on the kz reduction the korkine zolotareff kz reduction is one of the often used reduction strategies for lattice decoding in this paper we first investigate some important properties of kz reduced matrices specifically we present a linear upper bound on the hermit constant which is around 7 8 times of the existing sharpest linear upper bound and an upper bound on the kz constant which is polynomially smaller than the existing sharpest one we also propose upper bounds on the lengths of the columns of kz reduced matrices and an upper bound on the orthogonality defect of kz reduced matrices which are even polynomially and exponentially smaller than those of boosted kz reduced matrices respectively then we derive upper bounds on the magnitudes of the entries of any solution of a shortest vector problem svp when its basis matrix is lll reduced these upper bounds are useful for analyzing the complexity and understanding numerical stability of the basis expansion in a kz reduction algorithm finally we propose a new kz reduction algorithm by modifying the commonly used schnorr euchner search strategy for solving svps and the basis expansion method proposed by zhang et al simulation results show that the new kz reduction algorithm is much faster and more numerically reliable than the kz reduction algorithm proposed by zhang et al especially when the basis matrix is ill conditioned

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

new maximal functions and multiple weights for the multilinear calderon zygmund theory a multi sub linear maximal operator that acts on the product of m lebesgue spaces and is smaller than the m fold product of the hardy littlewood maximal function is studied the operator is used to obtain a precise control on multilinear singular integral operators of calderon zygmund type and to build a theory of weights adapted to the multilinear setting a natural variant of the operator which is useful to control certain commutators of multilinear calderon zygmund operators with bmo functions is then considered the optimal range of strong type estimates a sharp end point estimate and weighted norm inequalities involving both the classical muckenhoupt weights and the new multilinear ones are also obtained for the commutators

multilinear calderon zygmund theory abstract a systematic treatment of multilinear calderon zygmund operators is presented the theory developed includes strong type and endpoint weak type estimates interpolation a multilinear t 1 theorem and a variety of results regarding multilinear multiplier operators

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

lattice basis reduction improved practical algorithms and solving subset sum problems we report on improved practical algorithms for lattice basis reduction we present a variant of the l3 algorithm with deep insertions and a practical algorithm for blockwise korkine zolotarev reduction a concept extending l3 reduction that has been introduced by schnorr 1987 empirical tests show that the strongest of these algorithms solves almost all subset sum problems with up to 58 random weights of arbitrary bit length within at most a few hours on a unisys 6000 70 or within a couple of minutes on a sparc 2 computer

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

a lattice based public key cryptosystem ajtai recently found a random class of lattices of integer points for which he could prove the following worst case average case equivalence result if there is a probabilistic polynomial time algorithm which finds a short vector in a random lattice from the class then there is also a probabilistic polynomial time algorithm which solves several problems related to the shortest lattice vector problem svp in any n dimensional lattice ajtai and dwork then designed a public key cryptosystem which is provably secure unless the worst case of a version of the svp can be solved in probabilistic polynomial time however their cryptosystem suffers from massive data expansion because it encrypts data bit by bit here we present a public key cryptosystem based on similar ideas but with much less data expansion

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

hardness of approximating the shortest vector problem in lattices let p 1 be any fixed real we show that assuming np n rp there is no polynomial time algorithm that approximates the shortest vector problem svp in e p norm within a constant factor under the stronger assumption np n rtime 2poly log n we show that there is no polynomial time algorithm with approximation ratio 2 log n 1 2 e where n is the dimension of the lattice and e 0 is an arbitrarily small constant we first give a new randomized reduction from closest vector problem cvp to svp that achieves some constant factor hardness the reduction is based on bch codes its advantage is that the svp instances produced by the reduction behave well under the augmented tensor product a new variant of tensor product that we introduce this enables us to boost the hardness factor to 2 log n 1 2 e

