equations for the modular curve and models of elliptic curves with torsion points we describe an algorithm for constructing plane models of the modular curve x 1 n and discuss the resulting equations when n 51

polynomial factorization over f 2 we describe algorithms for polynomial factorization over the binary field f2 and their implementation they allow polynomials of degree up to 250 000 to be factored in about one day of cpu time distributing the work on two processors

sublinear root detection and new hardness results for sparse polynomials over finite fields we present a deterministic 2 o t q frac t 2 t 1 o 1 algorithm to decide whether a univariate polynomial f with t monomial terms and degree q has a root in the finite field math

on the finiteness of differential invariants abstract in this paper we show how weil s theory of near points yields a new light on the classical approaches to the study of the differential invariants of a sheaf of tangent vector fields we give conditions for the existence of invariant derivations for a sheaf of tangent vector fields which allows to apply lie s algorithm to obtain new differential invariants as quotients of jacobian determinants of known ones we give sufficient conditions for the asymptotic stability of the symbol of a sheaf of tangent vector fields and prove our main result a finiteness theorem for the differential invariants of a sheaf of lie algebras which simplifies and improves on the treatment given in j differential geom 10 1975 249 416

geometry of geometrically finite one dimensional maps we study the geometry of certain one dimensional maps as dynamical systems we prove the property of bounded and bounded nearby geometry of certainc 1 one dimensional maps with finitely many critical points this property enables us to give the quasisymmetric classification of geometrically finite one dimensional maps

a note on grobner bases and berlekamp s algorithm it will be shown that it is possible to factorize square free polynomials over finite fields using grobner bases and a maple implementation of this method will be described

fast algorithms for computing isogenies between elliptic curves we survey algorithms for computing isogenies between elliptic curves defined over a field of characteristic either 0 or a large prime we introduce a new algorithm that computes an isogeny of degree l l different from the characteristic in time quasi linear with respect to l this is based in particular on fast algorithms for power series expansion of the weierstrass function and related functions

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

the complexity of partial derivatives abstract let l denote the nonscalar complexity in k x 1 x n we prove l x 1 x n 3l using this we determine the complexity of single power sums single elementary symmetric functions the resultant and the discriminant as root functions up to order of magnitude also we linearly reduce matrix inversion to computing the determinant

symmetries of pde systems and correspondences between jet spaces abstract let m be a manifold a pde system r j m 1 m can be prolonged to another one r t m jimenez et al 2005 10 in analogy with the higher order symmetries symmetries of r will be called higher dimensional symmetries of r for a broad class of pde systems we prove that every infinitesimal or finite symmetry of r comes from another one of r we show that r does not have internal infinitesimal symmetries modulo trivial symmetries this fact allows us in the infinitesimal case to compute the internal symmetries of r as external symmetries of r we also give an algorithmic method to obtain solutions of r invariant by a given internal symmetry

geometry of cantor systems a cantor system is defined the geometry of a certain family of cantor systems is studied such a family arises in dynamical systems as hyperbolicity is created we prove that the bridge geometry of a cantor system in such a family is uniformly bounded and that the gap geometry is regulated by the size of the leading gap

