on the elastic fields of an elliptical inhomogeneity under plane deformation a generalized and unified approach is presented for the two dimensional plane problem of an elliptical inhomogeneity in an isotropic elastic medium the analysis is based upon the use of conformal mapping and laurent series expansion of muskhelishvili s complex potentials the resulting elastic fields are derived explicitly in both transformed and physical planes for the inhomogeneity and the surrounding matrix the associated expressions are universal in the sense of being applicable to generalized geometries and applied loads the application of the general solution is illustrated by several examples and the resulting solutions are verified with those existing in the literature the results advanced here can be used as building blocks for dealing with more complex problems involving a wide variety of geometry and loading conditions

a note on measuring in p we revisit the problem of generalising lutz s resource bounded measure rbm to small complexity classes and propose a definition of a random based rbm on p k n dtime o nk which we argue as being a good generalisation to p of lutz s rbm we cannot unconditionally prove the existence of such a measure but we give sufficient and necessary conditions for its existence we also revisit an rbm for p defined by strauss inform comput 136 1 1997 1 and correct an erroneous claim concerning the relations between and random sets a correction to this mistake is then proposed which is a less powerful but accurate relation between and random sets in order to obtain these results we introduce a mathematical structure called a measuring system which is a general setting that can be used to compare different rbms on any fixed complexity class through a partial ordering relation

the stress concentration factor for slightly roughened random surfaces analytical solution abstract we derive an analytical solution to the stress concentration factor k t for slightly roughened random surfaces topology is assumed to possess gaussian distribution of heights and auto correlation length acl for our development we combine gao s first order perturbation method the hilbert transform and an energy conservation principal related to the parseval theorem the root mean square rms value of k t results in a function of the ratio rms roughness to acl the derived formula agrees with experimental results previously reported the results provide insight for more efficient design

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

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

stress singularities along a cycloid rough surface abstract the stress concentration along a rough surface is of importance for understanding the nucleation of misfit dislocations and cracks in heteroepitaxial thin films and for general flaw initiation at material surfaces exposed to environmental corrosion in order to seek a basic understanding on this issue a cycloid wavy surface subject to a uniform bulk stress is adopted as a model problem the elastic stress and displacement fields are determined using muskhelishvili s conformal mapping method it is shown that a cusped cycloid surface generates a crack like singular stress field within a thin surface layer under uniform tension this singularity is found to have identical strength to a row of periodic parallel cracks the path independent j integral requires that the average surface strain energy density be identical to that in the bulk implying that any stress magnification along a rough surface must be compensated by unloading along some complementary portions of the surface along the cusped cycloid surface the strain energy distribution becomes dirac singular at the cusp tips the rest of the surface being completely stress free thus the effect of cycloid cusps is to redistribute and concentrate all the surface strain energy per wavelength at a single cusp point and because of that we can claim that the cycloid surface is the most efficient stress concentrator at a fixed wavelength even at a moderate bulk stress level this concentration may be sufficient to cause failure or nucleation of defects the full evolution of a rough surface under stress and other corrosion mechanisms must be solved by numerical methods our analytic solutions for a cycloid surface are of significant value for guiding numerical computations and gaining insights into essential features of the evolution process from a global point of view we show that a cusped cycloid surface becomes energetically favorable once the surface wavelength exceeds a critical value determined from the competition between the strain energy and the surface energy from a local point of view analysis of surface diffusion behaviors along an almost cusped cycloid surface indicates that the cusps are stable once they develop the critical condition for formation of a cusped cycloid corresponds to the griffith energy balance being exactly satisfied at the cusp tips while the chemical potential remains nearly constant along the rest of the surface this implies spontaneous griffith brittle fracture at tension cusps if plastic relaxation is not present to relieve the stress singularity

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

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

weak completeness in e and e 2 abstract the notions of weak m p completeness for the complexity classes e dtime 2 linear and e 2 dtime 2 polynomial are compared an element c of one of these classes is weakly m p complete for the class if the set p m c consisting of all languages a m p c does not have measure 0 in the class the following two results are proven 1 i every problem that is weakly m p complete for e is weakly m p complete for e 2 2 ii there is a problem in e that is weakly m p complete for e 2 but not for e

nontriviality for exponential time w r t weak reducibilities a set a is nontrivial for the linear exponential time class e dtime 2 l i n if email protected e and the sets from e which can be reduced to a are not from a single level dtime 2 k n of the linear exponential hierarchy similarly a set a is nontrivial for the polynomial exponential time class exp dtime 2 p o l y if email protected exp and the sets from exp which can be reduced to a are not from a single level dtime 2 n k of the polynomial exponential hierarchy see 2 here we compare the strength of the nontriviality notions with respect to the underlying reducibilities where we consider the polynomial time variants of many one bounded truth table truth table and turing reducibilities surprisingly the results obtained for e and exp differ while the above reducibilities yield a proper hierarchy of nontriviality notions for e nontriviality for exp under many one reducibility and truth table reducibility coincides