bounding the sum of square roots via lattice reduction let k and n be positive integers define r n k to be the minimum positive value of ei p s1 e2 p s2 ek p sk t where s1 s2 sk are positive integers no larger than n t is an integer and ei 2 1 0 1 for all 1 i k it is important in computational geometry to determine a good lower and upper bound of r n k in this paper we show that this problem is closely related to the shortest vector problem in certain integral lattices and present an algorithm to find lower bounds based on lattice reduction algorithms although we can only prove an exponential time upper bound for the algorithm it is ecient for large k when an exhaustive search for the minimum value is clearly infeasible it produces lower bounds much better than the root separation technique does based on numerical data we formulate a conjecture on the length of the shortest nonzero vector in the lattice whose validation implies that our algorithm runs in polynomial time and the problem of comparing two sums of square roots of small integers can be solved in polynomial time as a side result we obtain constructive upper bounds for r n k when n is much smaller than 2 2k

the hardness of the closest vector problem with preprocessing we give a new simple proof of the np hardness of the closest vector problem in addition to being much simpler than all previously known proofs the new proof yields new interesting results about the complexity of the closest vector problem with preprocessing this is a variant of the closest vector problem in which the lattice is specified in advance and can be preprocessed for an arbitrarily long amount of time before the target vector is revealed we show that there are lattices for which the closest vector problem remains hard regardless of the amount of preprocessing

convergence of numerical solutions to stochastic differential equations with markovian switching abstract the paper develops strong approximation schemes for solutions of stochastic differential equations with markovian switching sdewmss the convergent orders of existing strong numerical schemes all are 0 5 this paper provides a convergence theorem for the construction of strong approximations of any given order of convergence for sdewmss and then constructs the order 1 scheme the order 1 5 scheme the paper also develops a new way to simulate the markov chain and hence the order 1 scheme finally a numerical example is provided to illustrate the theoretical results

simultaneously good bases of a lattice and its reciprocal lattice it is well known that every lattice l has a basis consisting of relatively short vectors to state a quantitative version of this fact let 21 l denote the i th successive minimum of l which is the smallest value such that there are i linearly independent vectors in l in the ball x i x ii i l in particular let a be the minimal constant such that every lattice of rank n possesses a basis which satisfies

on lattices learning with errors random linear codes and cryptography our main result is a reduction from worst case lattice problems such as gapsvp and sivp to a certain learning problem this learning problem is a natural extension of the learning from parity with error problem to higher moduli it can also be viewed as the problem of decoding from a random linear code this we believe gives a strong indication that these problems are hard our reduction however is quantum hence an efficient solution to the learning problem implies a quantum algorithm for gapsvp and sivp a main open question is whether this reduction can be made classical i e nonquantum we also present a classical public key cryptosystem whose security is based on the hardness of the learning problem by the main result its security is also based on the worst case quantum hardness of gapsvp and sivp the new cryptosystem is much more efficient than previous lattice based cryptosystems the public key is of size o n2 and encrypting a message increases its size by a factor of o n in previous cryptosystems these values are o n4 and o n2 respectively in fact under the assumption that all parties share a random bit string of length o n2 the size of the public key can be reduced to o n

improved rounding for spline coefficients and knots when representing the coefficients and knots of a spline using only small integers independently rounding each infinite precision value is not the best strategy we show how to build an affine model for the error expanded about the optimal full precision free knot or parameterized spline then use the lovasz basis reduction algorithm to select a better rounding the technique could be used for other situations in which a quadratic error model can be computed

