new bounds in some transference theorems in the geometry of numbers the aim of this paper is to give new bounds in certain inequalities concerning mutually reciprocal lattices in r to formulate the problem we have to introduce some notation and terminology we shall treat ir as an n dimensional euclidean space with the norm ii ii and metric d the inner product of vectors u v will be denoted by uv we shall usually write u 2 instead of uu the closed and open unit balls in f will be denoted by b and b respectively if a c ir then span a and a z denote the linear subspace spanned over a and the orthogonal complement of a in p a lattice in ir is an additive subgroup of ir generated by n linearly independent vectors the family of all lattices in ir will be denoted by a given a lattice l e a we define the polar dual reciprocal lattice l in the usual way l u u v z for each v l

inapproximability results for the weight problems of subgroup permutation codes a subgroup permutation code is a set of permutations on n symbols with the property that its elements are closed under the operation of composition in this paper we give inapproximability results for the minimum and maximum weight problems of subgroup permutation codes under several well known metrics based on previous works we prove that under hamming lee cayley kendall s tau ulam s and lp distance metrics 1 there is no polynomial time 2log1 en approximation algorithm for the minimum weight problem for any constant e 0 unless np dtime 2polylog n quasi polynomial time and 2 there is no polynomial time r approximation algorithm for the minimum weight problem for any constant r 1 unless p np under l metric we prove that it is np hard to approximate the minimum weight problem within factor 2 e for any constant e 0 we also prove that for any constant e 0 it is np hard to approximate the maximum weight within p 3 2 e under lp distance metric and within 3 2 e under hamming lee cayley kendall s tau and ulam s distance metrics

parameterized and approximation complexity of partial vc dimension abstract we introduce the problem partial vc dimension that asks given a hypergraph h x e and integers k and l whether one can select a set c x of k vertices of h such that the set e c e e of distinct hyperedge intersections with c has size at least l the sets e c define equivalence classes over e partial vc dimension is a generalization of vc dimension which corresponds to the case l 2 k and of distinguishing transversal which corresponds to the case l e the latter is also known as test cover in the dual hypergraph we also introduce the associated fixed cardinality maximization problem max partial vc dimension that aims at maximizing the number of equivalence classes induced by a solution set of k vertices we study the algorithmic complexity of partial vc dimension and max partial vc dimension both on general hypergraphs and on more restricted instances in particular neighborhood hypergraphs of graphs

lower bounds of shortest vector lengths in random ntru lattices finding the shortest vector of a lattice is one of the most important problems in computational lattice theory for a random lattice one can estimate the length of the shortest vector using the gaussian heuristic however no rigorous proof can be provided for some classes of lattices as the gaussian heuristic may not hold for them in this paper we propose a general method to estimate lower bounds of the shortest vector lengths for random integral lattices in certain classes which is based on the incompressibility method from the theory of kolmogorov complexity as an application we can prove that for a random ntru lattice with an overwhelming probability the ratio between the length of the shortest vector and the length of the target vector which corresponds to the secret key is at least a constant independent of the rank of the lattice

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

a knapsack based probabilistic encryption scheme knapsack based cryptosystems had been viewed as the most attractive and the most promising asymmetric cryptographic algorithms for a long time due to their np completeness nature and high speed in encryption decryption unfortunately most of them are broken for the low density feature of the underlying knapsack problems in this paper we investigate a new easy compact knapsack problem and propose a novel knapsack based probabilistic public key cryptosystem in which the cipher text is non linear with the plaintext for properly chosen parameters the underlying knapsack problem enjoys a high density larger than 1 06 in the worst case hence it is secure against the low density subset sum attacks our scheme can also defeat other potential attacks such as the brute force attacks and the simultaneous diophantine approximation attack compared with previous knapsack based cryptosystems our scheme is efficient and practical