on elliptic curves with an isogeny of degree 7 on elliptic curves with an isogeny of degree 7 arxiv 1007 4617v3 math nt 16 oct 2012 r greenberg k rubin a silverberg and m stoll abstract we show that if e is an elliptic curve over q with a q rational isogeny of degree 7 then the image of the 7 adic galois representation attached to e is as large as allowed by the isogeny except for the curves with complex multiplication by q 7 the analogous result with 7 replaced by a prime p 7 was proved by the first author in 8 the present case p 7 has additional interesting complications we show that any exceptions correspond to the rational points on a certain curve of genus 12 we then use the method of chabauty to show that the exceptions are exactly the curves with complex multiplication as a by product of one of the key steps in our proof we determine exactly when there exist elliptic curves over an arbitrary field k of characteristic not 7 with a k rational isogeny of degree 7 and a specified galois action on the kernel of the isogeny and we give a parametric description of such curves 1 introduction suppose that e is an elliptic curve defined over q p is a rational prime t p e is the p adic tate module of e and g q gal q q then there is a natural homomorphism e p g q aut z p t p e giving the action of g q on t p e since t p e is a free z p module of rank 2 aut z p t p e can be identified non canonically with gl 2 z p serre 20 showed that if e does not have complex multiplication cm then e p g q has finite index in aut t p e and e p is surjective for all but finitely many primes p suppose now that e has an isogeny of degree p that is defined over q in other words e has a q rational p isogeny mazur 15 showed that p is then in the finite set 2 3 5 7 11 13 17 19 37 43 67 163 and if further e is non cm then p 2 3 5 7 11 13 17 37 the kernel of the isogeny is a q rational subgroup of order p since aut of e q f p the action of g q on is given by a homomorphism g q f p we refer to as the character of the isogeny the isogeny and the corresponding character put an obvious constraint on the image of the map e p in particular e p cannot be surjective if e has complex multiplication then the additional endomorphisms of e put another constraint on the image of e p in that case e p g q is a p adic lie group of dimension 2 we wish to understand whether these are the only constraints or equivalently whether there are any non cm elliptic curves over q for which e p g q does not contain 2000 mathematics subject classification 11g05 primary 11d45 11f80 11g18 11g30 14g05 secondary key words and phrases elliptic curves isogeny explicit families galois representations method of chabauty this material is based upon work supported by the national science foundation under grants dms 0200785 dms 0757807 and cns 0831004

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

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

a remark on goldschmidt s theorem on formal integrability abstract in this paper we apply weil bundle techniques to the study of formal integrability for systems of nonlinear partial differential equations we clarify the role of the curvature and show that both the statement and proof of goldschmidt s criterion on formal integrability are of algebraic nature this fact allows us to obtain a version of this theorem which holds for smooth algebraic or analytic manifolds finally we give a definition for the characteristic co vectors of a system of partial differential equations and how their relationship with the cauchy kowalevski normal form

isomorphisms between artin schreier towers we give a method for efficiently computing isomorphisms between towers of artin schreier extensions over a finite field we find that isomorphisms between towers of degree p over a fixed field fq can be computed composed and inverted in time essentially linear in p n the method relies on an approximation process

the black box niederreiter algorithm and its implementation over the binary field the most time consuming part of the niederreiter algorithm for factoring univariate polynomials over finite fields is the computation of elements of the nullspace of a certain matrix this paper describes the so called black box niederreiter algorithm in which these elements are found by using a method developed by wiedemann the main advantages over an approach based on gaussian elimination are that the matrix does not have to be stored in memory and that the computational complexity of this approach is lower the black box niederreiter algorithm for factoring polynomials over the binary field was implemented in the c programming language and benchmarks for factoring high degree polynomials over this field are presented these benchmarks include timings for both a sequential implementation and a parallel implementation running on a small cluster of workstations in addition the wan algorithm which was recently introduced is described and connections between implementation aspects of wan s and niederreiter s algorithm are given

on the asymptotic complexity of rectangular matrix multiplication abstract the numher of essential multiplications required to multiply matrices of size n x n and n x n is studied as a function f s 1 bounds to f x sharper than trivial ones are presented and the asymptotic behawour of f s 1 is studied an analogous investigation is performed for the problem of multiplying matrices of size n x n and n x n 1 introduction and preliminaries the complexity of square n x n matrix multiplication has been investigated by several authors l lo a lower bound to the number of operations is n and the best known upper bound is o n2 16h 8 some results on rectangular matrix multiplication have been obtained by brockett and dobkin a who have shown the upper bound n o n to the non scalar complexity of n x n by n x log n matrix multiplication recently coppersmith s found the upper bound o n log n for the n x n by n x n matrix multipli cation prnblem with n s 2 log 2 5 log 5 0 1722

subquadratic time factoring of polynomials over finite fields new probabilistic algorithms are presented for factoring univariate polynomials over finite fields the algorithms factor a polynomial of degree n over a finite field of constant cardinality in time o n 1 815 previous algorithms required time n 2 o 1 the new algorithms rely on fast matrix multiplication techniques more generally to factor a polynomial of degree n over the finite field f q with q elements the algorithms use o n 1 815 log q arithmetic operations in f q the new baby step giant step techniques used in our algorithms also yield new fast practical algorithms at super quadratic asymptotic running time and subquadratic time methods for manipulating normal bases of finite fields

