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

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

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

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

on mod p properties of siegel modular forms we prove two results on mod p properties of siegel modular forms first we use theta series in order to construct of a siegel modular form of weight p 1 which is congruent to 1 mod p second we define a theta operator vartheta on q expansions and show that the algebra of siegel modular forms mod p is stable under vartheta by exploiting the relation between vartheta and generalized rankin cohen brackets

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

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

note on mod p siegel modular forms ii the structure of the ring of mod p siegel modular forms of degree two is determined in the cases where the prime p is 2 or 3

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

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

universal relations on stable map spaces in genus zero we introduce a factorization for the map between moduli spaces of stable maps which forgets one marked point this leads to a study of universal relations in the cohomology of stable map spaces in genus zero

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

the ring of modular forms of degree two in characteristic three we determine the structure of the ring of siegel modular forms of degree 2 in characteristic 3

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

congruences between siegel modular forms using the moduli theory of abelian varieties and a recent result of bocherer nagaoka on lifting of the generalized hasse invariant we show congruences between the weights of siegel modular forms with congruent fourier expansions this result implies that the weights of p adic siegel modular forms are well defined

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

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

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

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

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

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

group representations and lattices this paper began as an attempt to understand the euclidean lattices that were recently constructed using the theory of elliptic curves by elkies e and shioda shl sh2 from the point of view of group representations the main idea appears in a note of thompson th2 if one makes strong irreducibility hypotheses on a rational representation v of a finite group g then the g stable euclidean lattices in v are severely restricted unfortunately these hypotheses are rarely satisfied when v is absolutely irreducible over q but there are many examples where the ring endg v is an imaginary quadratic field or a definite quaternion algebra these representations allow us to construct some of the mordell weil lattices considered by elkies and shioda as well as some interesting even unimodular lattices that do not seem to come from the theory of elliptic curves in 1 we discuss lattices and hermitian forms on y7 and in 2 4 the strong irreducibility hypotheses we wish to make in 5 we show how our hypotheses imply the existence of a finite number up to isomorphism of euclidean z g lattices l in v with endg l a maximal order in endg v we give some examples with dim v 8 in 6 and in 7 9 discuss the invariants of l such as the dual lattice and theta function the rest of the paper is devoted to examples in most g is a finite group of lie type and v is obtained as an irreducible summand of the weil representation of g some of the representation theoretic problems left open by this paper are to find all examples of pairs g v satisfying the strong irreducibility hypotheses of 2 4 and to determine the invariants shortest nonzero vector theta function thompson series of the g lattices l so effortlessly constructed inside v

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

tautological rings of stable map spaces we find a set of generators and relations for the system of extended tautological rings associated to the moduli spaces of stable maps in genus zero admitting a simple geometrical interpretation in particular when the target is p n these give a complete presentation for the cohomology and chow rings in the cases with without marked points

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

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

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

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

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

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

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

trace operators and theta series for γ 0 n q and γ n q we consider the action of suitable trace operators on non homogeneous theta series that are siegel modular forms for the principal congruence subgroups of the symplectic group of level q n q then we prove that modular forms for the hecke subgroup of level q 0 n q that are linear combination of such theta series can also be expressed as combination of homogeneous theta series that are modular forms with respect to 0 n q

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