moments of random matrices and hypergeometric orthogonal polynomials we establish a new connection between moments of n times n random matrices xn and hypergeometric orthogonal polynomials specifically we consider moments mathbb e rm tr x n s as a function of the complex variable s in mathbb c whose analytic structure we describe completely we discover several remarkable features including a reflection symmetry or functional equation zeros on a critical line in the complex plane and orthogonality relations an application of the theory resolves part of an integrality conjecture of cunden et al j math phys 57 111901 2016 on the time delay matrix of chaotic cavities in each of the classical ensembles of random matrix theory gaussian laguerre jacobi we characterise the moments in terms of the askey scheme of hypergeometric orthogonal polynomials we also calculate the leading order n rightarrow infty asymptotics of the moments and discuss their symmetries and zeroes we discuss aspects of these phenomena beyond the random matrix setting including the mellin transform of products and wronskians of pairs of classical orthogonal polynomials when the random matrix model has orthogonal or symplectic symmetry we obtain a new duality formula relating their moments to hypergeometric orthogonal polynomials

a strong type 2 2 estimate for a maximal operator associated to the schrodinger equation soit t f x sup t 0 t t f x ou t t f 1 exp it 2 f 1 4 on montre que etant donne un intervalle fini i i t f 2 dx c i r f x 2 dx

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

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

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

mellin transforms of a generalization of legendre polynomials abstract oberle m k s l scott g t gilbert r l hatcher and d f addis mellin transforms of a generalization of legendre polynomials journal of computational and applied mathematics 45 1993 367 369 we show that the zeros and poles of the mellin transforms of polynomials on 1 1 orthogonal with respect to the weight brvbar x 2r with r gt 1 2 are real and simple

regularity of solutions to nonlinear dispersive equations on considere le probleme de cauchy pour les equations dispersives non lineaires de la forme t u ip d u f u x r n t u x 0 u 0 x

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 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

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

on the cauchy problem for higher order nonlinear dispersive equations abstract we study the higher order nonlinear dispersive equation t u x 2 j 1 u 0 j 1 j 2 2 j a j 1 j 2 x j 1 u x j 2 u x t r where u is a real or complex valued function we show that the associated initial value problem is well posed in weighted besov and sobolev spaces for small initial data we also prove ill posedness results when a 0 k 0 for some k j in the sense that this equation cannot have its flow map c 2 at the origin in h s r for any s r the same technique leads to similar ill posedness results for other higher order nonlinear dispersive equation as higher order benjamin ono and intermediate long wave equations

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

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

the berry keating hamiltonian and the local riemann hypothesis the local riemann hypothesis states that the zeros of the mellin transform of a harmonic oscillator eigenfunction on a real or p adic configuration space have a real part 1 2 for the real case we show that the imaginary parts of these zeros are the eigenvalues of the berry keating hamiltonian projected onto the subspace of oscillator eigenfunctions of a lower level this gives a spectral proof of the local riemann hypothesis for the reals in the spirit of the hilbert polya conjecture the p adic case is also discussed

zeros of symmetric quasi definite orthogonal polynomials abstract it is well known that all zeros of real polynomials orthogonal with respect to a positive definite moment functional are real in the first section of this paper we show that the zeros of certain symmetric quasi definite orthogonal polynomials are real or purely imaginary in the second section we apply our finding to mellin transforms of laguerre polynomials generalizing a result of bump and ng 1

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

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

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

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

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

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

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

maximal functions associated to filtrations abstract let t be a bounded linear or sublinear operator from lp y to lq x a maximal operator t f x supj t f yj x is associated to any sequence of subsets yj of y under the hypotheses that q p and the sets yj are nested we prove that t is also bounded classical theorems of menshov and zygmund are obtained as corollaries multilinear generalizations of this theorem are also established these results are motivated by applications to the spectral analysis of schrodinger operators

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

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

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

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

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

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

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

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