morita theory for hopf algebroids and presheaves of groupoids comodules over hopf algebroids are of central importance in algebraic topology it is well known that a hopf algebroid is the same thing as a presheaf of groupoids on aff the opposite category of commutative rings we show in this paper that a comodule is the same thing as a quasi coherent sheaf over this presheaf of groupoids we prove the general theorem that internal equivalences of presheaves of groupoids with respect to a grothendieck topology t on aff give rise to equivalences of categories of sheaves in that topology we then show using faithfully flat descent that an internal equivalence in the flat topology gives rise to an equivalence of categories of quasi coherent sheaves the corresponding statement for hopf algebroids is that weakly equivalent hopf algebroids have equivalent categories of comodules we apply this to formal group laws where we get considerable generalizations of the miller ravenel and hovey sadofsky change of rings theorems in algebraic topology

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

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

analysis of the efficiency of the chor rivest cryptosystem implementation in a safe parameter range the chor rivest cryptosystem based on a high density knapsack problem on a finite field f q h was broken by vaudenay for q 200 h 24 and h admitting a factor s verifying a certain condition a new set of parameters q and h which prevent this cryptosystem against vaudenay s attack is presented and the computational aspects of its implementation in the magma computational algebra system are analyzed

on the equivalence of mceliece s and niederreiter s public key cryptosystems it is shown that mceliece s and niederreiter s public key cryptosystems are equivalent when set up for corresponding choices of parameters a security analysis for the two systems based on this equivalence observation is presented

fourier spectral method on ensemble architectures the direct simulation of three dimensional unsteady fluid flows is performed on some of the most powerful parallel computers presently available connection machine 2 intel i860 and the ncube 2 a fourier pseudospectral method and second order timestepping scheme is used implementation performance analysis and experimental timings are given results of fluid flow simulations on meshes up to 2563 show sustained rates of 520 mflops on the 64 k node cm 2 484 mflops on the 1 k node ncube 2 and 103 mflops on the 32 node intel i860

chosen ciphertext attacks on lattice based public key encryption and modern non quantum cryptography in a quantum environment modern cryptography is based on various building blocks such as one way functions with or without trapdoors pseudo random functions one way permutations with or without trapdoors etc in a quantum world some of the main candidates for these building blocks are broken for instance the security of the most popular public key cryptosystem rsa is related to the difficulty of factoring large numbers and is broken in principle by a quantum computer we investigate some of the remaining candidates and discuss the resulting post quantum cryptography namely the resulting modern cryptography in a quantum environment about half a decade ago ajtai and dwork and later on also goldreich goldwasser and halevi proposed a public key cryptosystem that has a proven security under a plausible complexity assumption the plausible assumption is that the so called unique shortest vector problem u svp is hard on the worst case this problem is potentially still hard also in a quantum environment recently regev introduced a new and much simpler public key cryptosystem based on the same u svp hardness assumption but with improved parameters in this paper we present chosen ciphertext attacks cca against all three cryptosystems our attack shows that these cryptosystems are totally insecure against cca because the private keys can be recovered in polynomial time we then discuss the possibility of making public key encryption pke secure against cca without adding stronger assumptions than the assumption that u svp is hard we conclude that the current understanding of modern cryptography in a quantum environment can only suggest cca secure interactive pke which is obviously weaker than cca secure pke finally we discuss the relation of our attack to the reaction attack of hall goldberg and schneier which we only recently became aware of

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

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

determinant bundles and virasoro algebras we consider the interplay of infinite dimensional lie algebras of virasoro type and moduli spaces of curves suggested by string theory we will see that the infinitesimal geometry of determinant bundles is governed by virasoro symmetries the mumford forms are just invariants of these symmetries the representations of virasoro algebra define twisted d modules on moduli spaces thesed modules are equations on correlators in conformal field theory

a simple deterministic reduction for the gap minimum distance of code problem we present a simple deterministic gap preserving reduction from sat to the minimum distance of code problem over f 2 we also show how to extend the reduction to work over any fixed finite field previously a randomized reduction was known due to dumer micciancio and sudan which was recently derandomized by cheng and wan these reductions rely on highly nontrivial coding theoretic constructions whereas our reduction is elementary as an additional feature our reduction gives hardness within a constant factor even for asymptotically good codes i e having constant positive rate and relative distance previously it was not known how to achieve a deterministic reduction for such codes

