existence of primitive divisors of lucas and lehmer numbers we prove that for n 30 every n th lucas and lehmer number has a primitive divisor this allows us to list all lucas and lehmer numbers without a primitive divisor

cryptanalysis of large rsa exponent by using the lll algorithm the study of paper cryptanalysis of rsa with private key d less than n 0 2 9 2 ieee trans information theory 46 2000 1339 which boneh and durfee published in ieee transactions on information theory in july 2000 supported that when d

higher identities for the ternary commutator we use computer algebra to study polynomial identities for the trilinear operation a b c abc acb bac bca cab cba in the free associative algebra it is known that a b c satisfies the alternating property in degree 3 no new identities in degree 5 a multilinear identity in degree 7 which alternates in 6 arguments and no new identities in degree 9 we use the representation theory of the symmetric group to demonstrate the existence of new identities in degree 11 the only irreducible representations of dimension 400 with new identities correspond to partitions 25 1 and 24 13 and have dimensions 132 and 165 we construct an explicit new multilinear identity for partition 25 1 and we demonstrate the existence of a new non multilinear identity in which the underlying variables are permutations of a2b2c2d2e2f

the optimal lll algorithm is still polynomial in fixed dimension in this paper we consider the open problem of the complexity of the lll algorithm in the case when the approximation parameter of the algorithm has as its extreme value 1 this case is of interest because the output is then the strongest lovasz reduced basis experiments reported by lagarias and odlyzko j acm 32 1 1985 229 seem to show that the algorithm remains polynomial in average however no bound better than a naive exponential order one is established for the worst case complexity of the optimal lll algorithm even for fixed small dimension higher than 2 here we prove that for any fixed dimension n the number of iterations of the lll algorithm is linear with respect to the size of the input it is easy to deduce from vallee j algorithms 12 1991 556 that the linear order is optimal moreover in 3 dimensions we give a tight bound for the maximum number of iterations and we characterize precisely the output basis our bound also improves the known one for the usual non optimal lll algorithm

fast multiple antenna differential decoding we present an algorithm based on lattice reduction for the fast decoding of diagonal differential modulation across multiple antenna while the complexity of the maximum likelihood ml algorithm is exponential both in the number of antenna and the rate the complexity of our approximate lattice algorithm is polynomial in the number of antennas and the rate we show that the error performance of our lattice algorithm is very close to the ml algorithm

arithmetic multivariate descartes rule let inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i be any number field or p adic field and consider f f 1 f k where f 1 f k inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i x 1 x n and no more than distinct exponent vectors occur in the monomial term expansions of the f i we prove that f has no more than 1 c n n 3 log n n geometrically isolated roots in inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i n where c is an explicit and effectively computable constant depending only on inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 04i this gives a significantly sharper arithmetic analogue of khovanski s theorem on real fewnomials and a higher dimensional generalization of an earlier result of hendrik w lenstra jr for the special case of a single univariate polynomial we also present some further refinements of our new bounds and a higher dimensional generalization of a bound of lipshitz on p adic complex roots connections to non archimedean amoebae and computational complexity including additive complexity and solving for the geometrically isolated rational roots are discussed along the way we thus provide the foundations for an effective arithmetic analogue of fewnomial theory

solving a system of linear diophantine equations with lower and upper bounds on the variables we develop an algorithm for solving a system of diophantine equations with lower and upper bounds on the variables the algorithm is based on lattice basis reduction it first finds a short vector satisfying the system of diophantine equations and a set of vectors belonging to the nullspace of the constraint matrix due to basis reduction all these vectors are relatively short the next step is to branch on linear combinations of the null space vectors which either yields a vector that satisfies the bound constraints or provides a proof that no such vector exists the research was motivated by the need for solving constrained diophantine equations as subproblems when designing integrated circuits for video signal processing our algorithm is tested with good results on real life data and on instances from the literature

the cross entropy method with patching for rare event simulation of large markov chains there are various importance sampling schemes to estimate rare event probabilities in markovian systems such as markovian reliability models and jackson networks in this work we present a general state dependent importance sampling method which partitions the state space and applies the cross entropy method to each partition we investigate two versions of our algorithm and apply them to several examples of reliability and queueing models in all these examples we compare our method with other importance sampling schemes the performance of the importance sampling schemes is measured by the relative error of the estimator and by the efficiency of the algorithm the results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator

importance sampling for weighted serve the longest queue this paper considers buffer overflow probabilities for stable queueing systems with one server and different classes of arrivals the service priority is given to the class of customers whose current weighted queue size is the largest weighted serve the longest queue policy we explicitly identify the exponential decay rate for the rare event probabilities of interest and construct asymptotically optimal importance sampling schemes for simulation