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

hard problems of algebraic geometry codes the minimum distance is one of the most important combinatorial characterizations of a code the maximum likelihood decoding problem is one of the most important algorithmic problems of a code while these problems are known to be hard for general linear codes the techniques used to prove their hardness often rely on the construction of artificial codes in general much less is known about the hardness of the specific classes of natural linear codes in this correspondence we show that both problems are np hard for algebraic geometry codes we achieve this by reducing a well known np complete problem to these problems using a randomized algorithm the family of codes in the reductions is based on elliptic curves they have positive rates but the alphabet sizes are exponential in the block lengths

a hyperelliptic smoothness test ii this series of papers presents and rigorously analyzes a probabilistic algorithm for finding small prime factors of an integer the algorithm uses the jacobian varieties of curves of genus 2 in the same way that the elliptic curve method uses elliptic curves this second paper in the series is concerned with the order of the group of rational points on the jacobian of a curve of genus 2 defined over a finite field we prove a result on the distribution of these orders 2000 mathematical subject classification 11y05 11g10 11m20 11n25

a stress concentration formula generating equation for arbitrary shallow surfaces abstract analytical understanding of how stress concentrates is invaluable an equation that generates stress concentration formulas is derived and shown to apply very well to a number of shallow irregularities on surfaces for the plane stress conditions and to a first order approximation under shallow conditions for any first order holder continuous surface function f x the derived equation is k t x 1 2 h f x where h is the hilbert transform and f x is the spatial derivative of f with respect to the independent variable it is shown that using this generating equation well known traditional results can be easily derived also a number of other stress concentration formulas for various cases are generated furthermore a second order approximation is introduced which shows the dependence of k t on not only the slope but also on the concavity of the surface the approach used herein can be extended to finding closed form solutions to other integral equations possessing similar kernels for applications such as the variation of the stress intensity factor due to an arbitrary crack front profile work in progress

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

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

almost complete sets we show that there is a set that is almost complete but not complete under polynomial time many one p m reductions for the class e of sets computable in deterministic time 2lin here a set in a complexity class c is almost complete for c under some given reducibility if the class of the problems in c that do not reduce to this set has measure 0 in c in the sense of lutz s resource bounded measure theory we also show that the almost complete sets for e under polynomial time bounded length increasing one one reductions and truth table reductions of norm 1 coincide with the almost p m complete sets for e moreover we obtain similar results for the class exp of sets computable in deterministic time 2poly

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

weakly hard problems a weak completeness phenomenon is investigated in the complexity class rm e rm dtime 2 rm linear according to standard terminology a language h is leq rm p m hard for e if the set rm p m h consisting of all languages a leq rm p m h contains the entire class e a language c is leq rm p m complete for e if it is leq rm p m hard for e and is also an element of e generalizing this a language h is weakly leq rm p m hard for e if the set rm p m h does not have measure 0 in e a language c is weakly leq rm p m complete for e if it is weakly leq rm p m hard for e and is also an element of e the main result of this paper is the construction of a language that is weakly leq rm p m complete but not leq rm p m complete for e the existence of such languages implies that previously known strong lower bounds on the complexity of weakly leq rm p m hard problems for e given by work of lutz mayordomo and juedes are indeed more general than the corresponding bounds for leq rm p m hard problems for e the proof of this result introduces a new diagonalization method called martingale diagonalization using this method one simultaneously develops an infinite family of polynomial time computable martingales betting strategies and a corresponding family of languages that defeat these martingales prevent them from winning too much money while also pursuing another agenda martingale diagonalization may be useful for a variety of applications

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 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

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 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

a boundary perturbation analysis for elastic inclusions and interfaces abstract a first order boundary perturbation method based on muskhelishvilli s complex variable representations is formulated for the two dimensional elasticity problem of a nearly circular inclusion embedded in an infinite dissimilar material universal relations which are independent of loading conditions are established among the solutions for a homogeneous infinite plane a perfectly circular inclusion and a slightly perturbed non circular inclusion it is shown that the solution to a circular inclusion can be constructed algebraically from that of an infinite homogeneous plane while the stress distribution along the interface circle leads to the perturbation solutions for a nearly circular inclusion explicit results arc given for inclusions with smooth polygon shapes by considering cosine wavy perturbations along a reference circle a similar analysts is carried out for a bimaterial interface whose shape deviates slightly from a straight line our perturbation results can be used to study elastically induced morphological perturbations of surfaces interfaces voids precipitates and inclusions in a stressed solid as an example we demonstrate that under sufficiently large stresses material surfaces become unstable against a range of diffusional perturbations bounded by two critical wavelengths also as suggested from our perturbation analysis and verified by a finite element calculation even slight surface undulation caused by an unstable morphological perturbation can result in substantial stress concentration along the surface which may cause plastic deformation or brittle fracture before the bulk stress reaches a critical level

