determining the small solutions to s unit equations in this paper we generalize the method of wildanger for finding small solutions to unit equations to the case of s unit equations the method uses a minor generalization of the lll based techniques used to reduce the bounds derived from transcendence theory followed by an enumeration strategy based on the fincke pohst algorithm the method used reduces the computing time needed from mips years down to minutes

an adaptive linear approximation algorithm for copositive programs we study linear optimization problems over the cone of copositive matrices these problems appear in nonconvex quadratic and binary optimization for instance the maximum clique problem and other combinatorial problems can be reformulated as such problems we present new polyhedral inner and outer approximations of the copositive cone which we show to be exact in the limit in contrast to previous approximation schemes our approximation is not necessarily uniform for the whole cone but can be guided adaptively through the objective function yielding a good approximation in those parts of the cone that are relevant for the optimization and only a coarse approximation in those parts that are not using these approximations we derive an adaptive linear approximation algorithm for copositive programs numerical experiments show that our algorithm gives very good results for certain nonconvex quadratic problems

the double base number system and its application to elliptic curve cryptography we describe an algorithm for point multiplication on generic elliptic curves based on a representation of the scalar as a sum of mixed powers of 2 and 3 the sparseness of this so called double base number system combined with some efficient point tripling formulae lead to efficient point multiplication algorithms for curves defined over both prime and binary fields side channel resistance is provided thanks to side channel atomicity

copositivity detection by difference of convex decomposition and ω subdivision we present three new copositivity tests based upon difference of convex d c decompositions and combine them to a branch and bound algorithm of subdivision type the tests employ lp or convex qp techniques but also can be used heuristically using appropriate test points we also discuss the selection of efficient d c decompositions and propose some preprocessing ideas based on the spectral d c decomposition we report on first numerical experience with this procedure which are very promising

computing the structure of a finite abelian group we present an algorithm that computes the structure of a finite abelian group g from a generating system m the algorithm executes o m g group operations and stores o g group elements

displacement operator based decompositions of matrices using circulants or other group matrices abstract we show how an arbitrary square matrix can be expressed as sums of products of circulant and upper or lower triangular toeplitz matrices and as sums of products of matrices derived from finite groups group matrices and matrices which are close to group matrices the results obtained are interesting from several points of view they lead to different methods for computing linear transforms using ffts or fast convolution algorithms to faster methods for solving toeplitz systems to potential methods for mapping linear transforms to parallel computer architectures with interconnection scheme given by the group graph of a finite group and possibly to matrix theoretic methods for expressing relationship between finite groups

fast elliptic scalar multiplication using new double base chain and point halving the fast implementation of elliptic curve cryptosystems relies on the efficient computation of scalar multiplication based on the double base chain representation of scalar using powers of 2 and 3 we propose a new representation with powers of 1 2 and 3 instead thus the efficient point halving operation can be incorporated in the new double base chain to achieve fast scalar multiplication experimental results show that our approach leads to a lower complexity which contributes to the efficient implementation of elliptic curve cryptosystems

structure of algebras of commutative matrices commutative spaces of matrices a nk 1 akjk are studied where jk is a set of 0 1 matrices with prescribed sum s and the ak s are complex parameters the great number of commutative spaces a obeying simple prescriptions on the first two rows of s can all be described in terms of two basic algebras of matrices denoted by tn and n a relationship between the structure of spaces a and the multiplicative complexity of the bilinear forms defined by the jk s is discussed

algorithms for determining the copositivity of a given symmetric matrix abstract andersson et al l e andersson g chang t elfving criteria for copositive matrices using simplices and barycentric coordinates linear algebra appl 220 1995 9 30 gives necessary and sufficient conditions for symmetric matrices of order n 5 to be copositive and yang and li s yang x li some simple criteria for copositive matrices in proceedings of the seventh international conference on matrix theory and applications advances in matrix theory and applications world academic union 2006 gives necessary and sufficient conditions for symmetric z matrices of any order to be strictly copositive and some simpler sufficient conditions for symmetric matrices of any order to be copositive or strictly copositive or to be not copositive based on these known results we will present six algorithms of determining whether a given symmetric matrix is strictly copositive or copositive or not copositive the algorithms for matrices of order 3 4 5 6 or 7 are efficient because they quickly give the complete answer for the test the algorithm for matrices of order n 8 is not guaranteed to produce an answer but usually does produce one

generalized circulants and class functions of finite groups abstract the regular representation of the ring of class functions of a finite group is used to construct families of matrices that enjoy many of the properties of circulant matrices for different choices of finite groups one gets different families of matrices the circulants block circulants block circulants with circulant blocks of all levels and many others the properties of these families are simple consequences of the character theory of finite groups many known properties of circulants and some of their generalizations are obtained as special cases of the properties of the families constructed some applications to character theory are also mentioned

