence length while the phase only approach yields a perfect cross correlation with zero sidelobes for a dna sequence that contains both irregularly distributed symbols and periodic patterns the difference in performance is less pronounced but still significant numerical experiments on synthesized and real data demonstrate that the phase only approach is robust to isolated symbol insertions and deletions and that it is capable of identifying positions of matching segments in the sequence

elliptic curve cryptosystems using curves of smooth order over the ring z sub n we present a new cryptosystem based on elliptic curves over the ring z sub n where n pq in which the message is held in the exponent and not the group element the security of this system is based on the difficulty of factoring n a digital signature scheme and id based key exchange system are also given

universal computation and physical dynamics abstract a dynamical system is said to be computationally universal if it can be programmed through its initial condition to perform and digital computation such systems include traditional abstract computers such as turing machines and cellular automata as well as more physical models such as hard sphere gases and even a single particle in an appropriately shaped box because of the ability of any two such systems to simulate one another they are all equivalent in the range of computations they can perform and the global dynamics of any one of them provides a microcosm of all cause effect relations that can be expressed by deductive logic or numerical simulation this allows universal computers to be used to define in an objective manner that sort of complexity which increases when a self organizing system organizes itself

a performance analysis of a simple prime testing algorithm this paper gives an empirical performance analysis of a prime proving program designed and implemented by the author and j l selfridge in 1974 the algorithm has been commonly referred to as the down algorithm because of its recursive characteristics it is shown among other things that of the 2270 primes tested 94 of them were proved in less than 3 seconds of ibm 360 67 time per number introduction there is a variety of algorithms available for using a computer to prove numbers prime see for example h c williams 6 the most straightforward method is to divide the number by each prime which is less than its square root if none of the remainders is zero the number is a prime this method has the advantage of being very easy to program but it uses a prohibitive amount of computer time for numbers exceeding 10 digits in length also the proof is nonconstructive in that the only way to verify the correctness of the proof is to repeat the calculation with another program perhaps on another computer recently michael 0 rabin 3 and r solovay and v strassen 5 have published probabilistic algorithms for testing primality these are quick relatively easy to program tests which can assert positively that a number n is composite but when it asserts the primality of n it does so with a very small probability of error that is n is asserted to be prime by a procedure that on the average will make no more than one mistake in 21 applications while these procedures can be useful in producing large collections of essentially composite free numbers they are not able to provide the mathematician with a conclusive proof of the primality of a single value of n in 1975 j l selfridge and the author 4 designed and implemented a program for prime proving which is not probabilistic and is effective for numbers between 10 and 35 decimal digits in length in this note we give a performance analysis of this algorithm based on the results of using the program on a collection of 2270 numbers in this range the algorithm for a detailed description of the program and the algorithm the reader should consult 4 but for completeness we will state the two relevant theorems their proofs can be found in 1 theorem 1 proth pocklington lehmer if n 1 2vp lpa2 pakc1 where cl has no prime factors less than b and 2vpalpa2 pakb nk then n is a prime if the following two conditions hold for any choices of bi received april 20 1980 revised february 8 1982 1980 mathematics subject classification primary 10a25 the research for this paper has been supported in part by the air force office of scientific research 1983 american mathematical society 0025 5718 82 oo0 0990 02 25

effective primality tests for some integers of the forms 5ⁿ 1 and 7ⁿ 1 etablissement de tests de caractere premier deterministes et rapides fonctionnant en temps polynomial pour des nombres de la forme ar n 1 r 5 7 a r n

symmetric phase only matched filtering of fourier mellin transforms for image registration and recognition presents a new method to match a 2d image to a translated rotated and scaled reference image the approach consists of two steps the calculation of a fourier mellin invariant fmi descriptor for each image to be matched and the matching of the fmi descriptors the fmi descriptor is translation invariant and represents rotation and scaling as translations in parameter space the matching of the fmi descriptors is achieved using symmetric phase only matched filtering spomf the performance of the fmi spomf algorithm is the same or similar to that of phase only matched filtering when dealing with image translations the significant advantage of the new technique is its capability to match rotated and scaled images accurately and efficiently the innovation is the application of spomf to the fmi descriptors which guarantees high discriminating power and excellent robustness in the presence of noise this paper describes the principle of the new method and its discrete implementation for either image detection problems or image registration problems practical results are presented for various applications in medical imaging remote sensing fingerprint recognition and multiobject identification