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

convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation the paper deals with convergence and stability of the semi implicit euler method for a linear stochastic differential delay equation it is proved that the semi implicit euler method is convergent with strong order p the conditions under which the method is ms stable and gms stable are determined and the numerical experiments are given

exponential stability of numerical solutions for a class of stochastic age dependent capital system with poisson jumps recently numerical solutions of stochastic differential equations have received a great deal of attention numerical approximation schemes are invaluable tools for exploring their properties in this paper we introduce a class of stochastic age dependent vintage capital system with poisson jumps we also give the discrete approximate solution with an implicit euler scheme in time discretization using gronwall s lemma and barkholder davis gundy s inequality some criteria are obtained for the exponential stability of numerical solutions to the stochastic age dependent capital system with poisson jumps it is proved that the numerical approximation solutions converge to the analytic solutions of the equations under the given conditions where information on the order of approximation is provided these error bounds imply strong convergence as the timestep tends to zero a numerical example is used to illustrate the theoretical results

algorithms to construct minkowski reduced and hermite reduced lattice bases abstract up to now the problem of constructing minkowski reduced lattice bases has been solved only for the two and three dimensional case this paper presents an algorithm to solve the problem for arbitrary dimension for fixed dimension the runtime is polynomial the algorithm hinges on the previous reduction algorithms of lenstra lenstra and lovasz 1982 and kannan 1983 moreover we shall improve kannan s algorithm to construct hermite reduced lattice bases

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

lattice problems in np conp we show that the problems of approximating the shortest and closest vector in a lattice to within a factor of n lie in np intersect conp the result almost subsumes the three mutually incomparable previous results regarding these lattice problems banaszczyk 1993 goldreich and goldwasser 2000 and aharonov and regev 2003 our technique is based on a simple fact regarding succinct approximation of functions using their fourier series over the lattice this technique might be useful elsewhere we demonstrate this by giving a simple and efficient algorithm for one other lattice problem cvpp improving on a previous result of regev 2003 an interesting fact is that our result emerged from a dequantization of our previous quantum result in aharonov and regev 2003 this route to proving purely classical results might be beneficial elsewhere

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

integer parameter estimation in linear models with applications to gps we consider parameter estimation in linear models when some of the parameters are known to be integers such problems arise for example in positioning using carrier phase measurements in the global positioning system gps where the unknown integers enter the equations as the number of carrier signal cycles between the receiver and the satellites when the carrier signal is initially phase locked given a linear model we address two problems 1 the problem of estimating the parameters and 2 the problem of verifying the parameter estimates we show that with additive gaussian measurement noise the maximum likelihood estimates of the parameters are given by solving an integer least squares problem theoretically this problem is very difficult computationally np hard verifying the parameter estimates computing the probability of estimating the integer parameters correctly requires computing the integral of a gaussian probability density function over the voronoi cell of a lattice this problem is also very difficult computationally however by using a polynomial time algorithm due to lenstra lenstra and lovasz 1982 the lll algorithm the integer least squares problem associated with estimating the parameters can be solved efficiently in practice sharp upper and lower bounds can be found on the probability of correct integer parameter estimation we conclude the paper with simulation results that are based on a synthetic gps setup

new korkin zolotarev inequalities korkin and zolotarev showed that if sum i a i big x i sum j i alpha ij x j big 2 is the lagrange expansion of a korkin zolotarev kz reduced positive definite quadratic form then a i 1 geq frac 3 4 a i and a i 2 geq frac 2 3 a i they showed that the implied bound a 5 geq frac 4 9 a 1 is not attained by any kz reduced form we propose a method to optimize numerically over the set of lagrange expansions of kz reduced quadratic forms using a semidefinite relaxation combined with a branch and bound process we use a rounding technique to derive exact results from the numerical data applying these methods we prove several new linear inequalities on the a i of any kz reduced form one of them being a i 4 geq frac 15 32 2 cdot 10 5 a i we also give a form with a 5 frac 15 32 a 1 these new inequalities are then used to study the cone of outer coefficients of kz reduced forms to find bounds on hermite s constant and to give better estimates on the quality of k block kz reduced lattice bases