solving low density subset sum problems the subset sum problem is to decide whether or not the 0 l integer programming problem s n i l a i x i m i x i 0 or 1 has a solution where the a i and m are given positive integers this problem is np complete and the difficulty of solving it is the basis of public key cryptosystems of knapsack type an algorithm is proposed that searches for a solution when given an instance of the subset sum problem this algorithm always halts in polynomial time but does not always find a solution when one exists it converts the problem to one of finding a particular short vector v in a lattice and then uses a lattice basis reduction algorithm due to a k lenstra h w lenstra jr and l lovasz to attempt to find v the performance of the proposed algorithm is analyzed let the density d of a subset sum problem be defined by d n log 2 max i a i then for almost all problems of density d d n it is proved that the lattice basis reduction algorithm locates v extensive computational tests of the algorithm suggest that it works for densities d d c n where d c n is a cutoff value that is substantially larger than 1 n this method gives a polynomial time attack on knapsack public key cryptosystems that can be expected to break them if they transmit information at rates below d c n as n

a large deviation principle for join the shortest queue we consider a join the shortest queue model which is as follows there are k single fifo servers and m arrival processes the customers from a given arrival process can be served only by the servers from a certain subset of all servers the actual destination is the server with the smallest weighted queue length the arrival processes are assumed to obey a large deviation principle while service is exponential a large deviation principle is established for the queue length process the action functional is expressed in terms of solutions to mathematical programming problems the large deviation limit point is identified as a weak solution to a system of idempotent equations uniqueness of the weak solution follows by trajectorial uniqueness

on the undecidability of computing stationary distributions and large deviation rates for constrained random walks we consider a constrained homogeneous random walk in z d such random walks are used to model various stochastic processes most importantly multiclass markovian queueing networks operating under state dependent scheduling policies these applications motivate the need to compute various performance measures including stationary probability distributions and large deviations rates in this paper we show that computing the stationary probability distributions exactly is an algorithmically undecidable problem that is no algorithm can possibly exist to solve this task the problem remains undecidable even if a linear lyapunov function that verifies positive recurrence of the underlying random walk is available as a part of the input we then prove that computing large deviation rates for this model is also an undecidable problem specifically we show that it is an undecidable problem to determine finiteness of large deviations rates

on the spectrum of markov semigroups via sample path large deviations the essential spectral radius of a sub markovian process is defined as the infimum of the spectral radiuses of all local perturbations of the process when the family of rescaled processes satisfies sample path large deviation principle the spectral radius and the essential spectral radius are expressed in terms of the rate function the paper is motivated by applications to reflected diffusions and jump markov processes describing stochastic networks for which the sample path large deviation principle has been established and the rate function has been identified while essential spectral radius has not been calculated

integer forcing linear receivers linear receivers are often used to reduce the implementation complexity of multiple antenna systems in a traditional linear receiver architecture the receive antennas are used to separate out the codewords sent by each transmit antenna which can then be decoded individually although easy to implement this approach can be highly suboptimal when the channel matrix is near singular this paper develops a new linear receiver architecture that uses the receive antennas to create an effective channel matrix with integer valued entries rather than attempting to recover transmitted codewords directly the decoder recovers integer combinations of the codewords according to the entries of the effective channel matrix the codewords are all generated using the same linear code which guarantees that these integer combinations are themselves codewords provided that the effective channel is full rank these integer combinations can then be digitally solved for the original codewords this paper focuses on the special case where there is no coding across transmit antennas and no channel state information at the transmitter s which corresponds either to a multiuser uplink scenario or to single user v blast encoding in this setting the proposed integer forcing linear receiver significantly outperforms conventional linear architectures such as the zero forcing and linear minimum mean squared error receiver in the high signal to noise ratio regime the proposed receiver attains the optimal diversity multiplexing tradeoff for the standard multiple input multiple output mimo channel with no coding across transmit antennas it is further shown that in an extended mimo model with interference the integer forcing linear receiver achieves the optimal generalized degrees of freedom

effective hilbert irreducibility in this paper we prove by entirely elementary means a very effective version of the hilbert irreducibility theorem we then apply our theorem to construct a probabilistic irreducibility test for sparse multivariate polynomials over arbitrary perfect fields for the usual coefficient fields the test runs in polynomial time in the input size