the complexity of class polynomial computation via floating point approximations we analyse the complexity of computing class polynomials that are an important ingredient for cm constructions of elliptic curves via complex floating point approximations of their roots the heart of the algorithm is the evaluation of modular functions in several arguments the fastest one of the presented approaches uses a technique devised by dupont to evaluate modular functions by newton iterations on an expression involving the arithmetic geometric mean it runs in time o d log 5 d log log d o d 1 epsilon o h 2 epsilon for any epsilon 0 where d is the cm discriminant and h is the degree of the class polynomial another fast algorithm uses multipoint evaluation techniques known from symbolic computation its asymptotic complexity is worse by a factor of log d up to logarithmic factors this running time matches the size of the constructed polynomials the estimate also relies on a new result concerning the complexity of enumerating the class group of an imaginary quadratic order and on a rigorously proven upper bound for the height of class polynomials

primality testing and jacobi sums we present a theoretically and algorithmically simplified version of a primality testing algorithm that was recently invented by adleman and rumely the new algorithm performs well in practice it is the first pnmality test in existence that can routinely handle numbers of hundreds of decimal digits

algorithms in algebraic number theory in this paper we discuss the basic problems of algorithmic algebraic number theory the emphasis is on aspects that are of interest from a purely mathematical point of view and practical issues are largely disregarded we describe what has been done and more importantly what remains to be done in the urea we hope to show that the study of algorithms not only increases our understanding of algebraic number fields but also stimulates our curiosity about them

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 infinite number of generalized dimensions of fractals and strange attractors abstract we show that fractals in general and strange attractors in particular are characterized by an infinite number of generalized dimensions dq q 0 to this aim we develop a rescaling transformation group which yields analytic expressions for all the quantities dq we prove that lim q 0 dq fractal dimension d limq 1dq information dimension and dq 2 correlation exponent v dq with other integer q s correspond to exponents associated with ternary quaternary and higher correlation functions we prove that generally dq dq for any q q for homogeneous fractals dq dq a particularly interesting dimension is dq for two examples feigenbaum attractor generalized baker s transformation we calculate the generalized dimensions and find that d is a non trivial number all the other generalized dimensions are bounded between the fractal dimension and d

generalised mersenne numbers revisited generalised mersenne numbers gmns were defined by solinas in 1999 and feature in the nist fips 186 2 and secg standards for use in elliptic curve cryptography their form is such that modular reduction is extremely efficient thus making them an attractive choice for modular multiplication implementation however the issue of residue multiplication efficiency seems to have been overlooked asymptotically using a cyclic rather than a linear convolution residue multiplication modulo a mersenne number is twice as fast as integer multiplication this property does not hold for prime gmns unless they are of mersenne s form in this work we exploit an alternative generalisation of mersenne numbers for which an analogue of the above property and hence the same efficiency ratio holds even at bitlengths for which schoolbook multiplication is optimal while also maintaining very efficient reduction moreover our proposed primes are abundant at any bitlength whereas gmns are extremely rare our multiplication and reduction algorithms can also be easily parallelised making our arithmetic particularly suitable for hardware implementation furthermore the field representation we propose also naturally protects against side channel attacks including timing attacks simple power analysis and differential power analysis which is essential in many cryptographic scenarios in constrast to gmns

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

the computational complexity of pattern formation the computational complexity of diffusion limited aggregation and fluid invasion in porous media is studied the time requirements on an idealized parallel computer for simulating the patterns formed by these models are investigated it is shown that these growth models are p complete these results provide strong evidence that such pattern formation processes are inherently sequential and cannot be simulated efficiently in parallel

the computational complexity of generating random fractals we examine a number of models that generate random fractals the models are studied using the tools of computational complexity theory from the perspective of parallel computation diffusion limited aggregation and several widely used algorithms for equilibrating the ising model are shown to be highly sequential it is unlikely they can be simulated efficiently in parallel this is in contrast to mandelbrot percolation which can be simulated in constant parallel time our research helps shed light on the intrinsic complexity of these models relative to each other and to different growth processes that have been recently studied using complexity theory in addition the results may serve as a guide to simulation physics

the structure factor of primes although the prime numbers are deterministic they can be viewed by some measures as pseudo random numbers in this article we numerically study the pair statistics of the primes using statistical mechanical methods particularly the structure factor in an interval with m large and l m smaller than unity we show that the structure factor of the prime number configurations in such intervals exhibits well defined bragg like peaks along with a small diffuse contribution this indicates that primes are appreciably more correlated and ordered than previously thought our numerical results definitively suggest an explicit formula for the locations and heights of the peaks this formula predicts infinitely many peaks in any non zero interval similar to the behavior of quasicrystals however primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational we also show numerically that the diffuse part decays slowly as m and l increases this suggests that the diffuse part vanishes in an appropriate infinite system size limit

