solving homogeneous linear equations over gf 2 via block wiedemann algorithm we propose a method of solving large sparse systems of homogeneous linear equations over g f 2 the field with two elements we modify an algorithm due to wiedemann a block version of the algorithm allows us to perform 32 matrix vector operations for the cost of one the resulting algorithm is competitive with structured gaussian elimination in terms of time and has much lower space requirements it may be useful in the last stage of integer factorization we address here the problem of solving a large sparse system of homogeneous linear equations over gf 2 the field with two elements one important application which motivates the present work arises in integer factorization during the last stage of most integer factorization algorithms we are presented with a large sparse integer matrix and are asked to find linear combinations of the columns of this matrix which vanish modulo 2 for example 7 the matrix may have 100 000 columns with an average of 15 nonzero entries per column for this application we would like to obtain several solutions because a given solution will lead to a nontrivial factorization with probability 112 with n independent solutions our probability of finding a factorization rises to 1 2 structured gaussian elimination can be used 7 but as problems get larger it may become infeasible to store the matrices obtained in the intermediate stages of gaussian elimination the wiedemann algorithm 9 71 has smaller storage requirements one need only store a few vectors and an encoding of a sparse matrix not a dense matrix as occurs in gaussian elimination after fillin and it may have fewer computational steps since one takes advantage of the sparseness of the matrix but its efficiency is hampered by the fact that the algorithm acts on only one bit at a time in the present paper we work with blocks of vectors at a single time by treating 32 vectors at a time on a machine with 32 bit words we can perform 32 matrix vector products at once thus considerably decreasing the cost of indexing this can be viewed as a block wiedemann algorithm the main technical difficulty is in obtaining the correct generalization of the berlekamp massey algorithm to a block version namely a multidimensional version of the extended euclidean algorithm received by the editor november 20 1991 and in revised form july 24 1992 1991 mathematics subject classijication primary 15a33 11y05 11 04 15 04 1994 amencan mathernat cal soc ety 0025 571 8 94 1 oo 25 per page

shift register synthesis modulo m the berlekamp algorithm takes a sequence of elements from a field and finds the shortest linear recurrence or linear feedback shift register that can generate the sequence we present an algorithm which generalizes berlekamp s to the case when the elements of the sequence are integers modulo m where m is an arbitrary but known integer details will be published elsewhere

computing the inertias in symmetric matrix pencils abstract we define and study generalized pencil eigenvalues for a pencil p s t of real symmetric matrices s and t this allows us to find all occurring inertias among pencil members as bt theoretically and numerically computed examples based on a matlab code are included

a fraction free matrix berlekamp massey algorithm abstract we describe a fraction free version of the matrix berlekamp massey algorithm the algorithm computes a minimal matrix generator of linearly generated square matrix sequences in an integral domain the algorithm performs all operations in the integral domain so all divisions performed are exact for scalar sequences the matrix algorithm specializes to a different algorithm than the algorithm currently in the literature this new scalar algorithm has smaller intermediate values than the known fraction free berlekamp massey algorithm

factoring polynomials over algebraic number fields we show that if f x is a polynomial in z alpha x where alpha satisfies a monic irreducible polynomial over z then f x can be factored over q alpha x in polynomial time we also show that the splitting field of f x can be determined in time polynomial in splitting field of f x q log x

the relationship between breaking the diffie hellman protocol and computing discrete logarithms both uniform and nonuniform results concerning the security of the diffie hellman key exchange protocol are proved first it is shown that in a cyclic group g of order g prod p i e i where all the multiple prime factors of g are polynomial in log g there exists an algorithm that reduces the computation of discrete logarithms in g to breaking the diffie hellman protocol in g and has complexity sqrt max nu p i cdot log g o 1 where nu p stands for the minimum of the set of largest prime factors of all the numbers d in the interval p 2 sqrt p 1 p 2 sqrt p 1 under the unproven but plausible assumption that nu p is polynomial in log p this reduction implies that the diffie hellman problem and the discrete logarithm problem are polynomial time equivalent in g second it is proved that the diffie hellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes there exists a polynomial time reduction algorithm that works for all those groups moreover it is shown that breaking the diffie hellman protocol for a small but nonnegligible fraction of the instances is equally difficult as breaking it for all instances finally efficient constructions of groups are described for which the algorithm reducing the discrete logarithm problem to the diffie hellman problem is efficiently constructible

