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

a survey of public key cryptosystems we give an overview of the most important public key cryptosystems and discuss the difficult task of evaluating the merit of such systems

gevrey solvability near the characteristic set for a class of planar complex vector fields of infinite type abstract we study the gevrey solvability of a class of complex vector fields defined on s 1 given by l t a x i b x x b 0 near the characteristic set 0 s 1 we show that the interplay between the order of vanishing of the functions a and b at x 0 plays a role in the gevrey solvability

refined analysis to the extended tower number field sieve abstract the hardness of discrete logarithm problem over finite fields is the security foundation of many cryptographic protocols when the characteristic of the finite field is medium or large the state of art algorithms for solving the corresponding problem are the number field sieve and its variants in 2016 kim and barbulescu presented the extended tower number field sieve which achieves a new complexity in the medium prime case and imposes a new estimation of the security of concrete parameters in certain cryptosystems such as pairing based cryptosystems in this paper a refined analysis to this algorithm is given as follows firstly a uniform formula is given for the total complexity of the extended tower number field sieve for a given polynomial selection method this formula can directly give the complexity in this case then a method is proposed to improve the computation in the smoothing phase by exploring subfield structures when the extension degree is composite at last the complexity of the descent phase is analyzed when sieving over degree one polynomials and high degree polynomials respectively and it is shown still negligible compared to the improved smoothing phase

normalization of complex valued planar vector fields which degenerate along a real curve abstract taking as a start point the recent article of meziani 7 we present several results concerning the normalization of a class of complex vector fields in the plane which degenerate along a real curve we mainly deal with operators with finite regularity and analyze both the local situation as well as the case of normalization near a circle some related questions e g on semi global solvability and on the normalization of a class of generalized mizohata operators are also discussed

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

geometric norm equality related to the harmonicity of the poisson kernel for homogeneous siegel domains in this paper we present a geometric norm equality involving an admissible linear form for the shilov boundary of a homogeneous siegel domain d we prove that the validity of this norm equality is equivalent to the symmetry of d and the reduction of essentially to the koszul form this in particular reveals a geometric reason that the poisson kernel is annihilated by the laplace beltrami operator if and only if d is symmetric a theorem due to hua look koranyi and xu

global solvability of real analytic complex vector fields in two variables abstract this paper deals with the global solvability of a complex vector field with real analytic coefficients in two real variables the vector field is assumed to satisfy the nirenberg treves condition p for local solvability normal forms for the vector field near the one dimensional orbits are obtained and a generalization of the riemann hilbert problem is considered

the mizohata structure on the sphere we prove that a compact surface that admits a mizohata structure is homeomorphic to a sphere and that there exists exactly one such structure l up to conjugation by diffeomorphisms we also characterize the range and the kernel of the operator 0 induced by l i e obtained from the exterior derivative on functions by passing to the quotient modulo l

representations of the affine transformation groups acting simply transitively on siegel domains let g be the split solvable lie group acting simply transitively on a siegel domain d we consider irreducible unitary representations of g realized on hilbert spaces of holomorphic functions on d we determine all such hilbert spaces by connecting them with positive riesz distributions on the dual cone and describe them through the fourier laplace transform moreover we classify the representations of g by making use of the orbit method

a rigorous time bound for factoring integers in this paper a probabilistic algorithm is exhibited that factors any positive integer n into prime factors in expected time at most ln 2 1 o for n oo where l a b exp b logx a loglogx l a many practical factoring algorithms including the quadratic sieve and the elliptic curve method are conjectured to have an expected running time that satisfies the same bound but this is the first algorithm for which the bound can be rigorously proved nevertheless this does not close the gap between rigorously established time bounds and merely conjectural ones for factoring algorithms this is due to the advent of a new factoring algorithm the number field sieve which is conjectured to factor any positive integer n in time ln i 0 1 the algorithm analyzed in this paper is a variant of the class group relations method which makes use of class groups of binary quadratic forms of negative discriminant this algorithm was first suggested by seysen and later improved by a k lenstra who showed that the algorithm runs in expected time at most ln 12 1 o 1 if one assumes the generalized riemann hypothesis the main device for removing the use of the generalized riemann hypothesis from the proof is the use of multipliers in addition a character sum estimate for algebraic number fields is used with an explicit dependence on possible exceptional zeros of the corresponding l functions another factoring algorithm using class groups that has been proposed is the random class groups method it is shown that there is a fairly large set of numbers that this algorithm cannot be expected to factor as efficiently as had previously been thought department of mathematics university of california berkeley california 94720 e mail address hwl math berkeley edu department of mathematics university of georgia athens georgia 30602 e mail address carl ada math uga edu this content downloaded from 157 55 39 186 on tue 12 apr 2016 08 54 17 utc all use subject to http about jstor org terms