user selection criteria for multiuser systems with optimal and suboptimal lr based detectors in this correspondence we investigate user selection criteria for various multiple input multiple output mimo detectors to exploit the multiuser diversity different user selection criteria are derived for various mimo detectors including the maximum likelihood ml detector and low complexity suboptimal detectors it is shown that the user selection criterion plays a crucial role in exploiting both multiuser and receive or spatial diversity we also show that the ml and even some low complexity suboptimal detectors based on the lattice reduction lr can achieve a full multiuser and receive diversity when the user selection criterion is properly chosen

differential operators on an affine curve let x denote an irreducible affine algebraic curve over an algebraically closed field k of characteristic zero denote by 3 x the sheaf of differential operators on x and 2 x f x 3 x the ring of global differential operators on x the following is established theorem 3 x is a finitely generated k algebra and a noetherian ring furthermore 2 x has a unique minimal non zero ideal j and 3 x j is a finite dimensional k algebra let x denoted the normalisation of x and n x x the projection map the main technique is to compare 2 x with 3 x

mad subalgebras of rings of differential operators on curves we study the maximal abelian ad nilpotent mad subalgebras of the domains morita equivalent to the first weyl algebra we give a complete description both of the individual mad subalgebras and of the space of all such a surprising consequence is that this last space is independent of our results generalize some classic theorems of dixmier about the weyl algebra

wishart planted ensemble a tunably rugged pairwise ising model with a first order phase transition we propose the wishart planted ensemble a class of zero field ising models with tunable algorithmic hardness and specifiable or planted ground state the problem class arises from a simple procedure for generating a family of random integer programming problems with specific statistical symmetry properties but turns out to have intimate connections to a sign inverted variant of the hopfield model the hamiltonian contains only 2 spin interactions with the coupler matrix following a type of wishart distribution the class exhibits a classical first order phase transition in temperature for some parameter settings the model has a locally stable paramagnetic state a feature which correlates strongly with difficulty in finding the ground state and suggests an extremely rugged energy landscape we analytically probe the ensemble thermodynamic properties by deriving the thouless anderson palmer equations and free energy and corroborate the results with a replica and annealed approximation analysis extensive monte carlo simulations confirm our predictions of the first order transition temperature the class exhibits a wide variation in algorithmic hardness as a generation parameter is varied with a pronounced easy hard easy profile and peak in solution time towering many orders of magnitude over that of the easy regimes by deriving the ensemble averaged energy distribution and taking into account finite precision representation we propose an analytical expression for the location of the hardness peak and show that at fixed precision the number of constraints in the integer program must increase with system size to yield truly hard problems the wishart planted ensemble is interesting for its peculiar physical properties and provides a useful and analytically transparent set of problems for benchmarking optimization algorithms

noncommutative bispectral darboux transformations we prove a general theorem establishing the bispectrality of noncommutative darboux transformations it has a wide range of applications that establish bispectrality of such transformations for differential difference and q difference operators with values in all noncommutative algebras all known bispectral darboux transformations are special cases of the theorem using the methods of quasideterminants and the spectral theory of matrix polynomials we explicitly classify the set of bispectral darboux transformations from rank one differential operators and airy operators with values in matrix algebras these sets generalize the classical calogero moser spaces and wilson s adelic grassmannian

automorphisms and ideals of the weyl algebra let a 1 be the first weyl algebra and let g be its automorphism group we study the natural action of g on the space of isomorphism classes of right ideals of a 1 equivalently of finitely generated rank 1 torsion free right a 1 modules we show that this space breaks up into a countable number of orbits each of which is a finite dimensional algebraic variety our results are strikingly similar to those for the commutative algebra of polynomials in two variables however we do not know of any general principle that would allow us to predict this in advance as a key step in the proof we obtain a new description of the bispectral involution of w1 we also make some comments on the group g from the viewpoint of shafarevich s theory of infinite dimensional algebraic groups