the space complexity analysis in the general number field sieve integer factorization the general number sieve is the most efficient algorithm for integer factorization it consists of polynomial selection sieving solving equations and finding square roots root lifting of polynomial is discussed in this paper the p adic evaluation provided by each root and the expected p value are also given then we gain the space complexity of sieving and building equations over the ring z 2 z

frobenius maps of abelian varieties and finding roots of unity in finite fields if twere done when tis done then twere well it were done quickly macbeth abstract we give a generalization to abelian varieties over finite fields of the algorithm of schoof for elliptic curves schoof showed that for an elliptic curve e over f given by a weierstrass equation one can compute the number of q f rational points of e in time 0 log our result is the following let a be an abelian variety over f then one can compute the characteristic polynomial of the frobenius endomorphism of a in time 0 o q where a and the implied constant depend only on the dimension of the embedding space of a the number of equations defining a and the addition law and their degrees the method generalizing that of schoof is to use the machinery developed by weil to prove the riemann hypothesis for abelian varieties by means of this theory the calculation is reduced to ideal theoretic computations in a ring of polynomials in several variables over f as applications we show how to count the rational points on the reductions modulo primes p of a fixed curve over q in time polynomial in logp we show also that for a fixed prime we can compute the th roots of unity mod p when they exist in polynomial time in logp this generalizes schoof s application of his algorithm to find square roots of a fixed integer x mod p in this paper we generalize to abelian varieties over finite fields the algorithm of schoof 19 for elliptic curves over finite fields and the application given by schoof for his algorithm schoof showed that for an elliptic curve e over a finite field f given by a weierstrass equation one can compute the number of f rational points of f in time polynomial in log q the algorithm computes the characteristic polynomial p t g z t of the frobenius endomorphism of f for an elliptic curve p t is a monic quadratic polynomial with constant term q and the number of fq rational points of e is f l our result is the following theorem a let a be an abelian variety over a finite field f given explicitly as a projective variety with an explicit addition law then one can compute

cantor versus harley optimization and analysis of explicit formulae for hyperelliptic curve cryptosystems hyperelliptic curves hec look promising for cryptographic applications because of their short operand size compared to other public key schemes the operand sizes seem well suited for small processor architectures where memory and speed are constrained however the group operation has been believed to be too complex and thus hec have not been used in this context so far in recent years a lot of effort has been made to speed up group operation of genus 2 hec in this paper we increase the efficiency of the genus 2 and genus 3 hyperelliptic curve cryptosystems hecc for certain genus 3 curves we can gain almost 80 percent performance for a group doubling this work not only improves gaudry and harley s algorithm but also improves the original algorithm introduced by cantor 1987 contrary to common belief we show that it is also practical for certain curves to use cantor s algorithm to obtain the highest efficiency for the group operation in addition we introduce a general reduction method for polynomials according to karatsuba we implemented our most efficient group operations on pentium and arm microprocessors

asymptotically fast group operations on jacobians of general curves abramovich d 1996 int math res notices p1005 aho a v 1975 design anal computer becker t 1993 graduate texts math v141 brawley j 2004 density primitixe el burgisserr p 1997 fundamentals pricnip v315 cantor dg 1987 math comput v48 p95 doi 10 2307 2007876 chow wl 1954 am j math v76 p453 doi 10 2307 2372585 coppersmith d 1990 j symb comput v9 p251 doi 10 1016 s0747 7171 08 80013 2 decker wolfram 1999 algorithmic algebra p187 eberly w 2000 j symb comput v29 p441 doi 10 1006 jsco 1999 0308 greg w 2002 adv math v172 p169 griffiths p 1994 principles algebraic hartshorne robin 1977 algebraic geometry hess f 1999 thesis technische u hess f 2002 j symb comput v33 p425 doi 10 1006 jsco 2001 0513 huang md 1994 j symb comput v18 p519 doi 10 1006 jsco 1994 1063 kemper g 2002 j symb comput v34 p229 doi 10 1006 jsco 2002 0560 kemper g 2004 math comput v73 p245 khurimakdisi k 2004 asymptotically fast lazarsfeld r 1989 sampling vector bund p500 pari group bord pari gp stein w modular forms databa volcheck e 1994 lect notes computer v877 p221