local solvability inlpof first order linear operators we prove that first order partial differential operators of principal type with smooth coefficients are locally solvable inlp 1 p if they satisfy condition p

global properties of a class of vector fields in the plane on etudie l existence des solutions globales au probleme lu o du o sur r 2 ou l est un champ vectoriel sans singularite on se restreint au cas d operateurs localement integrables et satisfaisant certaines proprietes

determinant type differential operators on homogeneous siegel domains abstract we study the differential operators whose symbols are determinant type polynomials associated to homogeneous cones in connection with unitary representations of the split solvable lie group acting simply transitively on a siegel domain

using number fields to compute logarithms in finite fields we describe an adaptation of the number field sieve to the problem of computing logarithms in a finite field we conjecture that the running time of the algorithm when restricted to finite fields of an arbitrary but fixed degree is l q 1 3 64 9 1 3 o 1 where q is the cardinality of the field l q s c exp c log q s log log q 1 s and the o 1 is for q the number field sieve factoring algorithm is conjectured to factor a number the size of q in the same amount of time

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

generic hardness of inversion on ring and its relation to self bilinear map abstract in this paper we study the generic hardness of the inversion problem on a ring which is a problem to compute the inverse of a given prime c by just using additions subtractions and multiplications on the ring if the characteristic of an underlying ring is public and coprime to c then it is easy to compute the inverse of c by using the extended euclidean algorithm on the other hand if the characteristic is hidden it seems difficult to compute it for discussing the generic hardness of the inversion problem we first extend existing generic ring models to capture a ring of an unknown characteristic then we prove that there is no generic algorithm to solve the inversion problem in our model when the underlying ring is isomorphic to z p for a randomly chosen prime p assuming the hardness of factorization of an unbalanced modulus we also study a relation between the inversion problem on a ring and a self bilinear map namely we give a construction of a self bilinear map based on a ring on which the inversion problem is hard and prove that natural complexity assumptions including the multilinear computational diffie hellman mcdh assumption hold w r t the resulting sef bilinear map

factoring polynomials with rational coefficients in this paper we present a polynomial time algorithm to solve the following problem given a non zero polynomial fe q x in one variable with rational coefficients find the decomposition of f into irreducible factors in q x it is well known that this is equivalent to factoring primitive polynomials fez x into irreducible factors in z x here we call f z x primitive if the greatest common divisor of its coefficients the content of f is 1 our algorithm performs well in practice cf 8 its running time measured in bit operations is o nl2 n9 log fd3

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

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

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

elliptic planar vector fields with degeneracies this paper deals with the normalization of elliptic vector fields in the plane that degenerate along a simple and closed curve the associated homogeneous equation lu 0 is studied and an application to a degenerate beltrami equation is given

elliptic curves over a finite field and the trace formula we prove formulas for power moments for point counts of elliptic curves over a finite field k such that the groups of k points of the curves contain a chosen subgroup these formulas express the moments in terms of traces of hecke operators for certain congruence subgroups of sl 2 z as our main technical input we prove an eichler selberg trace formula for a family of congruence subgroups of sl 2 z which include as special cases the groups 1 n and n our formulas generalize results of birch and ihara the case of the trivial subgroup and the full modular group and previous work of the authors the subgroups z 2z and z 2z 2 and congruence subgroups 0 2 0 4 we use these formulas to answer statistical questions about point counts for elliptic curves over a fixed finite field generalizing results of vladu gekeler howe and others

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

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

dual rsa and its security analysis we present new variants of an rsa whose key generation algorithms output two distinct rsa key pairs having the same public and private exponents this family of variants called dual rsa can be used in scenarios that require two instances of rsa with the advantage of reducing the storage requirements for the keys two applications for dual rsa blind signatures and authentication secrecy are proposed in addition we also provide the security analysis of dual rsa compared to normal rsa the security boundary should be raised when applying dual rsa to the types of small d small e and rebalanced rsa

computational complexity of sentences over fields hilbert s irreducibility theorem is applied to find the upper bounds of the time complexities of various decision problems in arithmetical sentences and the following results are proved 1 the decision problem of email protected sentences over an algebraic number field is in p 2 the decision problem of email protected sentences over the collection of all fields with characteristic 0 is in p 3 the decision problem of email protected sentences over a function field with characteristic p is polynomial time reducible to the factorization of polynomials over z p 4 the decision problem of email protected sentences over the collection of all fields with characteristic p is polynomial time reducible to the factorization of polynomials over z p 5 the decision problem of email protected sentences over the collection of all fields is polynomial time reducible to the factorization of integers over z and the factorization of polynomials over finite fields

a class of planar vector fields with homogeneous singular points solvability and boundary value problems abstract this paper deals with the solvability of planar complex vector fields with homogeneous degeneracies holder continuous solutions are obtained via a cauchy type integral operator associated to the vector field an associated boundary value problem of riemann hilbert type is also considered