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

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

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

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

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

baire categories on small complexity classes and meager comeager laws we introduce two resource bounded baire category notions on small complexity classes such as p quasipoly subexp and pspace and on probabilistic classes such as bpp which differ on how the corresponding finite extension strategies are computed we give an alternative characterization of small sets via resource bounded banach mazur games as an application of the first notion we show that for almost every language a i e all except a meager class computable in subexponential time p a bpp a we also show that almost all languages in pspace do not have small nonuniform complexity we then switch to the second baire category notion called locally computable and show that the class sparse is meager in p we show that in contrast to the resource bounded measure case meager comeager laws can be obtained for many standard complexity classes relative to locally computable baire category on bpp and pspace another topic where locally computable baire categories differ from resource bounded measure is regarding weak completeness we show that there is no weak completeness notion in p based on locally computable baire categories i e every p weakly complete set is complete for p we also prove that the class of complete sets for p under turing logspace reductions is meager in p if p is not equal to dspace logn and that the same holds unconditionally for quasipoly finally we observe that locally computable baire categories are incomparable with all existing resource bounded measure notions on small complexity classes which might explain why those two settings seem to differ so fundamentally

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

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

weakly useful sequences an infinite binary sequence x is defined to be i strongly useful if there is a computable time bound within which every decidable sequence is turing reducible to x and ii weakly useful if there is a computable time bound within which all the sequences in a non measure 0 subset of the set of decidable sequences are turing reducible to x juedes lathrop and lutz theorectical computer science 132 1994 37 proved that every weakly useful sequence is strongly deep in the sense of bennett the universal turing machine a half century survey 1988 227 and asked whether there are sequences that are weakly useful but not strongly useful the present paper answers this question affirmatively the proof is a direct construction that combines the martingale diagonalization technique of lutz siam journal on computing 24 1995 1170 with a new technique namely the construction of a sequence that is computably deep with respect to an arbitrary given uniform reducibility the abundance of such computably deep sequences is also proven and used to show that every weakly useful sequence is computably deep with respect to every uniform reducibility

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

compressibility and resource bounded measure we give a new definition of resource bounded measure based on compressibility of infinite binary strings we prove that the new definition is equivalent to the one commonly used this new characterization offers us a different way to look at resource bounded measure shedding more light on the meaning of measure zero results and providing one more tool to prove such results the main contribution of the paper is the new definition and the proofs leading to the equivalence result we then show how this new characterization can be used to prove that the class of linear autoreducible sets has p measure 0 we also prove that the class of sets that are truth table reducible to a p selective set has p measure 0 and that the class of sets that turing reduce to a subpolynomial dense set has p measure 0 this strengthens various results

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

infinite sets of primes with fast primality tests and quick generation of large primes infinite sets p and q of primes are described p c q for any natural number n it can be decided if n e p in deterministic time o logn 9 this answers affirmatively the question of whether there exists an infinite set of primes whose membership can be tested in polynomial time and is a main result of the paper also for every n e q we show how to randomly produce a proof of the primality of n the expected time is that needed for 1 exponentiations mod n we also show how to randomly generate k digit integers which will be in q with probability proportional to k 1 combined with the fast verification of n e q just mentioned this gives an 0 k4 expected time algorithm to generate and certify primes in a given range and is probably the fastest method to generate large certified primes known to belong to an infinite subset finally it is important that p and q are relatively dense at least cn2 3 log n elements less than n elements of q in a given range may be generated quickly but it would be costly for an adversary to search q in this range a property that could be useful in cryptography

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

robust structural topology optimization considering boundary uncertainties in classical deterministic topology optimization the effect of the possible boundary variations on the performance of the structure is not taken into account which may lead to designs that are very sensitive to manufacturing errors as a consequence the performance of the real structure may be far from optimal and even not meet the design requirements in the present paper structural topology optimization considering the uncertainty of boundary variations is considered via level set approach in order to make the optimal designs less sensitive to the possible boundary variations we choose the compliance and fundamental frequency of structure enduring the worst case perturbation as the objective function for ensuring the robustness of the optimal solution with use of the schwarz inequality the original bi level optimization problem is transformed to a single level optimization problem which can be solved efficiently numerical examples demonstrate the effectiveness of the proposed approach

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

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

singularities interacting with interfaces and cracks abstract solutions to singularities such as point force point moment edge dislocation and transformation strain spot embedded in bonded elastic blocks of dissimilar materials are found to relate to the solutions to the same singularities in an infinite homogeneous plane by a formula independent of the nature of the singularities this universal result is then used to analyze the interactions between singularities and interface cracks the complete solutions and stress intensity factors are presented for two important interface crack configurations

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

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