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

nearly optimal algorithms for canonical matrix forms a las vegas type probabilistic algorithm is presented for finding the frobenius canonical form of an n times n matrix t over any field kk the algorithm requires softo mm n mm n cdot log n o 1 operations in kk where o mm n operations in kk are sufficient to multiply two n times n matrices over kk this nearly matches the lower bound of omega mm n operations in kk for this problem and improves on the o n 4 operations in kk required by the previously best known algorithms a fast parallel implementation of the algorithm is also demonstrated for the frobenius form which is processor efficient on a pram as an application we give an algorithm to evaluate a polynomial g in kk x at t which requires only softo mm n operations in kk when deg g leq n 2 other applications include sequential and parallel algorithms for computing the minimal and characteristic polynomials of a matrix the rational jordan form of a matrix for testing whether two matrices are similar and for matrix powering which are substantially faster than those previously known

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

a new algorithm for factoring polynomials over finite fields in f t there are two standard methods one is due to e berlekamp the other appears to be a folk method both are described by d knuth in 3 pp 381 397 further improvements for special values of q are given by r moenck in 5j see also e berlekamp 1 note that both methods as described in the references apply only when d 1 however straightforward modifications described below allow d 1 both methods use the calculation of resultants or equivalently the solution of linear equations to reduce the problem to finding the roots of a polynomial which has all of its roots in f when p is very small probabilistic methods are used an improvement to the folk method method along with a more explicit calculation of the work required has recently been given by m rabin 6 we present here a new probabilistic method which when combined with the above algorithms avoids the need for both resultants and linear equations it leads to algorithms which are conceptually simpler than previous methods moreover it works equally well for all finite fields f regardless of the magnitude of q when used for factoring a quadratic x2 a it reduces to berlekamp s algorithm other standard algorithms for factoring quadratics are due to d h lehmer 4 and d shanks 7 our algorithm is also suitable for finding solutions of polynomial equations over finite fields

a shorter proof of the existence of the feigenbaum fixed point we use the leray schauder fixed point theorem to prove the existence of an analytic fixed point for the period doubling accumulation renormalization operator our argument does not however show that the linearization of the renormalization operator at this fixed point is hyperbolic

a note on elliptic curves over finite fields let e be an elliptic curve over a finite field k and let e k be the group of k rational points on e we evaluate all the possible groups e k where e runs through all the elliptic curves over a given fixed finite field k

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

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

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

computing isogenies between elliptic curves over f p n using couveignes s algorithm the heart of the improvements by elkies to schoof s algorithm for computing the cardinality of elliptic curves over a finite field is the ability to compute isogenies between curves elkies approach is well suited for the case where the characteristic of the field is large couveignes showed how to compute isogenies in small characteristic the aim of this paper is to describe the first successful implementation of couveignes s algorithm in particular we describe the use of fast algorithms for performing incremental operations on series we also insist on the particular case of the characteristic 2

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

factoring polynomials over arbitrary finite fields abstract we analyse an extension of shoup s inform process lett 33 1990 261 267 deterministic algorithm for factoring polynomials over finite prime fields to arbitrary finite fields in particular we prove the existence of a deterministic algorithm which completely factors all monic polynomials of degree n over f q q odd except possibly o n 2 log 2 q q polynomials using o n 2 e log 2 q arithmetical operations in f q

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

primality testing using elliptic curves we present a primality proving algorithm a probablistic primality test that produces short certificates of primality on prime inputs we prove that the test runs in expected polynomial time for all but a vanishingly small fraction of the primes as a corollary we obtain an algorithm for generating large certified primes with distribution statistically close to uniform under the conjecture that the gap between consecutive primes is bounded by some polynomial in their size the test is shown to run in expected polynomial time for all primes yielding a las vegas primality test our test is based on a new methodology for applying group theory to the problem of prime certification and the application of this methodology using groups generated by elliptic curves over finite fields we note that our methodology and methods have been subsequently used and improved upon most notably in the primality proving algorithm of adleman and huang using hyperelliptic curves and in practical primality provers using elliptic curves

