a distinguisher for high rate mceliece cryptosystems the goppa code distinguishing gd problem consists in distinguishing the matrix of a goppa code from a random matrix the hardness of this problem is an assumption to prove the security of code based cryptographic primitives such as mceliece s cryptosystem up to now it is widely believed that the gd problem is a hard decision problem we present the first method allowing to distinguish alternant and goppa codes over any field our technique can solve the gd problem in polynomial time provided that the codes have sufficiently large rates the key ingredient is an algebraic characterization of the key recovery problem the idea is to consider the rank of a linear system which is obtained by linearizing a particular polynomial system describing a key recovery attack it appears that this dimension depends on the type of code considered explicit formulas derived from extensive experimentations for the rank are provided for generic random alternant and goppa codes over any field finally we give theoretical explanations of these formulas in the case of random codes alternant codes over any field of characteristic two and binary goppa codes

centers and isochronous centers for generalized quintic systems in this paper we classify the centers and the isochronous centers of certain polynomial differential systems in r 2 of degree d 5 odd that in complex notation are z i z z z d 5 2 a z 5 b z 4 z c z 3 z 2 d z 2 z 3 e z z 4 f z 5 where z x i y r and a b c d e f c note that if d 5 we obtain the class of polynomial differential systems in the form of a linear system with homogeneous polynomial nonlinearities of degree 5 due to the huge computations required for computing the necessary and sufficient conditions for the characterization of the centers and isochronous centers our study uses algorithms of computational algebra based on the grobner basis theory and on modular arithmetics

implementation of a new algorithm of computation of the poincare liapunov constants in the last years many papers giving different methods to compute the poincare liapunov constants have been published in appl math warsaw 28 2002 17 a new method to compute recursively all the poincare liapunov constants as a function of the coefficients of the system for an arbitrary analytic system which has a perturbed linear center at the origin was given and thus a theoretical answer to the classical center problem was given the method also computes the coefficients of the poincare series as a function of the same coefficients we describe its implementation in two different ways by means of a computer algebra system and an algorithm in any computer language if this second alternative is used later it is necessary to translate the results so that they can be manipulated with a computer algebra system we describe also how the availability of symbolic manipulation procedures has recently led to a significant progress in the resolution of the different problems related with the poincare liapunov constants as they are the central problems like the small amplitude limit cycles

the number of limit cycles of certain polynomial differential equations two dimensional differential systems are considered where p and q are polynomials the question of interest is the maximum possible numberof limit cycles of such systems in terms of the degree of p and q an algorithm is described for determining a so called focal basis this can be implemented on a computer estimates can then be obtained for the number of small amplitude limit cycles the technique is applied to certain cubic systems a class of examples with exactly five small amplitude limit cycles is constructed quadratic systems are also considered

folding alternant and goppa codes with non trivial automorphism groups the main practical limitation of the mceliece public key encryption scheme is probably the size of its key a famous trend to overcome this issue is to focus on subclasses of alternant goppa codes with a non trivial automorphism group such codes display then symmetries allowing compact parity check or generator matrices for instance a key reduction is obtained by taking quasi cyclic qc or quasi dyadic qd alternant goppa codes we show that the use of such symmetric alternant goppa codes in cryptography introduces a fundamental weakness it is indeed possible to reduce the key recovery on the original symmetric public code to the key recovery on a much smaller code that has no symmetry anymore this result is obtained thanks to an operation on codes called folding that exploits the knowledge of the automorphism group this operation consists in adding the coordinates of codewords which belong to the same orbit under the action of the automorphism group the advantage is twofold the reduction factor can be as large as the size of the orbits and it preserves a fundamental property folding the dual of an alternant respectively goppa code provides the dual of an alternant respectively goppa code a key point is to show that all the existing constructions of alternant goppa codes with symmetries follow a common principal of taking codes whose support is globally invariant under the action of affine transformations by building upon prior works of berger and dur this enables not only to present a unified view but also to generalize the construction of qc qd and even quasi monoidic goppa codes finally our results can be harnessed to boost up any key recovery attack on mceliece systems based on symmetric alternant or goppa codes and in particular algebraic attacks

parameters of goppa codes revisited we discuss parameters of goppa 1970 codes such as minimum distance covering radius distance distribution and generalized hamming weights by a variation on the exponential sums method and combinatorial arguments we sharpen known bounds on these parameters

a method for characterizing nilpotent centers to characterize when a nilpotent singular point of an analytic differential system is a center is of particular interest first for the problem of distinguishing between a focus and a center and after for studying the bifurcation of limit cycles from it or from its period annulus we give an effective algorithm in the search of necessary conditions for detecting nilpotent centers based in recent developments moreover we survey the last results on this problem and illustrate our approach by means of examples

polynomial first integrals via the poincare series in appl math 28 2001 17 a method to compute the poincare liapunov constants for an arbitrary analytic differential system which has a linear center at the origin in function of the coefficients of the system was given this method also computes the coefficients of the poincare series in function of the same coefficients in this work we use this method to determine polynomial differential systems which have a polynomial first integral

limit cycle bifurcations from a non degenerate center in this work we discuss the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number moreover we calculate the minimum number of bautin ideal generators which give the number of small limit cycles under certain hypothesis about the generators in particular we consider polynomial systems of the form x y pn x y y x qn x y where pn and qn are a homogeneous polynomial of degree n we use center bifurcation rather than multiple hopf bifurcations used a previous work 19 to estimate the cyclicity of a unique singular point of focus center type for n 4 5 6 7 and compare with the results given by the conjecture presented in 18

algorithmic derivation of centre conditions the conditions for a critical point of a polynomial differential system to be a centre are of particular significance because of the frequency with which they are required in applications we demonstrate how computer algebra can be effectively employed in the search for necessary and sufficient conditions for critical points of such systems to be centres we survey recent developments and illustrate our approach by means of examples