a rapid method of evaluating the regulator and class number of a pure cubic field let 2 0 be the algebraic number field formed by adjoining 9 to the rationals 2 let r and h be respectively the regulator and class number of shanks has described a method of evaluating r for 2 vd where d is a positive integer his technique improved the speed of the usual continued fraction algorithm for finding r by allowing one to proceed almost directly from the nth to the mth step where m is approximately in in the continued fraction expansion of id this paper shows how shanks idea can be extended to the voronoi algorithm which is used to find r in cubic fields of negative discriminant it also discusses at length an algorithm for finding r and h for pure cubic fields 1 fd d an integer under a certain generalized riemann hypothesis the ideas developed here will provide a new method which will find r and h in 0 02 5 e operations when h is small this is an improvement over the 0 d h operations required by voronoi s algorithm to find r for example with d 200171999 it required only 5 minutes for an amdahl 470 vi computer to find that r 518594546 969083 and h 1 this same calculation would require about 8 days of computer time if it used only the standard voronoi algorithm a 0 and gcd w m2 m3 n2 n3 a 1 also if e m2n3 m3n2 then e a if u e we denote its conjugates by w and we also write the norm of as nia and define it as nico oco w from the simple fact that

computing discrete logarithms in high genus hyperelliptic jacobians in provably subexponential time we provide a subexponential algorithm for solving the discrete logarithm problem in jacobians of high genus hyperelliptic curves over finite fields its expected running time for instances with genus g and underlying finite field fq satisfying g log q for a positive constant is given by o e f 1 3 2 3 2 o 1 g log q log g log q the algorithm works over any finite field and its running time does not rely on any unproven assumptions

improved n term karatsuba like formulas in gf 2 it is well known that chinese remainder theorem crt can be used to construct efficient algorithms for multiplication of polynomials over gf 2 in this note we show how to select an appropriate set of modulus polynomials to obtain minimal number of multiplications

f2 lanczos revisited we present a new variant of the block lanczos algorithm for finding vectors in the kernel of a symmetric matrix over f2 our algorithm is at least as efficient as that of montgomery peter l montgomery a block lanczos algorithm for finding dependencies over gf 2 in advances in cryptology eurocrypt 95 saint malo 1995 lecture notes in comput sci vol 921 springer berlin 1995 pp 106 120 while the sequence of matrices wi constructed here have different algebraic properties that may be useful in eventually providing a provable upper bound on the time required to solve this problem namely our wi satisfy witwj 0 for i j as opposed to witawj 0 in 6

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

linear algebra algorithms for divisors on an algebraic curve we use an embedding of the symmetric dth power of any algebraic curve c of genus g into a grassmannian space to give algorithms for working with divisors on c using only linear algebra in vector spaces of dimension o g and matrices of size o g2 o g when the base field k is finite or if c has a rational point over k these give algorithms for working on the jacobian of c that require o g4 field operations arising from the gaussian elimination our point of view is strongly geometric and our representation of points on the jacobian is fairly simple to deal with in particular none of our algorithms involves arithmetic with polynomials we note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in florian hess 1999 ph d thesis which works with function fields as extensions of k x however for special classes of curves hess algorithms are asymptotically more efficient than ours generalizing other known efficient algorithms for special classes of curves such as hyperelliptic curves cantor 1987 superelliptic curves galbraith paulus and smart 2002 and cab curves harasawa and suzuki 2000 in all those cases one can attain a complexity of o g2

a generic approach to searching for jacobians we consider the problem of finding cryptographically suitable jacobians by applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family we can search for jacobians containing a large subgroup of prime order for a suitable distribution of curves the complexity is subexponential in genus 2 and o n1 12 in genus 3 we give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size improving previous results our approach is particularly effective over low degree extension fields where in genus 2 we find jacobians over fp2 and trace zero varieties over fp3 with near prime orders up to 372 bits in size for p 261 1 the average time to find a group with 244 bit near prime order is under an hour on a pc

solving sparse linear equations over finite fields a coordinate recurrence method for solving sparse systems of linear equations over finite fields is described the algorithms discussed all require o n 1 omega n 1 log k n 1 field operations where n 1 is the maximum dimension of the coefficient matrix omega is approximately the number of field operations required to apply the matrix to a test vector and the value of k depends on the algorithm a probabilistic algorithm is shown to exist for finding the determinant of a square matrix also probabilistic algorithms are shown to exist for finding the minimum polynomial and rank with some arbitrarily small possibility of error