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

introduction to the numerical analysis of stochastic delay differential equations abstract we consider the problem of the numerical solution of stochastic delay differential equations of ito form d x t f x t x t d t g x t x t d w t t 0 t and x t t for t 0 with given f g wiener noise w and given 0 with a prescribed initial function we indicate the nature of the equations of interest and give a convergence proof for explicit single step methods some illustrative numerical examples using a strong euler maruyama scheme are provided

mixed a1 a bounds for fractional integrals abstract for a fractional integral operator i we show that i f l q w q c w a q 1 p w a 1 q 1 q f l p w p where w a q sup q 1 w q q q m q w q d x is wilson a q s constant which is smaller than the a 1 q constant we also prove a weak type 1 1 bound for a couple of weights of the form w m w assuming that w a

closest point search in lattices in this semitutorial paper a comprehensive survey of closest point search methods for lattices without a regular structure is presented the existing search strategies are described in a unified framework and differences between them are elucidated an efficient closest point search algorithm based on the schnorr euchner 1995 variation of the pohst 1981 method is implemented given an arbitrary point x spl isin spl ropf sup m and a generator matrix for a lattice spl lambda the algorithm computes the point of spl lambda that is closest to x the algorithm is shown to be substantially faster than other known methods by means of a theoretical comparison with the kannan 1983 1987 algorithm and an experimental comparison with the pohst 1981 algorithm and its variants such as the viterbo boutros see ibid vol 45 p 1639 42 1999 decoder modifications of the algorithm are developed to solve a number of related search problems for lattices such as finding a shortest vector determining the kissing number computing the voronoi 1908 relevant vectors and finding a korkine zolotareff 1873 reduced basis

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

integer relations among algebraic numbers a vector m m 1 m n z n 0 is called an integer relation for the real numbers 1 n if i m i 0 holds we present an algorithm that when given algebraic numbers 1 n and a parameter e either finds an integer relation for 1 a n or proves that no relation of euclidean length shorter than 1 e exists each algebraic number is assumed to be given by its minimal polynomial and by a sufficiently precise rational approximation

iterated chvatal gomory cuts and the geometry of numbers chvatal gomory cutting planes cg cuts for short are a fundamental tool in integer programming given any single cg cut one can derive an entire family of cg cuts by iterating its multiplier vector modulo one this leads naturally to two questions first which iterates correspond to the strongest cuts and second can we find such strong cuts efficiently we answer the first question empirically by showing that one specific approach for selecting the iterate tends to perform much better than several others the approach essentially consists of solving a nonlinear optimization problem over a special lattice associated with the cg cut we then provide a partial answer to the second question by presenting a polynomial time algorithm that yields an iterate that is strong in a certain well defined sense the algorithm is based on results from the algorithmic geometry of numbers

trellis complexity and minimal trellis diagrams of lattices this paper presents results on trellis complexity and low complexity trellis diagrams of lattices we establish constructive upper bounds on the trellis complexity of lattices these bounds both improve and generalize the similar results of tarokh and vardy see ibid vol 43 p 1294 1300 1997 we also construct trellis diagrams with minimum number of paths for some important lattices such trellises are called minimal the constructed trellises which are novel in many cases can be employed to efficiently decode the lattices via the viterbi algorithm in particular a general structure for minimal trellis diagrams of d sub n lattices is obtained this structure corresponds to a new code formula for d sub n moreover we develop some important duality results which are used in both deriving the upper bounds and finding the minimal trellises all the discussions are based on a universal approach to the construction and analysis of trellis diagrams of lattices using their bases

disproof of the mertens conjecture the mertens conjecture states that m x 1 where m x n x n and n is the mo bius function this conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the riemann hypothesis this paper disproves the mertens conjecture by showing that x lim sup m x x 2 1 06 the disproof relies on extensive computations with the zeros of the zeta function and does not provide an explicit counterexample disproof of the mertens conjecture a m odlyzko at t bell laboratories murray hill new jersey 07974 usa and h j j te riele centre for mathematics and computer science kruislaan 413 1098 sj amsterdam the netherlands

exponential stability of numerical solutions to sddes with markovian switching abstract the main aim of this paper is to investigate the exponential stability of the euler method for a class of autonomous stochastic delay differential equations with markovian switching the definition of exponential mean square stability of numerical method is introduced it is proved that the euler scheme is exponentially stable in mean square sense several examples are given for illustration

numerical solutions of stochastic differential delay equations under local lipschitz condition in this paper a variant of the euler maruyama method is used to define the numerical solutions for stochastic differential delay equations sddes with variable delay the key aim is to show that the numerical solutions will converge to the true solutions of the sddes under the local lipschitz condition