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

jordan quadruple systems abstract we define jordan quadruple systems by the polynomial identities of degrees 4 and 7 satisfied by the jordan tetrad a b c d a b c d d c b a as a quadrilinear operation on associative algebras we find further identities in degree 10 which are not consequences of the defining identities we introduce four infinite families of finite dimensional jordan quadruple systems and construct the universal associative envelope for a small system in each family we obtain analogous results for the anti tetrad a b c d a b c d d c b a our methods rely on computer algebra especially linear algebra on large matrices the lll algorithm for lattice basis reduction representation theory of the symmetric group noncommutative grobner bases and wedderburn decompositions of associative algebras

integer programming with a fixed number of variables it is shown that the integer linear programming problem with a fixed number of variables is polynomially solvable the proof depends on methods from geometry of numbers

sentences over integral domains and their computational complexities letrbe a hilbertian domain and letkbe its fraction field let x1 xn y be a quantifier free arithmetical formula overr we may also take x1 xn y to be an arithmetical formula overkx1 xn and write it as y in this paper we show that ifrhas enough non units and x1 xn y x1 xn y called an n sentence is true inr then y y is true inkx1 xn also ifr kt wherekis an infinite integral domain and x1 xn y x1 xn y is true inr then y y is true inrx1 xn these results are applied to find the upper and lower bounds of the time complexities of various decision problems on diophantine equations with parameters and arithmetical sentences some of the results are 1 the decision problem of sentences and diophantine equations with parameters over the ring of integers of a global field are co np complete 2 the decision problem of sentences over the ring of integers of a global field is np complete 3 letkbe an infinite domain the time complexities of the decision problems of equations with parameters and sentences over the polynomial ringkt are polynomial time reducible to factoring polynomials overk 4 the decision problem of sentences over all algebraic integer rings is in p 5 the decision problem of sentences over all integral domains with characteristic 0 is in p 6 the time complexity of the decision problem of sentences over all integral domains is polynomial time reducible to factoring integers overzand factoring polynomials over finite fields

large deviations for processes with discontinuous statistics this paper is devoted to the problem of sample path large deviations for the markov processes on r n having a constant but different transition mechanism on each boundary set x x i 0 for i f a x i 0 for i the global sample path large deviation principle and an integral representation of the rate function are derived from local large deviation estimates our results complete the proof of dupuis and ellis of the sample path large deviation principle for markov processes describing a general class of queueing networks

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

a multidimensional continued fraction based on a high order recurrence relation the paper describes and studies an iterative algorithm for finding of a set of linear forms over vectors of integers the algorithm small values of a set of linear forms over vectors of integers the algorithm uses a linear recurrence relation to generate a vector sequence the basic idea being to choose the integral coefficients in the recurrence relation in such a way that the linear forms take small values subject to the requirement that the integers should not become too large the problem of choosing good coefficients for the recurrence relation is thus related to the problem of finding a good approximation of a given vector by a vector in a certain one parameter family of lattices the novel feature of our approach is that practical formulae for the coefficients are obtained by considering the limit as the parameter tends to zero the paper discusses two rounding procedures to solve the underlying inhomogeneous diophantine approximation problem the first which we call naive rounding leads to a multidimensional continued fraction algorithm with suboptimal asymptotic convergence properties in particular when it is applied to the familiar problem of simultaneous rational approximation the algorithm reduces to the classical jacobi perron algorithm the second rounding procedure is babai s nearest plane procedure we compare the two rounding procedures numerically our experiments suggest that the multidimensional continued fraction corresponding to nearest plane rounding converges at an optimal asymptotic rate

nonhomogeneous subalgebras of lie and special jordan superalgebras abstract we consider polynomial identities satisfied by nonhomogeneous subalgebras of lie and special jordan superalgebras we ignore the grading and regard the superalgebra as an ordinary algebra the lie case has been studied by volichenko and baranov they found identities in degrees 3 4 and 5 which imply all the identities in degrees 6 we simplify their identities in degree 5 and show that there are no new identities in degree 7 the jordan case has not previously been studied we find identities in degrees 3 4 5 and 6 which imply all the identities in degrees 6 and demonstrate the existence of further new identities in degree 7 our proofs depend on computer algebra we use the representation theory of the symmetric group the hermite normal form of an integer matrix the lll algorithm for lattice basis reduction and the chinese remainder theorem

how to compute the volume in high dimension in some areas of theoretical computer science we feel that randomized algorithms are better and in some others we can prove that they are more efficient than the deterministic ones

tile invariants new horizons let t be a finite set of tiles the group of invariants g t introduced by pak trans ams 352 2000 5525 is a group of linear relations between the number of copies of tiles in tilings of the same region we survey known results about g the height function approach the local move property various applications and special cases