euclid s algorithm and the lanczos method over finite fields this paper shows that there is a close relationship between the euclidean algorithm for polynomials and the lanczos method for solving sparse linear systems especially when working over finite fields it uses this relationship to account rigorously for the appearance of self orthogonal vectors arising in the course of the lanczos algorithm it presents an improved lanczos method which overcomes problems with self orthogonality and compares this improved algorithm with the euclidean algorithm

linearizing torsion classes in the picard group of algebraic curves over finite fields we address the problem of computing in the group of ell k torsion rational points of the jacobian variety of algebraic curves over finite fields with a view toward computing modular representations

the group law on the jacobian of a curve of genus 2 an explicit description is given of the group law on the jacobian of a curve c of genus 2 the kummer surface provides a useful intermediary stage bilinear forms relating to the kummer surface imply that the global group law may be given projectively by biquadratic forms defined over the same ring as the coefficients of c it is not assumed that c has a rational weierstrass point and the theory presented applies over an arbitrary ground field

solving sparse linear systems of equations over finite fields using bit flipping algorithm let fq be the finite field with q elements we give an algorithm for solving sparse linear systems of equations over fq when the coefficient matrix of the system has a specific structure here called relatively connected this algorithm is based on a well known decoding algorithm for low density parity check codes called bit flipping algorithm we modify and extend this hard decision decoding algorithm the complexity of this algorithm is linear in terms of the number of columns n and the number of nonzero coefficients of the matrix per iteration the maximum number of iterations is bounded above by m the number of equations

on the discrete logarithm problem in class groups of curves we study the discrete logarithm problem in degree 0 class groups of curves over finite fields with particular emphasis on curves of small genus we prove that for every fixed g 2 the discrete logarithm problem in degree 0 class groups of curves of genus g can be solved in an expected time of o q 2 2 g where f q is the ground field this result generalizes a corresponding result for hyperelliptic curves given in imaginary quadratic representation with cyclic degree 0 class group and just as this previous result it is obtained via an index calculus algorithm with double large prime variation generalizing this result we prove that for fixed g 0 2 the discrete logarithm problem in class groups of all curves c f q of genus g g 0 can be solved in an expected time of o q g 2 g 0 1 1 g 0 and in an expected time of o cl 0 c 2 g 0 1 1 g0 as a complementary result we prove that for any fixed n with n 2 the discrete logarithm problem in the groups of rational points of elliptic curves over finite fields f q n q a prime power can be solved in an expected time of o q 2 2 n furthermore we give an algorithm for the efficient construction of a uniformly randomly distributed effective divisor of a specific degree given the curve and its l polynomial

computing discrete logarithms in real quadratic congruence function fields of large genus the discrete logarithm problem in various finite abelian groups is the basis for some well known public key cryptosystems recently real quadratic congruence function fields were used to construct a public key distribution system the security of this public key system is based on the difficulty of a discrete logarithm problem in these fields in this paper we present a probabilistic algorithm with subexponential running time that computes such discrete logarithms in real quadratic congruence function fields of sufficiently large genus this algorithm is a generalization of similar algorithms for real quadratic number fields

an l 1 3 algorithm for ideal class group and regulator computation in certain number fields we analyse the complexity of the computation of the class group structure regulator and a system of fundamental units of a certain class of number fields our approach differs from buchmann s who proved a complexity bound of l 1 2 o 1 when the discriminant tends to infinity with fixed degree we achieve a subexponential complexity in o l 1 3 o 1 when both the discriminant and the degree of the extension tend to infinity by using techniques due to enge and gaudry in the context of algebraic curves over finite fields

dynamical study while searching equilibrium solutions in n body problem the theory of complex dynamics is usually applied to compare the global convergence properties of different iterative methods by obtaining the attraction basins for simple polynomial equations in the complex domain however in this work we use it in quite another context the study of a nontrivial nonlinear system that describes the motion of interacting bodies in celestial mechanics namely newtonian planar circular restricted four body problem and its relative equilibrium solutions these have been investigated from a dynamical point of view new properties of the solutions of this system have been obtained practical guidelines for efficient search of relative equilibrium solutions of n body problem have been given