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

aurifeuillian factorizations and the period of the bell numbers modulo a prime we show that the minimum period modulo p of the bell exponential integers is p p 1 p 1 for all primes p 102 and several larger p our proof of this result requires the prime factorization of these periods for some primes p the factoring is aided by an algebraic formula called an aurifeuillian factorization we explain how the coefficients of the factors in these formulas may be computed

a deterministic version of pollard s p 1 algorithm in this article we present applications of smooth numbers to the unconditional derandomization of some well known integer factoring algorithms we begin with pollard s p 1 algorithm which finds in random polynomial time the prime divisors p of an integer n such that p 1 is smooth we show that these prime factors can be recovered in deterministic polynomial time we further generalize this result to give a partial derandomization of the k th cyclotomic method of factoring k 2 devised by bach and shallit we also investigate reductions of factoring to computing euler s totient function we point out some explicit sets of integers n that are completely factorable in deterministic polynomial time given n these sets consist roughly speaking of products of primes p satisfying with the exception of at most two certain conditions somewhat weaker than the smoothness of p 1 finally we prove that o ln n oracle queries for values of are sufficient to completely factor any integer n in less than exp 1 o 1 ln n 1 3 ln ln n 2 3 deterministic time

operator valued bessel functions on schrodinger fock spaces and siegel domains of type ii in this paper a general theory of operator valued bessel functions on mixed schrodinger fock spaces and siegel domains of type ii is presented in general bessel functions j are associated to the partial fourier transform on a mixed schrodinger fock space j can be realized as a reduced bessel function k which is a generalized inverse laplace transform on a siegel domain of type ii the application of this theory to representation theory will be studied in the next paper

the probability that the number of points on the jacobian of a genus 2 curve is prime in 2000 galbraith and mckee heuristically derived a formula that estimates the probability that a randomly chosen elliptic curve over a fixed finite prime field has a prime number of rational points we show how their heuristics can be generalized to jacobians of curves of higher genus we then elaborate this in genus g 2 and study various related issues such as the probability of cyclicity and the probability of primality of the number of points on the curve itself finally we discuss the asymptotic behavior for g

a characterization of bounded symmetric domains by curvature this paper will prove that a bounded homogeneous domain is symmetric if and only if in the bergman metric all sectional curvatures are nonpositive introduction it is well known that a bounded symmetric domain has nonpositive sectional curvature in the bergman metric this paper is devoted to the converse namely that a bounded homogeneous domain equivalently homogeneous siegel domain which has nonpositive sectional curvature in the bergman metric must be a symmetric domain the proof uses the techniques of normal j algebras 13 as well as results of vinberg 16 17 the first author 1 3 and indirectly dorfmeister 5 6 the body of the paper is in two sections in the first we derive certain relations between the dimensions of the root spaces in an irreducible normal j algebra which in the presence of the additional curvature assumption show the equality of the dimensions of all root spaces corresponding to certain roots for which no multiple is a root and the equality of the dimensions of all root spaces corresponding to roots which are half of another root in the second section we show that the first dimension result implies that the cone of the corresponding siegel domain is self dual while the two dimension results together imply that the domain is quasi symmetric the proof is then finished because 3 a quasi symmetric domain with nonpositive sectional curvature in the bergman metric is known to be symmetric we remark that some of the subsidiary results mentioned above are not essentially new however for consistency of presentation it is necessary to have them in the language of normal jalgebras and for that reason they are included there is also overlap here with work of zelow lundquist 21 1 throughout this paper s j will denote a normal j algebra with admissible form w this means that s is a finite dimensional real split solvable lie algebra with almost complex structure j s s such that x y j jx y j x jy jx jy and w is a linear form on s such that the bilinear form o jx y is symmetric positive definite and j invariant note that the assumption that s is real split was incorrectly omitted in 2 let n s s and let a be the orthogonal complement of n in s by the basic structure theorem of pyatetskii shapiro 13 a is a commutative received by the editors july 13 1981 1980 mathematics subiect classification primary 32mi0 53c30 1983 american mathematical society 0002 9947 82 000 0759 09 00

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