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

thue equations and the method of chabauty coleman let ok be any domain with field of fractions k let f x y ok x y be a homogeneous polynomial of degree n coprime to y and assumed to have unit content i e the coefficients of f generate the unit ideal in ok assume that gcd n char k 1 let h ok and assume that the polynomial hzn f x y is irreducible in k x y z we denote by x f h k the nonsingular complete model of the projective plane curve cf h k defined by the equation hzn f x y 0 we shall assume in this article that g x f h 2 when k is a number field mordell s conjecture now faltings theorem implies that x f h k caporaso harris and mazur chm 1 1 have shown that if lang s conjecture for varieties of general type is true then for any number field k the size x k of the set of k rational points of any curve x k of genus g x 2 can be bounded by a constant depending only on g x prior to the paper chm mazur and others had asked whether x k can be bounded by a constant depending only on g x and the mordell weil rank of x k over k that is the rank of the group j k of k rational points of the jacobian j k of x k these farreaching questions are totally open as we shall recall in sect 1 the method of chabauty coleman sometimes yields a bound for x f h k depending only on g x f h when it is known in advance that the mordell weil rank of x f h k is small unfortunately the chabauty coleman method does not yield a bound for x f h k independent of the coefficients of hzn f x y for all curves of the form x f h it does however produce such a nice bound for the number of primitive integral solutions of f x y h as we now explain let k q and ok z a classical thue equation is an equation f x y h where f x 1 does not have repeated roots thue showed in 1909 that such an equation has finitely many solutions x y z2 if n 3

morphisms of berkovich curves and the different function abstract given a generically etale morphism f y x of quasi smooth berkovich curves we define a different function f y 0 1 that measures the wildness of the topological ramification locus of f this provides a new invariant for studying f which cannot be obtained by the usual reduction techniques we prove that f is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical riemann hurwitz formula and show that f can be used to explicitly construct the simultaneous skeletons of x and y as another application we use our results to completely describe the topological ramification locus of f when its degree equals to the residue characteristic p

most odd degree hyperelliptic curves have only one rational point consider the smooth projective models c of curves y 2 f x with f x 2 z x monic and separable of degree 2g 1 we prove that for g 3 a positive fraction of these have only one rational point the point at innity we prove a lower bound on this fraction that tends to 1 as g 1 finally we show that c q can be algorithmically computed for such a fraction of the curves the method can be summarized as follows using p adic analysis and an idea of mccallum we develop a reformulation of chabauty s method that shows that certain computable conditions imply c q 1 on the other hand using further p adic analysis the theory of arithmetic surfaces a new result on torsion points on hyperelliptic curves and crucially the bhargava gross theorems on the average number and equidistribution of nonzero 2 selmer group elements we prove that these conditions are often satised for p 2

a hierarchy of polynomial time lattice basis reduction algorithms we present a hierarchy of polynomial time lattice basis reduction algorithms that stretch from lenstra lenstra lovasz reduction to korkine zolotareff reduction let l be the length of a shortest nonzero element of a lattice l we present an algorithm which for k n finds a nonzero lattice vector b so that b 2 6k2 nk l 2 this algorithm uses o n2 kk o k n2 log b arithmetic operations on o n log b bit integers this holds provided that the given basis vectors b1 bn zn are integral and have the length bound b this algorithm successively applies korkine zolotareff reduction to blocks of length k of the lattice basis we also improve kannan s algorithm for korkine zolotareff reduction

how many rational points does a random curve have a large part of modern arithmetic geometry is dedicated to or motivated by the study of rational points on varieties for an elliptic curve over q the set of rational points forms a finitely generated abelian group the ranks of these groups when ranging over all elliptic curves are conjectured to be evenly distributed between rank 0 and rank 1 with higher ranks being negligible we will describe these conjectures and discuss some results on bounds for average rank highlighting the recent work of bhargava and shankar

security of the most significant bits of the shamir message passing scheme boneh and venkatesan have recently proposed a polynomial time algorithm for recovering a hidden element of a finite field fp of p elements from rather short strings of the most significant bits of the remainder modulo p of t for several values of t selected uniformly at random from fp unfortunately the applications to the computational security of most significant bits of private keys of some finite field exponentiation based cryptosystems given by boneh and venkatesan are not quite correct for the diffie hellman cryptosystem the result of boneh and venkatesan has been corrected and generalized in our recent paper here a similar analysis is given for the shamir message passing scheme the results depend on some bounds of exponential sums

mesures et equidistribution sur les espaces de berkovich the proof by ullmo and zhang of bogomolov s conjecture about points of small height in abelian varieties made a crucial use of an equidistribution property for small points in the associated complex abelian variety we study the analogous equidistribution property at p adic places our results can be conveniently stated within the framework of the analytic spaces defined by berkovich the first one is valid in any dimension but is restricted to algebraic metrics the second one is valid for curves but allows for more general metrics in particular to the normalized heights with respect to dynamical systems

hidden number problem with hidden multipliers timed release crypto and noisy exponentiation we consider a generalisation of the hidden number problem recently introduced by boneh and venkatesan the initial problem can be stated as follows recover a number a fp such that for many known random t fp approximations to the values of at p are known here we study a version of the problem where the multipliers t are not known but rather certain approximations to them are given we present a probabilistic polynomial time solution when the error is small enough and we show that the problem cannot be solved if the error is sufficiently large we apply the result to the bit security of timed release crypto introduced by rivest shamir and wagner to noisy exponentiation black boxes and to the bit security of the inverse exponentiation we also show that it implies a certain bit security result for weil pairing on elliptic curves