hilbert class polynomials and traces of singular moduli where q e the values of j z at imaginary quadratic arguments in the upper half of the complex plane are known as singular moduli singular moduli are algebraic integers which play prominent roles in classical and modern number theory see c bcsh for example hilbert class fields of imaginary quadratic fields are generated by singular moduli furthermore isomorphism classes of elliptic curves with complex multiplication are distinguished by singular moduli throughout let d 0 3 mod 4 be a positive integer so that d is the discriminant of an order in an imaginary quadratic field and let h d be the hurwitz kronecker class number for the discriminant d let qd be the set of positive definite integral binary quadratic forms note including imprimitive forms if there are any

the interpretation and measurement of the f α spectrum of a multifractal measure the authors analyse the growth probability distribution for differential limited aggregates dla in terms of its f curve and f and computed independently as generalized entropies using data from 10 clusters of 10000 to 100000 particles the growth probability is obtained by solving the laplace equations numerically with a hierarchical method which improves accuracy and speed two additional methods are used to compare results for dla clusters and for a few objects with simple geometry and known f spectrum these comparisons reinforce their results for the part of the dla spectrum corresponding to q0 they also bring to light some unexpected features for q 0 1 which suggest a non universality of the scaling exponents in this region in two dimensions

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

ramanujan s class invariants and their use in elliptic curve cryptography the complex multiplication cm method is a method frequently used for the generation of elliptic curves ecs over a prime field f p the most demanding and complex step of this method is the computation of the roots of a special type of class polynomials called hilbert polynomials however there are several polynomials called class polynomials which can be used in the cm method having much smaller coefficients and fulfilling the prerequisite that their roots can be easily transformed to the roots of the corresponding hilbert polynomials in this paper we propose the use of a new class of polynomials which are derived from ramanujan s class invariants t n we explicitly describe the algorithm for the construction of the new polynomials and give the necessary transformation of their roots to the roots of the corresponding hilbert polynomials we provide a theoretical asymptotic bound for the bit precision requirements of all class polynomials and also using extensive experimental assessments we compare the efficiency of using the new polynomials against the use of the other class polynomials our comparison shows that the new class of polynomials clearly surpass all of the previously used polynomials when they are used in the generation of prime order elliptic curves

parallel algorithm and dynamic exponent for diffusion limited aggregation a parallel algorithm for diffusion limited aggregation dla is described and analyzed from the perspective of computational complexity the dynamic exponent z of the algorithm is defined with respect to the probabilistic parallel random access machine pram model of parallel computation according to t sim l z where l is the cluster size t is the running time and the algorithm uses a number of processors polynomial in l it is argued that z d d 2 2 where d is the fractal dimension and d 2 is the second generalized dimension simulations of dla are carried out to measure d 2 and to test scaling assumptions employed in the complexity analysis of the parallel algorithm it is plausible that the parallel algorithm attains the minimum possible value of the dynamic exponent in which case z characterizes the intrinsic history dependence of dla

the computational complexity of sandpiles given an initial distribution of sand in an abelian sandpile what final state does it relax to after all possible avalanches have taken place in d 3 we show that this problem is p complete so that explicit simulation of the system is almost certainly necessary we also show that the problem of determining whether a sandpile state is recurrent is p complete in d 3 and briefly discuss the problem of constructing the identity in d 1 we give two algorithms for predicting the sandpile on a lattice of size n both faster than explicit simulation a serial one that runs in time mathcal o n log n and a parallel one that runs in time mathcal o log3n i e the class nc3 the latter is based on a more general problem we call additive ranked generability this leaves the two dimensional case as an interesting open problem

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

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

structural and computational depth of diffusion limited aggregation diffusion limited aggregation dla is studied from the perspective of computational complexity a parallel algorithm is exhibited that requires a number of steps that scales as the depth of the tree defined by the cluster the existence of this algorithm suggests a connection between a fundamental computational and structural property of dla

computing isogenies between elliptic curves over f p n using couveignes s algorithm the heart of the improvements by elkies to schoof s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves elkies approach is well suited for the case where the characteristic of the field is large couveignes showed how to compute isogenies in small characteristic the aim of this paper is to describe the first successful implementation of couveignes s algorithm in particular we describe the use of fast algorithms for performing incremental operations on series we also insist on the particular case of the characteristic 2

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

finding strong pseudoprimes to several bases define m to be the smallest strong pseudoprime to all the first m prime bases if we know the exact value of m we will have for integers n m a deterministic primality testing algorithm which is not only easier to implement but also faster than either the jacobi sum test or the elliptic curve test thanks to pornerance et al and jaeschke m are known for 1 m 8 upper bounds for 9 10 and 1 were given by jaeschke in this paper we tabulate all strong pseudoprimes spsp s n 10 24 to the first ten prime bases 2 3 29 which have the form n pq with p q odd primes and q 1 k p 1 k 2 3 4 there are in total 44 such numbers six of which are also spsp 31 and three numbers are spsp s to both bases 31 and 37 as a result the upper bounds for 10 and 11 are lowered from 28 and 29 decimal digit numbers to 22 decimal digit numbers and a 24 decimal digit upper bound for 12 is obtained the main tools used in our methods are the biquadratic residue characters and cubic residue characters we propose necessary conditions for n to be a strong pseudoprime to one or to several prime bases comparisons of effectiveness with both jaeschke s and arnault s methods are given