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

optimized spectrum permutation for the multidimensional sparse fft a multidimensional sparse fast fourier transform algorithm is introduced via generalizations of key concepts used in the one dimensional 1 d sparse fourier transform algorithm it is shown that permutation parameters are of key importance and should not be chosen randomly but instead can be optimized a connection is made between the sparse fourier transform algorithm and lattice theory thus establishing a rigorous understanding of the effect of the permutations on the algorithm performance lattice theory is then used to optimize the set of parameters to achieve a more robust and better performing algorithm other algorithms using pseudorandom spectrum permutation can also benefit from the methods developed in this paper the contributions address the case of the exact k sparse fourier transform but the underlying concepts can be applied to the general case of finding a k sparse approximation of the fourier transform of an arbitrary signal simulations illustrate the efficiency and accuracy of the proposed algorithm the optimizations of the parameters and the improved performance are shown in simulations for the 2 d case where worst case and average case peak signal to noise ratio psnr improves by several decibels

on the proximity factors of lattice reduction aided decoding lattice reduction aided decoding lrad features reduced decoding complexity and near optimum performance in multiinput multioutput communications in this paper a quantitative error rate analysis of lrad is presented to this aim the proximity factors are defined to measure the worst case losses in decoding distances associated with the decision region relative to infinite lattice decoding ild namely closest point search in an infinite lattice upper bounds on the proximity factors are derived which are functions of the dimension n of the lattice alone the study is then extended to the dual basis reduction it is found that the bounds for dual basis reduction may be smaller reasonably good bounds are derived in many cases proximity factors not only imply the same diversity order in fading channels but also relate the error probabilities of ild and lrad

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

on the complexity of decoding lattices using the korkin zolotarev reduced basis upper and lower bounds are derived for the decoding complexity of a general lattice l the bounds are in terms of the dimension n and the coding gain spl gamma of l and are obtained based on a decoding algorithm which is an improved version of kannan s 1983 method the latter is currently the fastest known method for the decoding of a general lattice for the decoding of a point x the proposed algorithm recursively searches inside an n dimensional rectangular parallelepiped cube centered at x with its edges along the gram schmidt vectors of a proper basis of l we call algorithms of this type recursive cube search rcs algorithms it is shown that kannan s algorithm also belongs to this category the complexity of rcs algorithms is measured in terms of the number of lattice points that need to be examined before a decision is made to tighten the upper bound on the complexity we select a lattice basis which is reduced in the sense of korkin zolotarev 1873 it is shown that for any selected basis the decoding complexity using rcs algorithms of any sequence of lattices with possible application in communications spl gamma spl ges 1 grows at least exponentially with n and spl gamma it is observed that the densest lattices and almost all of the lattices used in communications e g barnes wall lattices and the leech lattice have equal successive minima esm for the decoding complexity of esm lattices a tighter upper bound and a stronger lower bound result are derived

hkz and minkowski reduction algorithms for lattice reduction aided mimo detection recently lattice reduction has been widely used for signal detection in multiinput multioutput mimo communications in this paper we present three novel lattice reduction algorithms first using a unimodular transformation a significant improvement on an existing hermite korkine zolotareff reduction algorithm is proposed then we present two practical algorithms for constructing minkowski reduced bases to assess the output quality we compare the orthogonality defect of the reduced bases produced by lll algorithm and our new algorithms and find that in practice minkowski reduced basis vectors are the closest to being orthogonal an error rate analysis of suboptimal decoding algorithms aided by different reduction notions is also presented to this aim the proximity factor is employed as a measurement we improve some existing results and derive upper bounds for the proximity factors of minkowski reduction aided decoding mrad to show that mrad can achieve the same diversity order with infinite lattice decoding ild