improving the efficiency of the branch and bound algorithm for integer programming based on flatness information abstract this paper describes a strategy for defining priorities for the branching variables in a branch and bound algorithm the strategy is based on shape information about the polyhedron over which we are optimizing this information is related to measures of the integer width as provided by the so called flatness theorem our selection rule uses that knowledge to define branching priorities on the variables computational results with simulated small to medium size integer problems are presented as well with multi knapsack problems these show savings of about 40 in cpu time as well as in nodes generated in the search tree and compare favorably with respect to other standard branching rules

integer sets with distinct subset sums in section 1 we introduce the problem of finding minimal height sets of n natural numbers with distinct subset sums ssd and in section 2 review the well known conway guy sequence u conjectured to yield a minimal ssd set for every n we go on section 3 to prove that u certainly cannot be improved upon by any greedy sequence to verify numerically section 4 that it does yield ssd sets for n 80 and section 5 by direct search to show that these are minimal for n s 8 there is a brief interlude section 6 on the problem of decoding the subset from its sum in section 7 generalizations of u are constructed which are asymptotically smaller defining the limit ratio of a sequence w to be a 1imn wn 2l the atkinson negro santoro sequence v known to give ssd sets has a 0 6334 conwayguy conjectured to has a 0 4703 and our best generalization has a 0 4419 we also section 8 discuss when such sequences have the same a and section 9 how a may efficiently be computed to high accuracy 1 the distinct subset sum problem a well known problem in combinatorial number theory 2 pp 64 65 involves the construction for given n of a set p pi i 1 n of natural numbers possessing the property we shall call subset sum distinctness or ssd for short that is l l f pi p e pi s t c l n i s i t or distinct subsets of p have distinct sums evidently choosing pi 21 satisfies 1 1 the interest lies in how much the maximum element pn can be reduced below 2n 1 in order to explore the ssd property it is convenient to consider a more general concept we say that x has a representation by p with length k and signature 1 when for some s and t as before 1 2 x e pe po ies iet where s n t 0 si it k isi iti 1 that is x is the difference of two disjoint subset sums of which the first has 1 terms more than the second and the two together have k terms alternatively n 1 3 x eip i l received june 23 1986 revised october 20 1986 1980 mathematics subject classification 1985 revision primary 11b13 05 04 05a99 11j13 94a60 65b05

the exponential accuracy of fourier and chebyshev differencing methods it is shown that when differencing analytic functions using the pseudospectral fourier or chebyshev methods the error committed decays to zero at an exponential rate

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

a deterministic reduction for the gap minimum distance problem determining the minimum distance of a linear code is one of the most important problems in algorithmic coding theory the exact version of the problem was shown to be np complete by vardy the gap version of the problem was shown to be np hard for any constant factor under a randomized reduction in an earlier work it was shown in the same paper that the minimum distance problem is not approximable in randomized polynomial time to the factor 2log1 n unless np rtime 2polylog n in this paper we derandomize the reduction and thus prove that there is no deterministic polynomial time algorithm to approximate the minimum distance to any constant factor unless p np we also prove that the minimum distance is not approximable in deterministic polynomial time to the factor 2log1 n unless np dtime 2polylog n as the main technical contribution for any constant 2 3 1 we present a deterministic algorithm that given a positive integer s runs in time poly s and constructs a code c of length poly s with an explicit hamming ball of radius d c such that the projection at the first s coordinates sends the codewords in the ball surjectively onto a linear subspace of dimension s where d c denotes the minimum distance of c the codes are obtained by concatenating reed solomon codes with hadamard codes