on the equivalence between berlekamp s and euclid s algorithms corresp it is shown that berlekamp s iterative algorithm can be derived from a normalized version of euclid s extended algorithm simple proofs of the results given recently by cheng are also presented

computing the sign or the value of the determinant of an integer matrix a complexity survey computation of the sign of the determinant of a matrix and the determinant itself is a challenge for both numerical and exact methods we survey the complexity of existing methods to solve these problems when the input is an n n matrix a with integer entries we study the bit complexities of the algorithms asymptotically in n and the norm of a existing approaches rely on numerical approximate computations on exact computations or on both types of arithmetic in combination

a polynomial time algorithm to find the shortest cycle basis of a graph define the length of a basis of the cycle space of a graph to be the sum of the lengths of all cycles in the basis an algorithm is given that finds a cycle basis with the shortest possible length in o m 3 n operations where m is the number of edges and n is the number of vertices this is the first known polynomial time algorithm for this problem edges may be weighted or unweighted also the shortest cycle basis is shown to have at most 3 n 1 n 2 2 edges for the unweighted case o mn 2 algorithm to obtain a suboptimal cycle basis of length o n 2 for unweighted graphs is also given

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

classifying quadratic maps from plane to plane let v be a 2 dimensional vector space over an algebraically closed field f of characteristic 2 3 the elements in s 2 v v under the natural action of gl 2 f are classified generi

computing discrete logarithms in the jacobian of high genus hyperelliptic curves over even characteristic finite fields we describe improved versions of index calculus algorithms for solving discrete logarithm problems in jacobians of high genus hyperelliptic curves de ned over even characteristic elds our rst improvement is to incorporate several ideas for the low genus case by gaudry and theriault including the large prime variant and using a smaller factor base into the large genus algorithm of enge and gaudry we extend the analysis in 24 to our new algorithm allowing us to predict accurately the number of random walk steps required to nd all relations and to select optimal degree bounds for the factor base our second improvement is the adaptation of sieving techniques from flassenberg and paulus and jacobson to our setting the new algorithms are applied to concrete problem instances arising from the weil descent attack methodology for solving the elliptic curve discrete logarithm problem demonstrating signi cant improvements in practice

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

stratified economic exchange on networks abstract we investigate a model of stratified economic interactions between agents when the notion of spatial location is introduced the agents are placed on a network with near neighbor connections interactions between neighbors can occur only if the difference in their wealth is less than a threshold value that defines the width of the economic classes by employing concepts from spatiotemporal dynamical systems three types of patterns can be identified in the system as parameters are varied laminar intermittent and turbulent states the transition from the laminar state to the turbulent state is characterized by the activity of the system a quantity that measures the average exchange of wealth over long times the degree of inequality in the wealth distribution for different parameter values is characterized by the gini coefficient high levels of activity are associated to low values of the gini coefficient it is found that the topological properties of the network have little effect on the activity of the system but the gini coefficient increases when the clustering coefficient of the network is increased

additive preconditioning and aggregation in matrix computations we combine our novel svd free additive preconditioning with aggregation and other relevant techniques to facilitate the solution of a linear system of equations and other fundamental matrix computations our analysis and experiments show the power of our algorithms guide us in selecting most effective policies of preconditioning and aggregation and provide some new insights into these and related subjects compared to the popular svd based multiplicative preconditioners our additive preconditioners are generated more readily and for a much larger class of matrices furthermore they better preserve matrix structure and sparseness and have a wider range of applications e g they facilitate the solution of a consistent singular linear system of equations and of the eigenproblem

fast matrix decomposition in f2 in this work an efficient algorithm to perform a block decomposition for large dense rectangular matrices with entries in f 2 is presented matrices are stored as column blocks of row major matrices in order to facilitate rows operation and matrix multiplications with blocks of columns one of the major bottlenecks of matrix decomposition is the pivoting involving both rows and column exchanges since row swaps are cheap and column swaps are order of magnitude slower the number of column swaps should be reduced as much as possible here an algorithm that completely avoids the column permutations is presented an asymptotically fast algorithm is obtained by combining the four russian algorithm and the recursion with the strassen algorithm for matrix matrix multiplication moreover optimal parameters for the tuning of the algorithm are theoretically estimated and then experimentally verified a comparison with the state of the art public domain software sage shows that the proposed algorithm is generally faster