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

table errata new primality criteria and factorizations of 2 1 math comp 29 1975 620 647 mr 52 5546 by the author d h lehmer and j l selfridge

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

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

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

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

plane dla is not self similar is it a fractal that becomes increasingly compact as it grows abstract using two new methods of geometric analysis this paper establishes that dla clusters are definitely not self similar compared to small clusters the morphology of large clusters of sizes up to 30 million particles can be characterized both visually and quantitatively as being far more compact or less lancunar qualitatively the number of arms increases during growth the evidence does not exclude that the cluster remains fractal and that its fractal dimension remains constant however new pitfalls in the estimation of d are revealed the gradual change in the morphology of dla opens the possibility that there is continuity between the standard morphology observed for small to medium computer generated dla clusters and the compact morphology observed in many actual physical phenomena

majority vote cellular automata ising dynamics and p completeness we study cellular automata where the state at each site is decided by a majority vote of the sites in its neighborhood these are equivalent for a restricted set of initial conditions to nonzero probability transitions in single spin flip dynamics of the ising model at zero temperature we show that in three or more dimensions these systems can simulate boolean circuits of and and or gates and are therefore p complete that is predicting their state t time steps in the future is at least as hard as any other problem that takes polynomial time on a serial computer therefore unless a widely believed conjecture in computer science is false it is impossible even with parallel computation to predict majority vote cellular automata or zero temperature single spin flip ising dynamics qualitatively faster than by explicit simulation

primality testing of large numbers in maple primality testing of large numbers is very important in many areas of mathematics computer science and cryptography and in recent years many of the modern primality testing algorithms have been incorporated in computer algebra systems cas such as axiom and maple as a standard in this paper we discuss primality testing of large numbers in maple v release 3 a maple version newly released in 1994 our computation experience shows that the maple primality testing facility isprime based on a combined use of a strong pseudoprimality test and a lucas test is efficient and reliable

choosing the correct elliptic curve in the cm method we give an elementary way to distinguish between the twists of an ordinary elliptic curve e over f p in order to identify the one with p 1 2u points when p u 2 dv 2 with 2u 2v and e is constructed using the cm method for finding elliptic curves with a prescribed number of points our algorithms consist in most cases of reading off simple congruence conditions on u and v modulo 4

implementing the asymptotically fast version of the elliptic curve primality proving algorithm the elliptic curve primality proving ecpp algorithm is one of the current fastest practical algorithms for proving the primality of large numbers its running time cannot be proven rigorously but heuristic arguments show that it should run in time o log n 5 to prove the primality of n an asymptotically fast version of it attributed to j o shallit runs in time o log n 4 the aim of this article is to describe this version in more details leading to actual implementations able to handle numbers with several thousands of decimal digits

modularity of galois traces of class invariants zagier showed that the galois traces of the values of j invariant at cm points are fourier coefficients of a weakly holomorphic modular form of weight 3 2 and bruinier funke expanded his result to the sums of the values of arbitrary modular functions at heegner points in this paper we identify the galois traces of real valued class invariants with modular traces of the values of certain modular functions at heegner points so that they are fourier coefficients of weight 3 2 weakly holomorphic modular forms

multiscaling analysis of large scale off lattice dla abstract an improved algorithm has been used to simulate off lattice diffusion limited aggregation dla in two dimensions this enables us to grow clusters of up to 6 106 particles which are the largest such clusters made up to now the density profile in two dimensions is analyzed for the scaling form multiscaling g r r r d d x c x with x r r where r is the radius of gyration and r is the distance from the cluster origin in contrast with previous studies the dimension d x shows no symmetric tendency to drop at values of x but only fluctuates around the global fractal dimension d 1 712 0 003

implementation of a new primality test an implementation of the cohen lenstra version see ibid vol 42 p 297 330 1984 of the adleman pomerance rumely primality test see l m adleman c pomerance and r s rumely ann of math vol 117 p 173 206 1983 is presented primality of prime numbers of up to 213 decimal digits can now routinely be proved within approximately ten minutes

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

modular functions and the uniform distribution of cm points 1 zd i d 2 if d 0 mod 4 1 i d 2 if d 3 mod 4 the j function has the remarkable property that j zd is an algebraic integer of degree h d the class number of k q zd in fact k j zd is the hilbert class field of k 4 the first few values of j zd are j z3 0 j z4 12 j z7 153 j z8 20 j z11 323 j z15 191025 85995 5 2 j z19 963 j z20 632000 282880 5