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 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

differential operators on rational projective curves fix a singular projective rational curve 3 over an algebraically closed field k of characteristic zero with normalisation map a p 3 and write s 5 r 9 for its ring of global differential operators we prove theorem 1 the ring g a is a noerherian domain finirely generated as a k algebra with a unique minimal non zero ideal j z moreover f y 9 j f is finite dimensional over k in the case when n is injective we give complete description of the structure of 2 x

an improved balanced algorithm for the subset sum problem abstract we present balsublast which is an improved balanced algorithm for the subset sum problem that was designed to solve benchmarks that require the exhaustion of the search space and where there are many subsets with the same sum this new algorithm which spends time o n 2 w max and space o n w max where w max is the largest of n items has obtained excellent performance in comparison to other known algorithms

the computational complexity of simultaneous diophantine approximation problems simultaneous diophantine approximation in d dimensions deals with the approximation of a vector bf alpha alpha 1 cdots alpha d of d real numbers by vectors of rational numbers all having the same denominator this paper considers the computational complexity of algorithms to find good simultaneous approximations to a given vector bf alpha of d rational numbers we measure the goodness of an approximation using the sup norm we show that a result of h w lenstra jr produces polynomial time algorithms to find sup norm best approximations to a given vector bf alpha when the dimension d is fixed we show that a recent algorithm of a k lenstra h w lenstra jr and l lovasz to find short vectors in an integral lattice can be used to find a good approximation to a given vector bf alpha in d dimensions with a denominator q satisfying 1 leqq q leqq 2 d 2 n which is within a factor sqrt 5d 2 d 1 2 of the best approximation with denominato

limits of compactified jacobians and d modules on smooth projective curves abstract letsbe a smooth projective curve and d sits sheaf of differential operators this paper classifies the rank one torsion free d s modules up to isomorphism such a module e has a degree which depends on the homological properties of e furthermore the set of isomorphism classes with fixed degreed say bijects to lim j d x x the set of points of the limit of compactified jacobians of curves with injective normalization s x here if f is a rank one torsion free o x module of degreed x representing the point of the limit corresponding to a torsion free d s module e then e d o s f the sheaf of differential operators from o sto f as an application we classify the domains morita equivalent to d p 1 we also study d f the ring of globally defined differential operators on a rank one torsion free sheaf over a curve with injective normalization p 1 x it is shown that the finite dimensional simple d f modules are preciselyhi x f when they are non zero and that the non vanishing ofhi x f determines the morita equivalence class of d f

polynomially solvable instances of the shortest and closest vector problems with applications to compute and forward a particular instance of the shortest vector problem svp appears in the context of compute and forward despite the np hardness of the svp we will show that this certain instance can be solved in complexity order o n psi log n psi where psi sqrt p text h 2 1 depends on the transmission power and the norm of the channel vector we will then extend our results to integer forcing and finally introduce a more general class of lattices for which the svp and the closest vector problem can be approximated within a constant factor

a new paradigm for public key identification the present paper investigates the possibility of designing zero knowledge identification schemes based on hard problems from coding theory zero knowledge proofs were introduced by goldwasser micali and rackoff 1985 their practical significance was soon demonstrated in the work of fiat and shamir 1986 who turned zero knowledge proofs of quadratic residuosity into efficient means of establishing user identities in the present paper we propose a new identification scheme based on error correcting codes which is zero knowledge and seems of practical value furthermore we describe several variants including one which has an identity based character the security of our schemes depends on the hardness of finding a word of given syndrome and prescribed small weight with respect to some randomly generated binary linear error correcting code this is of course not the first attempt to design a cryptographic scheme using tools from coding theory the difference is that identification protocols do not follow the public key paradigm based on trap door functions and described in the seminal diffie hellman paper 1976 rather they only require one way functions which opens the way to using in a rather direct manner simple combinatorial problems of the kind provided by coding theory the resulting schemes compare favorably to their number theoretic analogs

on the lagarias odlyzko algorithm for the subset sum problem we give a simple analysis of an algorithm for solving subset sum problems proposed by lagarias and odlyzko 2