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

stability of solution to a class of investment system abstract the investment model of nonlinear stochastic neutral technical progress with time delay is given in this paper adopting the theory of stochastic functional differential equation and using ito formula gronwall s lemma and barkholder davis gundy s lemma exponential stability of strong solution is proved for the system of nonlinear stochastic neutral technical progress assets with time delay on hilbert space also a sufficient condition of the exponential stability is obtained the results are the improvement and extension of the existed results of this field

improved inapproximability of lattice and coding problems with preprocessing we show that the closest vector problem with preprocessing cvpp is np hard to approximate to within spl radic 3 spl epsi for any spl epsi 0 in addition we show that the nearest codeword problem with preprocessing ncpp is np hard to approximate to within 3 spl epsi these results improve previous results of feige and micciancio we also present the first inapproximability result for the relatively nearest codeword problem with preprocessing rncp finally we describe an n approximation algorithm to cvpp

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

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

on some pointwise inequalities in a recent paper studia math 138 2000 285 291 we proved pointwise estimates relating some classical maximal and singular integral operators here we show that in a sense there are more flexible inequalities which not only imply the previously known results but also give something new in particular they hold for the multilinear calderon zygmund operators this result gives a new approach to a recent work by grafakos and torres and unifies some classical inequalities by cotlar and coifman and fefferman

exponential stability of numerical solutions to a stochastic age structured population system with diffusion the main aim of this paper is to investigate the exponential stability of the euler method for a stochastic age structured population system with diffusion the definition of exponential mean square stability of numerical method is introduced it is proved that the euler scheme is exponentially stable in mean square sense an example is given for illustration

an efficient optimal algorithm for the successive minima problem in many applications including integer forcing linear multiple input and multiple output mimo receiver design one needs to solve a successive minima problem smp on an n dimensional lattice to get an optimal integer coefficient matrix boldsymbol a star in z n times n in this paper we first propose an efficient optimal smp algorithm with an mathcal o n 2 memory complexity the main idea behind the new algorithm is it first initializes with a suitable suboptimal solution which is then updated via a novel algorithm with only mathcal o n 2 flops in each updating until boldsymbol a star is obtained different from existing algorithms which find boldsymbol a star column by column through using a sphere decoding search strategy n times the new algorithm uses a search strategy once only we then rigorously prove the optimality of the proposed algorithm furthermore we theoretically analyze its complexity in particular we not only show that the new algorithm is omega n times faster than the most efficient existing algorithm with polynomial memory complexity but also assert that it is even more efficient than the most efficient existing algorithm with exponential memory complexity finally numerical simulations are presented to illustrate the optimality and efficiency of our novel smp algorithm

random lattices threshold phenomena and efficient reduction algorithms two new lattice reduction algorithms are presented and analyzed these algorithms called the schmidt reduction and the gram reduction are obtained by relaxing some of the constraints of the classical lll algorithm by analyzing the worst case behavior and the average case behavior in a tractable model we prove that the new algorithms still produce good reduced basis while requiring fewer iterations on average in addition we provide empirical tests on random lattices coming from applications that confirm our theoretical results about the relative behavior of the different reduction algorithms

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

fast lll type lattice reduction we modify the concept of lll reduction of lattice bases in the sense of lenstra lenstra lovasz factoring polynomials with rational coefficients math ann 261 1982 515 534 towards a faster reduction algorithm we organize lll reduction in segments of the basis our slll bases approximate the successive minima of the lattice in nearly the same way as lll bases for integer lattices of dimension n given by a basis of length 2 o n slll reduction runs in o n 5 e bit operations for every e 0 compared to o n 7 e for the original lll and to o n 6 e for the lll algorithms of schnorr a more efficient algorithm for lattice reduction journal of algorithm 9 1988 47 62 and storjohann faster algorithms for integer lattice basis reduction tr 249 swiss federal institute of technology eth zurich department of computer science zurich switzerland july 1996 we present an even faster algorithm for slll reduction via iterated subsegments running in o n 3log n arithmetic steps householder reflections are shown to provide better accuracy than gram schmidt for orthogonalizing lll bases in floating point arithmetic