modular polynomials via isogeny volcanoes we present a new algorithm to compute the classical modular polynomial l in the rings z x y and z mz x y for a prime l and any positive integer m our approach uses the graph of l isogenies to eciently compute l mod p for many primes p of a suitable form and then applies the chinese remainder theorem crt under the generalized riemann hypoth esis grh we achieve an expected running time of o l 3 logl 3 log logl and compute l mod m using o l 2 logl 2 l 2 logm space we have used the new algorithm to compute l with l over 5000 and l mod m with l over 20000 we also consider several modular functions g for which g is smaller than l allowing us to handle l over 60000

counting points on elliptic curves over f2m in 1985 schoof 136 presented a polynomial time algorithm for computing e f q the number of fq rational points on an elliptic curve e defined over the field f q the algorithm has a running time of 0 log8 q bit operations and is rather cumbersome in practice buchmann and muller 20 combined schoof s algorithm with shanks baby step giant step algorithm and were able to compute orders of curves over fp where p is a 27 decimal digit prime the algorithm took 4 5 hours on a sun 1 sparc station

chinese match and alternative to atkin s match and sort metjhod used in the sea algorithm a classical way to compute the number of points of elliptic curves defined over finite fields from partial data obtained in sea schoof elkies atkin algorithm is a so called match and sort method due to atkin this method is a baby step giant step way to find the number of points among c candidates with o sqrt c elliptic curve additions observing that the partial information modulo atkin s primes is redundant we propose to take advantage of this redundancy to eliminate the usual elliptic curve algebra in this phase of the sea computation this yields an algorithm of similar complexity but the space needed is smaller than what atkin s method requires in practice our technique amounts to an acceleration of atkin s method allowing us to count the number of points of an elliptic curve defined over mathbb f 2 1663 as far as we know this is the largest point counting computation to date furthermore the algorithm is easily parallelized

integrability of lie equations and pseudogroups in this paper the theory of jets based on weil s near points is applied to lie equations and pseudogroups linear systems of partial differential equations are interpreted in a canonical way as distributions on the fibre bundles of invertible jets invariant under translations we prove the two fundamental theorems for lie equations and generalize the results of rodrigues a geometric correspondence between linear and nonlinear lie equations is given and the symbols of a linear lie equation and its prolongations are canonically identified with the symbols of their attached nonlinear equations from this fact we deduce that a linear lie equation verifies the conditions of goldsmichmidt s criterion on formal integrability if and only if its attached nonlinear lie equation satisfies them locally finally we define the cartan 1 form on the fibre bundle of invertible jets and give a global form to the equivalence between the lie and cartan definitions of continuous groups

internal external and generalized symmetries backlund s theorem which characterizes contact transformations is generalized to give an analogous characterization of internal symmetries of systems of differential equations for a wide class of systems of differential equations we prove that every internal symmetry comes from a first order generalized symmetry and conversely every first order generalized symmetry satisfying certain explicit contact conditions determines an internal symmetry we analyze the contact conditions in detail deducing powerful necessary conditions for a system of differential equations that admit genuine internal symmetries i e ones which do not come from classical external symmetries applications include a direct proof that both the internal symmetry group and the first order generalized symmetries of a remarkable differential equation due to hilbert and cartan are the noncompact real form of the exceptional simple lie group g2

computing the l power torsion of an elliptic curve over a finite field the algorithm we develop outputs the order and the structure including generators of the l sylow subgroup of the group of rational points of an elliptic curve defined over a finite field to do this we do not assume any knowledge of the group order we are able to choose points in such a way that a linear number of successive l divisions leads to generators of the subgroup under consideration after the computation of a couple of polynomials each division step relies on finding rational roots of polynomials of degree l we specify in complete detail the case l 3 when the complexity of each trisection is given by the computation of cubic roots in finite fields

hensel lifting and bivariate polynomial factorisation over finite fields this paper presents an average time analysis of a hensel lifting based factorisation algorithm for bivariate polynomials over finite fields it is shown that the average running time is almost linear in the input size this explains why the hensel lifting technique is fast in practice for most polynomials