symplectic reflection algebras calogero moser space and deformed harish chandra homomorphism to any finite group sp v of automorphisms of a symplectic vector space v we associate a new multi parameter deformation h of the algebra v smash product of with the polynomial algebra on v the parameter runs over points of r where r number of conjugacy classes of symplectic reflections in the algebra h called a symplectic reflection algebra is related to the coordinate ring of a poisson deformation of the quotient singularity v this leads to a symplectic analogue of mckay correspondence which is most complete in case of wreath products if is the weyl group of a root system in a vector space and v then the algebras h are certain rational degenerations of the double affine hecke algebra introduced earlier by cherednik

quadratic compact knapsack public key cryptosystem knapsack type cryptosystems were among the first public key cryptographic schemes to be invented their np completeness nature and the high speed in encryption decryption made them very attractive however these cryptosystems were shown to be vulnerable to the low density subset sum attacks or some key recovery attacks in this paper additive knapsack type public key cryptography is reconsidered we propose a knapsack type public key cryptosystem by introducing an easy quadratic compact knapsack problem the system uses the chinese remainder theorem to disguise the easy knapsack sequence the encryption function of the system is nonlinear about the message vector under the relinearization attack model the system enjoys a high density we show that the knapsack cryptosystem is secure against the low density subset sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions it is shown that the proposed cryptosystem is also secure against some brute force attacks and some known key recovery attacks including the simultaneous diophantine approximation attack and the orthogonal lattice attack

on the hardness of approximating minimum vertex cover we prove the minimum vertex cover problem to be np hard to approximate to within a factor of 1 3606 extending on previous pcp and hardness of approximation technique to that end one needs to develop a new proof framework and to borrow and extend ideas from several fields

cusps and d modules 1 1 v modules on singular varieties let y denote a variety over a field and let vy denote the full sheaf of differential operators on y in characteristic zero vy is the familiar sheaf of differential operators and in characteristic p the sheaf vy will contain all divided powers of operators d it is well known that on a general singular variety y the ring of differential operators is badly behaved in particular it need not be noetherian nor if y is affine a simple ring see bgg following grothendieck there is an alternative notion of a sheaf with infinitesi mal parallel transport called a co stratification see be2 be3 bdi or section 2 which agrees with the notion of left respectively right py module when y is smooth but will not agree in general when y is singular our first result presents a simple vanishing condition that guarantees that the notions of x module and co stratification on a singular variety coincide

the parallel fourier pseudospectral method abstract parallel algorithms of the fourier pseudospectral method are presented for the solution of the unsteady incompressible navier stokes equations the only major operation that requires parallelization is the multidimensional fft in tests performed on a 1024 node hypercube computer an efficiency of about 83 is obtained for a three dimensional problem with mesh size 128 3 the all fortran code requires 17 s per timestep rivalling rates obtained from optimized codes on current supercomputers

minkowski s convex body theorem and integer programming the paper presents an algorithm for solving integer programming problems whose running time depends on the number n of variables as non this is done by reducing an n variable problem to 2n5i 2 problems in n i variables for some i greater than zero chosen by the algorithm the factor of on5 2 per variable improves the best previously known factor which is exponential in n minkowski s convex body theorem and other results from geometry of numbers play a crucial role in the algorithm several related algorithms for lattice problems are presented the complexity of these problems with respect to polynomial time reducibilities is studied

approximating svp to within almost polynomial factors is np hard abstract we show svp and cvp to be np hard to approximate to within n c log log n for some constant c 0 we show a direct reduction from sat to these problems that combines ideas from arora et al proc 34th ieee symp on foundations of computer science 1993 p 724 and dinur et al approximating cvp to within almost polynomial factors is np hard manuscript 1999 along with some modifications our result is obtained without relying on the pcp characterization of np although some of our techniques are derived from the proof of the pcp characterization itself stoc acm symposium on theory of computing stoc 1999