fractal sets attached to homogeneous quadratic maps in two variables abstract the fractal set attached to the iteration of a generic homogeneous quadratic map from the plane to itself is studied and depicted by using a two parametric family of normal forms obtained from the theory of invariants of symmetric bilinear maps f r r r under the full linear group of the plane while invariant theory classifies maps f c c c on the complex plane we confine ourselves to consider maps on the real plane in order to include the results obtained from the theory of two dimensional discrete dynamical systems a discrete number of topological types for such fractals is conjectured to exist

gauss composition for p1 and the universal jacobian of the hurwitz space of double covers abstract in this paper we give an explicit description of the moduli space of line bundles on hyperelliptic curves including singular curves we study the universal jacobian j 2 g n of degree n line bundles over the hurwitz stack of double covers of p 1 by a curve of genus g our main results are the construction of a smooth irreducible universally closed but not separated moduli compactification j bd 2 g n of j 2 g n whose points we describe simply and explicitly as sections of certain vector bundles on p 1 a description of the global geometry and moduli properties of these stacks and a computation of the picard groups of j bd 2 g n and j 2 g n in the cases when n g is even an important ingredient of our work is the parametrization of line bundles on double covers by binary quadratic forms this parametrization generalizes the classical number theoretic correspondence between ideal classes of quadratic rings and integral binary quadratic forms which in particular gives the group law on integral binary quadratic forms first discovered by gauss

the index calculus method using non smooth polynomials we study a generalized version of the index calculus method for the discrete logarithm problem in f q when q p n p is a small prime and n the database consists of the logarithms of all irreducible polynomials of degree between given bounds the original version of the algorithm uses lower bound equal to one we show theoretically that the algorithm has the same asymptotic running time as the original version the analysis shows that the best upper limit for the interval coincides with the one for the original version the lower limit for the interval remains a free variable of the process we provide experimental results that indicate practical values for that bound we also give heuristic arguments for the running time of the waterloo variant and of the coppersmith method with our generalized database

efficient maximum likelihood decoding of ldpc codes over the binary erasure channel we propose an efficient maximum likelihood ml decoding algorithm for decoding low density parity check ldpc codes over the binary erasure channel bec we also analyze the computational complexity of the proposed algorithm

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

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

explict bounds and heuristics on class numbers in hyperelliptic function fields in this paper we provide tight estimates for the divisor class number of hyperelliptic function fields we extend the existing methods to any hyperelliptic function field and improve the previous bounds by a factor proportional to g with the help of new results we thus obtain a faster method of computing regulators and class numbers furthermore we provide experimental data and heuristics on the distribution of the class number within the bounds on the class number these heuristics are based on recent results by katz and sarnak our numerical results and the heuristics imply that our approximation is in general far better than the bounds suggest

arithmetic on superelliptic curves this paper is concerned with algorithms for computing in the divisor class group of a nonsingular plane curve of the form yn c x which has only one point at infinity divisors are represented as ideals and an ideal reduction algorithm based on lattice reduction is given we obtain a unique representative for each divisor class and the algorithms for addition and reduction of divisors run in polynomial time an algorithm is also given for solving the discrete logarithm problem when the curve is defined over a finite field

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

modular equations for hyperelliptic curves we dene modular equations describing the torsion subgroups of the jacobian of a hyperelliptic curve over a nite base eld we prove factorization properties that extend the well known results used in atkin s improvement of schoof s genus 1 point counting algorithm

hermite normal form computation using modulo determinant arithmetic this paper describes a new class of hermite normal form solution procedures which perform modulo determinant arithmetic throughout the computation this class of procedures is shown to possess a polynomial time complexity bound which is a function of the length of the input string computational results are also given

the arithmetic of certain cubic function fields in this paper we discuss the properties of curves of the form y3 f x over a given field k of characteristic different from 3 if f x satisfies certain properties then the jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field we seek to make this connection concrete and then use it to develop an explicit arithmetic for the jacobian of such curves from a purely mathematical perspective this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory at the same time it provides another source of groups which are suitable for diffie hellman type protocols in cryptographic applications