an analytic numerical method for computation of the liapunov and period constants derived from their algebraic structure we consider the problem of computing the liapunov and the period constants for a smooth differential equation with a nondegenerate critical point first we investigate the structure of both constants when they are regarded as polynomials on the coefficients of the differential equation second we take advantage of this structure to derive a method to obtain the explicit expression of the above mentioned constants although this method is based on the use of the runge kutta fehlberg methods of orders 7 and 8 and the use of richardson s extrapolation it provides the real expression for these constants

bifurcations of limit cycles in complex networks of hamiltonian systems and lloyd s conjecture abstract this paper is motivated by hilbert s sixteenth problem concerning the number and distribution of limit cycles in planar polynomial vector fields we propose an original method inspired by the complex systems framework in order to construct a family of hamiltonian systems admitting non degenerate centers located on the vertices of a planar network we introduce two novel polynomial perturbations adapted to those systems with the aim to produce a high number of limit cycles the first one is oriented along the gradient of the unperturbed system and the second one is again constructed with a complex systems approach using the melnikov integral we reach in a small number of particular cases the lower bound of cubical order for the hilbert number h n conjectured by lloyd in 1988

a method of constructing cycles without contact around a weak focus has a weak focus then limit cycles can be made to appear in its neighborhood by varying slightly the right sides 11 frommer 2 bautin 3 chin et al 4 and tung 5 are examples of excellent expositions of poincare s idea it is necessary to go from numerical examples to general theory in the first part of this paper we present a method of constructing cycles without contact around a weak focus which makes clear poincart s idea used in the above work 2 51 in the second part another method to analyze global phase portraits is used quadratic systems with at least four limit cycles are produced by means of both the above methods this contradicts the results of petrovskii and landis 6

minimum number of ideal generators for a linear center perturbed by homogeneous polynomials abstract using the algorithm presented in j gine x santallusia on the poincare liapunov constants and the poincare series appl math warsaw 28 1 2001 17 30 the poincare liapunov constants are calculated for polynomial systems of the form x y p n x y y x q n x y where p n and q n are homogeneous polynomials of degree n the objective of this work is to calculate the minimum number of ideal generators i e the number of functionally independent poincare liapunov constants through the study of the highest fine focus order for n 4 and n 5 and compare it with the results that give the conjecture presented in j gine on the number of algebraically independent poincare liapunov constants appl math comput 188 2 2007 1870 1877 moreover the computational problems which appear in the computation of the poincare liapunov constants and the determination of the number of functionally independent ones are also discussed

polynomial time attack on wild mceliece over quadratic extensions we present a polynomial time structural attack against the mceliece system based on wild goppa codes defined over a quadratic finite field extension we show that such codes can be efficiently distinguished from random codes the attack uses this property to compute a filtration that is to say a family of nested subcodes which will reveal their secret algebraic description

conditions for the existence of a center for the kukles homogeneous systems abstract in this work we study necessary and sufficient conditions for the existence of centers in two families of linear centers with homogeneous quartic and quintic nonlinearities systems of this class are called kukles homogeneous systems systems of this type were studied for the first time by kukles who studied a linear center with cubic nonhomogeneous nonlinearities

on the number of algebraically independent poincare liapunov constants abstract in this paper an upper bound for the number of algebraically independent poincare liapunov constants in a certain basis for planar polynomial differential systems is given finally it is conjectured that an upper bound for the number of functionally independent poincare liapunov quantities would be m 2 3 m 7 where m is the degree of the polynomial differential system moreover the computational problems which appear in the computation of the poincare liapunov constants and in the determination of the center cases are also discussed

on the structure of poincare lyapunov constants for the weak focus of polynomial vector fields in this paper generalized poincare lyapunov constants vi i 1 2 are defined and an explicit formula is given for the lyapunov functions it is shown that vi is determined modulo v1 vi 1 this is used for the investigation and creation of limit cycles

a new research into default logic in previous papers some important properties of extensions of general default theories were given in order to dedicate further research to default logic a characterization of extensions is presented again and some new algorithms for reasoning tasks in default logic are presented in this paper a class of default theories auto compatible default theory is also developed we show that the characterization and notions of compatibility and auto compatibility suitably applied to logic programs yield some sufficient conditions of existence of answer sets all these essentially develop the theories of reiter and his followers

higher order limit cycle bifurcations from non degenerate centers abstract the computational problems which appear in the computation of the poincare liapunov constants and the determination of their functionally independent number has led in recent works to consider only the lowest terms of these constants in this work we improve the results obtained in this direction for polynomials systems of the form x y p n x y y x q n x y where p n and q n are a homogeneous polynomial of degree n we use center bifurcation to estimate the cyclicity of a unique singular point of focus center type for different values of n and compare with the results given by the conjecture presented in 15

squares of random linear codes given a linear code c one can define the d th power of c as the span of all componentwise products of d elements of c a power of c may quickly fill the whole space our purpose is to answer the following question does the square of a code typically fill the whole space we give a positive answer for codes of dimension k and length roughly frac 1 2 k 2 or smaller moreover the convergence speed is exponential if the difference k k 1 2 n is at least linear in k the proof uses random coding and combinatorial arguments together with algebraic tools involving the precise computation of the number of quadratic forms of a given rank and the number of their zeros

centers for a class of generalized quintic polynomial differential systems abstract we characterize the centers of the planar polynomial differential systems of degree d 5 odd that in complex notation writes as z iz z z d 5 2 az 5 bz 4 z cz 3 z 2 dz 2 z 3 ez z 4 f z 5 where a b c d e f c and either a re e 0 or a im e 0 or re a e 0 or im a e 0