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 transformation of pde systems related to the drach theorem we will define a new transformation of pde systems as follows given a particular pde system f there is a new system f whose solutions are the spaces of elements attached to the solutions of f we will show that the system f is a second order pde system in a single unknown as an application we will derive as well a global version of the drach theorem

rapid multiplication of rectangular matrices the number of essential multiplications required to multiply matrices of size n times n and n times n 0 172 is bounded by cn 2 log 2 n

connections between the algorithms of berlekamp and niederreiter for factoring polynomials over fq abstract in this paper we determine an explicit isomorphism between the solution spaces of berlekamp s and niederreiter s algorithm for factoring polynomials over finite fields this can be used for the implementation of a procedure combining the benefits of both methods

fast polynomial factorization and modular composition we obtain randomized algorithms for factoring degree n univariate polynomials over mathbb f q requiring o n 1 5 o 1 rm log 1 o 1 q n 1 o 1 rm log 2 o 1 q bit operations when rm log q n this is asymptotically faster than the best previous algorithms j von zur gathen and v shoup comput complexity 2 1992 pp 187 224 e kaltofen and v shoup math comp 67 1998 pp 1179 1197 for rm log q ge n it matches the asymptotic running time of the best known algorithms the improvements come from new algorithms for modular composition of degree n univariate polynomials which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields the best previous algorithms for modular composition use o n omega 1 2 field operations where omega is the exponent of matrix multiplication r p brent and h t kung j assoc comput mach 25 1978 pp 581 595 with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication x huang and v y pan j complexity 14 1998 pp 257 299 we show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent in the sense that an algorithm for one achieving exponent alpha implies an algorithm for the other with exponent alpha o 1 and vice versa we then give two new algorithms that solve the problem near optimally an algebraic algorithm for fields of characteristic at most n o 1 and a nonalgebraic algorithm that works in arbitrary characteristic the latter algorithm works by lifting to characteristic 0 applying a small number of rounds of multimodular reduction and finishing with a small number of multidimensional ffts the final evaluations are reconstructed using the chinese remainder theorem as a bonus this algorithm produces a very efficient data structure supporting polynomial evaluation queries which is of independent interest our algorithms use techniques that are commonly employed in practice in contrast to all previous subquadratic algorithms for these problems which relied on fast matrix multiplication

constructive recognition of finite alternating and symmetric groups acting as matrix groups on their natural permutation modules we present a las vegas algorithm which for a given matrix group known to be isomorphic modulo scalars to a finite alternating or symmetric group acting on the fully deleted permutation module produces an explicit isomorphism with the standard permutation representation of the group this algorithm exploits information available from the matrix representation and thereby is faster than existing black box recognition algorithms applied to these groups in particular it uses the fact that certain types of elements in these groups can be identified and constructed from the structure of their characteristic polynomials the algorithm forms part of a large scale program for computing with groups of matrices over finite fields when combined with existing black box recognition algorithms the results of this paper prove that any d dimensional absolutely irreducible matrix representation of a finite alternating or symmetric group over a finite field can be recognised with o d1 2 random group elements and o d1 2 matrix multiplications up to some logarithmic factors

generators of function fields of the modular curves x_1 5 and x_1 6 we show that the modular functions j 1 5 and j 1 6 generate function fields of the modular curves x 1 n n 5 6 respectively and find some number theoretic properties of these modular functions

trading grh for algebra algorithms for factoring polynomials and related structures in this paper we develop techniques that eliminate the need of the generalized riemann hypothesis grh from various almost all known results about deterministic polynomial factoring over finite fields our main result shows that given a polynomial f x of degree n over a finite field k we can find in deterministic poly n log n log k time either a nontrivial factor of f x or a nontrivial automorphism of k x f x of order n this main tool leads to various new grh free results most striking of which are 1 given a noncommutative algebra a of dimension n over a finite field k there is a deterministic poly n logn log k time algorithm to find a zero divisor in a this is the best known deterministic grh free result since r 1990 first studied the problem of finding zero divisors in finite algebras and showed that this problem has the same complexity as factoring polynomials over finite fields 2 given a positive integer r 4 such that either 4 r or r has two distinct prime factors there is a deterministic polynomial time algorithm to find a nontrivial factor of ther th cyclotomic polynomial over a finite field this is the best known deterministic grh free result since evdokimov 1989 showed that cyclotomic polynomials can be factored over finite fields in deterministic polynomial time assuming grh in this paper following the seminal work of lenstra 1991 on constructing isomorphisms between finite fields we further generalize classical galois theory constructs