exploring the structural regularities in networks in this paper we consider the problem of exploring structural regularities of networks by dividing the nodes of a network into groups such that the members of each group have similar patterns of connections to other groups specifically we propose a general statistical model to describe network structure in this model a group is viewed as a hidden or unobserved quantity and it is learned by fitting the observed network data using the expectation maximization algorithm compared with existing models the most prominent strength of our model is the high flexibility this strength enables it to possess the advantages of existing models and to overcome their shortcomings in a unified way as a result not only can broad types of structure be detected without prior knowledge of the type of intrinsic regularities existing in the target network but also the type of identified structure can be directly learned from the network moreover by differentiating outgoing edges from incoming edges our model can detect several types of structural regularities beyond competing models tests on a number of real world and artificial networks demonstrate that our model outperforms the state of the art model in shedding light on the structural regularities of networks including the overlapping community structure multipartite structure and several other types of structure which are beyond the capability of existing models

computing the mordell weil rank of jacobians of curves of genus two we derive the equations necessary to perform a two descent on the jacobians of curves of genus two with rational weierstrass points we compute the mordell weil rank of the jacobian of some genus two curves defined over the rationals and discuss the practicality of using this method

the arithmetic of jacobian groups of superelliptic cubics we present two algorithms for the arithmetic of cubic curves with a totally ramified prime at infinity the first algorithm inspired by cantor s reduction for hyperelliptic curves is easily implemented with a few lines of code making use of a polynomial arithmetic package we prove explicit reducedness criteria for superelliptic curves of genus 3 and 4 which show the correctness of the algorithm the second approach quite general in nature and applicable to further classes of curves uses the fglm algorithm for switching between grobner bases for different orderings carrying out the computations symbolically we obtain explicit reduction formulae in terms of the input data

analysis of coppersmith s block wiedemann algorithm for the parallel solution of sparse linear systems by using projections by a block of vectors in place of a single vector it is possible to parallelize the outer loop of iterative methods for solving sparse linear systems we analyze such a scheme proposed by coppersmith for wiedemann s coordinate recurrence algorithm which is based in part on the krylov subspace approach we prove that by use of certain randomizations on the input system the parallel speed up is roughly by the number of vectors in the blocks when using as many processors our analysis is valid for fields of entries that have sufficiently large cardinality our analysis also deals with an arising subproblem of solving a singular block toeplitz system by use of the theory of toeplitz like matrices

factoring multivariate polynomials via partial differential equations a new method is presented for factorization of bivariate polynomials over any field of characteristic zero or of relatively large characteristic it is based on a simple partial differential equation that gives a system of linear equations as in berlekamp s and niederreiter s algorithms for factoring univariate polynomials the dimension of the solution space of the linear system is equal to the number of absolutely irreducible factors of the polynomial to be factored and any basis for the solution space gives a complete factorization by computing gcd s and by factoring univariate polynomials over the ground field the new method finds absolute and rational factorizations simultaneously and is easy to implement for finite fields local fields number fields and the complex number field the theory of the new method allows an effective hilbert irreducibility theorem thus an efficient reduction of polynomials from multivariate to bivariate

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

superintegrable systems and riemann roch theorem in algebraic geometry there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor which existence is guaranteed by the riemann roch theorem we discuss application of this algorithm to construction of finite dimensional superintegrable systems with n degrees of freedom identifying coordinates of the reduced divisor with integrals of motion

efficient matrix preconditioners for black box linear algebra abstract the main idea of the black box approach in exact linear algebra is to reduce matrix problems to the computation of minimum polynomials in most cases preconditioning is necessary to obtain the desired result here good preconditioners will be used to ensure geometrical algebraic properties on matrices rather than numerical ones so we do not address a condition number we offer a review of problems for which algebraic preconditioning is used provide a bestiary of preconditioning problems and discuss several preconditioner types to solve these problems we present new conditioners including conditioners to preserve low displacement rank for toeplitz like matrices we also provide new analyses of preconditioner performance and results on the relations among preconditioning problems and with linear algebra problems thus improvements are offered for the efficiency and applicability of preconditioners the focus is on linear algebra problems over finite fields but most results are valid for entries from arbitrary fields

alignment and integration of complex networks by hypergraph based spectral clustering complex networks possess a rich multiscale structure reflecting the dynamical and functional organization of the systems they model often there is a need to analyze multiple networks simultaneously to model a system by more than one type of interaction or to go beyond simple pairwise interactions but currently there is a lack of theoretical and computational methods to address these problems here we introduce a framework for clustering and community detection in such systems using hypergraph representations our main result is a generalization of the perron frobenius theorem from which we derive spectral clustering algorithms for directed and undirected hypergraphs we illustrate our approach with applications for local and global alignment of protein protein interaction networks between multiple species for tripartite community detection in folksonomies and for detecting clusters of overlapping regulatory pathways in directed networks