reduced and fixed complexity variants of the lll algorithm for communications the lenstra lenstra lovasz lll algorithm is a popular lattice reduction algorithm in communications in this paper variants of the lll algorithm with either reduced or fixed complexity are proposed and analyzed specifically the use of effective lll reduction for lattice decoding is presented where size reduction is only performed for pairs of consecutive basis vectors its average complexity measured by the number of floating point operations and averaged over i i d standard normal lattice bases is shown to be o n 3 log n where n is the lattice dimension this average complexity is an order lower than previously thought to address the issue of variable complexity of the lll algorithm two fixed complexity approximations are proposed one is fixed complexity effective lll for which the first vector of the basis is proven to be bounded in length the other is fixed complexity lll with deep insertion which is shown to be closely related to the well known v blast algorithm such fixed complexity structures are much desirable in hardware implementation since they allow straightforward constant throughput implementation

martin boundary of a reflected random walk on a half space the complete representation of the martin compactification for reflected random walks on a half space mathbb z d times mathbb n is obtained it is shown that the full martin compactification is in general not homeomorphic to the radial compactification obtained by ney and spitzer for the homogeneous random walks in mathbb z d convergence of a sequence of points z n in mathbb z d 1 times mathbb n to a point of on the martin boundary does not imply convergence of the sequence zn zn on the unit sphere sd our approach relies on the large deviation properties of the scaled processes and uses pascal s method combined with the ratio limit theorem the existence of non radial limits is related to non linear optimal large deviation trajectories

fundamental limits of linear equalizers diversity capacity and complexity linear equalizers les have been widely adopted for practical systems due to their low computational complexity however it is also well known that les provide inferior performance relative to a maximum likelihood equalizer mle or other near mles because les usually cannot collect the diversity order enabled by the transmitter and at the same time they lose mutual information more importantly unlike mle or near mles the performance of les has not been well quantified this hinders more general applications of les in wireless systems in this paper we reveal a fundamental link between a channel parameter orthogonality deficiency od of the channel matrix and the diversity and capacity of les we identify that when the od of channel matrix has an upper bound strictly less than 1 the same diversity order as that of mles is collected by les and the outage capacity loss relative to mles is also a constant over signal to noise ratio snr these results can be applied to designing a framework for hybrid equalizers furthermore by studying the statistical properties of the od and comparing the complexity of different equalizers we show that hybrid equalizers can trade off between performance and complexity by tuning the channel matrix od the theoretical analysis is corroborated by computer simulations

a primal all integer algorithm based on irreducible solutions this paper introduces an exact primal augmentation algorithm for solving general linear integer programs the algorithm iteratively substitutes one column in a tableau by other columns that correspond to irreducible solutions of certain linear diophantine inequalities we prove that various versions of our algorithm are finite it is a major concern in this paper to show how the subproblem of replacing a column can be accomplished effectively an implementation of the presented algorithms is given computational results for a number of hard 0 1 integer programs from the miplib demonstrate the practical power of the method

hard equality constrained integer knapsacks we consider the following integer feasibility problem given positive integer numbersa0 a1 a n with gcd a1 a n 1 anda a1 a n does there exist a vectorx z n 0satisfyinga x a0 we prove that if the coefficientsa1 a nhave a certain decomposable structure then the frobenius number associated witha1 a n i e the largest value ofa0 for whicha x a0 does not have a nonnegative integer solution is close to a known upper bound in the instances we consider we takea0 to be the frobenius number furthermore we show that the decomposable structure ofa1 a nmakes the solution of a lattice reformulation of our problem almost trivial since the number of lattice hyperplanes that intersect the polytope resulting from the reformulation in the direction of the last coordinate is going to be very small for branch and bound such instances are difficult to solve since they are infeasible and have large values ofa0 a i 1 i n we illustrate our results by some computational examples

an efficient low complexity detector for spatially multiplexed mc cdm we propose a novel suboptimum detection method for spatially multiplexed multicarrier code division multiplexing sm mc cdm communications compared to the spatially multiplexed ofdm sm ofdm the frequency domain spreading in sm mc cdm systems results in an additional diversity gain to take advantage of diversity and multiplexing while mitigating interference we design a low complexity efficient detector called unified successive interference cancellation sic detector for sm mc cdm communications further performance improvement is achieved by adopting in conjunction with the unified sic the iterative subcarrier reconstruction detection algorithm originally proposed for single antenna systems the results demonstrate significant performance improvement over other existing methods of comparable complexity a close approximation for the probability density function pdf of the proposed detector s output signal to interference plus noise ratio sinr is found and used to obtain error bounds for the bit error rate ber performance of the coded sm mc cdm transmission is also discussed in the paper

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