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

imposing structure on smith form decompositions of rational resampling matrices imposing structure on the smith form of an integer periodicity matrix n u spl lambda v leads to efficient multidimensional m d dft implementations guessoum 1984 for nonsingular integer and rational matrices the authors introduce algorithms to generate u or v matrices with minimum norm and spl lambda matrices whose diagonal elements exhibit minimum variance such structure simplifies nonuniform m d filter bank design gardos et al 1992

stochastic functional differential equations with infinite delay abstract the stability and boundedness of the solution for stochastic functional differential equation with finite delay have been studied by several authors but there is almost no work on the stability of the solutions for stochastic functional differential equations with infinite delay the main aim of this paper is to close this gap we establish criteria of pth moment t bounded for neutral stochastic functional differential equations with infinite delay and exponentially stable criteria for stochastic functional differential equations with infinite delay and we also illustrate the result with an example

improvements in the analysis of kannan s cvp algorithm finding a closest lattice vector to a given target vector is a widely studied problem in computational mathematics and computer science the classical enumeration algorithm was given by kannan in 1983 together with an algorithm for the shortest vector problem the time complexity of the closest vector enumeration algorithm based on hkz reduced bases was proven to within 2 o n n n 2 by hanrot and stehle in 2007 using their point counting method the constant hidden in o n is strictly bigger than 1 9 in this paper we give a new counting method which we call sequence counting method to bound the number of loop iterations and show that the time complexity is actually upper bounded by 2 0 401 n o n n n 2 our paper revisits kannan s cvp algorithm and points out a new method sequence counting to upper bound the time complexity by 2 0 401 n o n n n 2 we connects the loops of the enumeration algorithm with lattice points by a one to one mapping using the sequence as a tool we use different methods to upper bound the number of lattice points near a target vector according to the dimension of the lattice and then combine them to conclude our result

sampling methods for shortest vectors closest vectors and successive minima we study four problems from the geometry of numbers the shortest vector problem svp the closest vector problem cvp the successive minima problem smp and the shortest independent vectors problem sivp extending and generalizing results of ajtai kumar and sivakumar we present probabilistic single exponential time algorithms for all four problems for all p norms the results on smp and sivp are new for all norms the results on svp and cvp generalize previous results of ajtai et al for the euclidean 2 norm to arbitrary p norms we achieve our results by introducing a new lattice problem the generalized shortest vector problem gsvp we describe a single exponential time algorithm for gsvp we also describe polynomial time reductions from svp cvp smp and sivp to gsvp establishing single exponential time algorithms for the four classical lattice problems this approach leads to a unified algorithmic treatment of the lattice problems svp cvp smp and sivp

convergence of numerical solutions to stochastic pantograph equations with markovian switching in this paper a class of stochastic pantograph equations with markovian switching is considered the main purpose is to investigate the convergence of the euler method of the equations it is proved that the euler approximation solution converge to the analytic solution in probability under weaker conditions an example is provided to illustrate our theory

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

the role of integer matrices in multidimensional multirate systems various theoretical issues in multidimensional m d multirate signal processing are formulated and solved in the problems considered the decimation matrix and the expansion matrix are nondiagonal so that extensions of 1 d results are nontrivial the m d polyphase implementation technique for rational sampling rate alterations the perfect reconstruction properties for the m d delay chain systems and the periodicity matrices of decimated m d signals both deterministic and statistical are treated the discussions are based on several key properties of integer matrices including greatest common divisors and least common multiples these properties are reviewed