aspects of pairing inversion in this paper we discuss some applications of the pairing inversion problem and outline some potential approaches for solving it our analysis of these approaches gives further evidence that pairing inversion is a difficult problem

factorization of polynomials and some linear algebra problems over finite fields abstract the recently developed algorithm of niederreiter for the factorization of polynomials over finite fields provides a linearization of the factorization problem via differential equations in rational function fields we show that the range of applicability of this algorithm can be extended by various techniques such as the application of normal bases and the use of hasse teichmuller derivatives we also discuss topics from pseudorandom vector generation and combinatorial linear algebra that lead to interesting linear algebra problems over finite fields

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 canonical heights using arithmetic intersection theory the canonical height on an abelian variety a defined over a global field k is an object of fundamental importance in the study of the arithmetic of a for many applications it is required to compute p for a given point p a k for instance given generators of a subgroup of the mordell weil group a k of finite index this is necessary for most known approaches to the computation of generators of the mordell weil group a k furthermore the regulator of a k which appears in the statement of the conjecture of birch and swinnerton dyer is defined in terms of the canonical height and thus we need the ability to compute canonical heights in order to gather numerical evidence for the conjecture in the case of positive rank

population stratification using a statistical model on hypergraphs population stratification is a problem encountered in several areas of natural science engineering and public health we tackle this problem by mapping a population and its element attributes onto a hypergraph a natural extension of the concept of graph or network to encode associations among any number of elements on this hypergraph we construct a statistical model reflecting our intuition about how the element attributes can emerge from a postulated population structure finally we introduce the concept of stratification representativeness as a mean to identify the simplest stratification already containing most of the information about the population structure we demonstrate the power of this framework stratifying an animal and a human population based on phenotypic and genotypic properties respectively

factoring with two large primes the study of integer factoring algorithms and the design of faster factoring algorithms is a subject of great importance in cryptology cf 1 and a constant concern for cryptographers in this paper we present a new technique that proved to be extremely useful not only to achieve a considerable speed up of an older and widely studied factoring algorithm but also and more importantly to make practical application of a new factoring algorithm feasible while this first application does not pose serious threats to factorization based cryptosystems the consequences of the second application could be very encouraging from the cryptanalysts point of view

a double large prime variation for small genus hyperelliptic index calculus in this article we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation two algorithms are presented the first algorithm is a rather natural adaptation of the double large prime variation to the intended context on heuristic and experimental grounds it seems to perform quite well but lacks a complete and precise analysis our second algorithm is a considerably simplified variant which can be analyzed easily the resulting complexity improves on the fastest known algorithms computer experiments show that for hyperelliptic curves of genus three our first algorithm surpasses pollard s rho method even for rather small field sizes

information based clustering in an age of increasingly large data sets investigators in many different disciplines have turned to clustering as a tool for data analysis and exploration existing clustering methods however typically depend on several nontrivial assumptions about the structure of data here we reformulate the clustering problem from an information theoretic perspective that avoids many of these assumptions in particular our formulation obviates the need for defining a cluster prototype does not require an a priori similarity metric is invariant to changes in the representation of the data and naturally captures nonlinear relations we apply this approach to different domains and find that it consistently produces clusters that are more coherent than those extracted by existing algorithms finally our approach provides a way of clustering based on collective notions of similarity rather than the traditional pairwise measures

from picard groups of hyperelliptic curves to class groups of quadratic fields let c be a hyperelliptic curve defined over mathbb q whose weierstrass points are defined over extensions of mathbb q of degree at most three and at least one of them is rational generalizing a result of r soleng in the case of elliptic curves we prove that any line bundle of degree 0 on c which is not torsion can be specialised into ideal classes of imaginary quadratic fields whose order can be made arbitrarily large this gives a positive answer for such curves to a question by agboola and pappas

the complexity of the word problems for commutative semigroups and polynomial ideals abstract any decision procedure for the word problems for commutative semigroups and polynomial deals inherently requires computational storage space growing exponentially with the size of the problem instance to which the procedure is applied this bound is achieved by a simple procedure for the semigroup problem