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

calogero moser hierarchy and kp hierarchy identification of the hierarchy of calogero moser dynamical systems with the dynamical systems of poles of certain solutions to the kp hierarchy is achieved

the shortest vector in a lattice is hard to approximate to within some constant we show that approximating the shortest vector problem in any ell p norm to within any constant factor less than sqrt p 2 is hard for np under reverse unfaithful random reductions with inverse polynomial error probability in particular approximating the shortest vector problem is not in rp random polynomial time unless np equals rp we also prove a proper np hardness result i e hardness under deterministic many one reductions under a reasonable number theoretic conjecture on the distribution of square free smooth numbers as part of our proof we give an alternative construction of ajtai s constructive variant of sauer s lemma that greatly simplifies ajtai s original proof

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

an lll algorithm with quadratic complexity the lenstra lenstra lovasz lattice basis reduction algorithm called lll or rm l 3 is a fundamental tool in computational number theory and theoretical computer science which can be viewed as an efficient algorithmic version of hermite s inequality on hermite s constant given an integer d dimensional lattice basis with vectors of euclidean norm less than b in an n dimensional space the rm l 3 algorithm outputs a reduced basis in o d 3n rm log b cdot mathcal m d rm log b bit operations where mathcal m k denotes the time required to multiply k bit integers this worst case complexity is problematic for applications where d or and rm log b are often large as a result the original rm l 3 algorithm is almost never used in practice except in tiny dimension instead one applies floating point variants where the long integer arithmetic required by gram schmidt orthogonalization is replaced by floating point arithmetic unfortunately this is known to be unstable in the worst case the usual floating point rm l 3 algorithm is not even guaranteed to terminate and the output basis may not be rm l 3 reduced at all in this article we introduce the rm l 2 algorithm a new and natural floating point variant of the rm l 3 algorithm which provably outputs rm l 3 reduced bases in polynomial time o d 2n d rm log b rm log b cdot mathcal m d this is the first rm l 3 algorithm whose running time without fast integer arithmetic provably grows only quadratically with respect to rm log b like euclid s gcd algorithm and lagrange s two dimensional algorithm

decoding square free goppa codes over bbf_ p we propose a new efficient nondeterministic decoding algorithm for square free goppa codes over fp for any prime p if the code in question has degree t and the average distance to the closest codeword is at least 4 p t 1 the proposed decoder can uniquely correct up to 2 p t errors with high probability the correction capability is higher if the distribution of error magnitudes is not uniform approaching or reaching t errors when any particular error value occurs much more often than others or exclusively this makes the method interesting for semantically secure cryptosystems based on the decoding problem for permuted and punctured goppa codes

cryptanalytic attacks on the multiplicative knapsack cryptosystem and on shamir s fast signature scheme the basic merkle hellman additive trapdoor knapsack public key cryptosystem was recently shown to be insecure and attacks have also been developed on stronger variants of it such as the graham shamir system and the iterated knapsack cryptosystem it is shown that some simple variants of another merkle hellman system the multiplicative knapsack cryptosystem are insecure it is also shown that the shamir fast signature scheme can be broken quickly similar attacks can also be used to break the schobi massey authentication scheme these attacks have not been rigorously proved to succeed but heuristic arguments and empirical evidence indicate that they work on systems of practical size

kp hierarchy for the cyclic quiver we introduce a generalisation of the kp hierarchy closely related to the cyclic quiver and the cherednik algebra hk zm this hierarchy depends on m parameters one of which can be eliminated with the usual kp hierarchy corresponding to the m 1 case generalising the result of wilson invent math 133 1 1 41 1998 we show that our hierarchy admits solutions parameterised by suitable quiver varieties the pole dynamics for these solutions is shown to be governed by the classical calogero moser system for the wreath product zm sn and its new spin version these results are further extended to the case of the multi component hierarchy

safer parameters for the chor rivest cryptosystem vaudenay s cryptanalysis against chor rivest cryptosystem is applicable when the parameters p and h originally proposed by the authors are used nevertheless if p and h are both prime integers then vaudenay s attack is not applicable in this work a choice of these parameters resistant to the existing cryptanalytic attacks is presented the parameters are determined in a suitable range guaranteeing its security and the computational feasibility of implementation regrettably the obtained parameters are scarce in practice