extended elliptic curve montgomery ladder algorithm over binary fields with resistance to simple power analysis in this paper we propose a scalar multiplication algorithm on elliptic curves over gf 2 m the proposed algorithm is an extended version of the montgomery ladder algorithm with the quaternary representation of the scalar in addition in order to improve performance we have developed new composite operation formulas and apply them to the proposed scalar multiplication algorithm the proposed composite formulas are 2p 1 2p 2 3p 1 p 2 and 4p 1 where p 1 and p 2 are points on an elliptic curve they can be computed using only the x coordinate of a point p x y in the affine coordinate system however the proposed scalar multiplication algorithm is vulnerable to simple power analysis attacks because different operations are performed depending on the bits of the scalar unlike the original montgomery ladder algorithm therefore we combine the concept of the side channel atomicity with the proposed composite operation formulas to prevent simple power analysis furthermore to optimize the computational cost we use the montgomery trick which can reduce the number of finite field inversion operations used in the affine coordinate system as the result the proposed scalar multiplication algorithm saves at least 26 of running time with small storage compared to the previous algorithms such as window based methods and comb based methods

a generic approach to searching for jacobians we consider the problem of finding cryptographically suitable jacobians by applying a probabilistic generic algorithm to compute the zeta functions of low genus curves drawn from an arbitrary family we can search for jacobians containing a large subgroup of prime order for a suitable distribution of curves the complexity is subexponential in genus 2 and o n1 12 in genus 3 we give examples of genus 2 and genus 3 hyperelliptic curves over prime fields with group orders over 180 bits in size improving previous results our approach is particularly effective over low degree extension fields where in genus 2 we find jacobians over fp2 and trace zero varieties over fp3 with near prime orders up to 372 bits in size for p 261 1 the average time to find a group with 244 bit near prime order is under an hour on a pc

a nonlinear elliptic curve cryptosystem based on matrices we propose a new mathematical problem that is applicable to public key cryptography based on the discrete logarithm problem dlp it uses certain elements formed by two matrices with elements in a finite field and a matrix whose elements are points of an elliptic curve with this system we get a larger key space without increasing the underlying elliptic curve and consequently without the computational requirements inherent to the set up of elliptic curves at random also we expose the diffie hellman key agreement protocol with this system acting as the underlying mathematical problem

evaluation of discrete logarithms in a group of p torsion points of an elliptic curve in characteristic p we show that to solve the discrete log problem in a subgroup of order p of an elliptic curve over the finite field of characteristic p one needs o ln p operations in this field

linear and sublinear time algorithms for the basis of abelian groups it is well known that every finite abelian group g can be represented as a direct product of cyclic groups email protected g 1xg 2x xg t where each g i is a cyclic group of order p j for some prime p and integer j 1 if a i generates the cyclic group of g i i 1 2 t then the elements a 1 a 2 a t are called a basis of g we show a randomized algorithm such that given a set of generators m x 1 x k for an abelian group g and the prime factorization of order ord x i i 1 k it computes a basis of g in o m logn email protected i 1 tn ip i n i 2 time where n g has prime factorization p 1 n 1p 2 n 2 p t n t which is not a part of input this generalizes buchmann and schmidt s algorithm that takes o m g time in another model all elements in an abelian group are put into a list as a part of input we obtain an o n time deterministic algorithm and a sublinear time randomized algorithm for computing a basis of an abelian group

algorithmic copositivity detection by simplicial partition we present new criteria for copositivity of a matrix i e conditions which ensure that the quadratic form induced by the matrix is nonnegative over the nonnegative orthant these criteria arise from the representation of the quadratic form in barycentric coordinates with respect to the standard simplex and simplicial partitions thereof we show that as the partition gets finer and finer the conditions eventually capture all strictly copositive matrices we propose an algorithmic implementation which considers several numerical aspects as an application we present results on the maximum clique problem we also briefly discuss extensions of our approach to copositivity with respect to arbitrary polyhedral cones

construction of cm picard curves in this article we generalize the cm method for elliptic and hyperelliptic curves to picard curves we describe the algorithm in detail and discuss the results of our implementation

smooth ideals in hyperelliptic function fields recently several algorithms have been suggested for solving the discrete logarithm problem in the jacobians of high genus hyperelliptic curves over finite fields some of them have a provable subexponential running time and are using the fact that smooth reduced ideals are sufficiently dense we explicitly show how these density results can be derived all proofs are purely combinatorial and do not exploit analytic properties of generating functions

the montgomery modular inverse revisited we modify an algorithm given by kaliski to compute the montgomery inverse of an integer modulo a prime number we also give a new definition of the montgomery inverse and introduce efficient algorithms for computing the classical modular inverse the kaliski montgomery inverse and the new montgomery inverse the proposed algorithms are suitable for software implementations on general purpose microprocessors

structure computation and discrete logarithms in finite abelian p groups we present a generic algorithm for computing discrete logarithms in a finite abelian p group h improving the pohlig hellman algorithm and its generalization to noncyclic groups by teske we then give a direct method to compute a basis for h without using a relation matrix the problem of computing a basis for some or all of the sylow p subgroups of an arbitrary finite abelian group g is addressed yielding a monte carlo algorithm to compute the structure of g using o g 1 2 group operations these results also improve generic algorithms for extracting pth roots in g

approximation of the stability number of a graph via copositive programming lovasz and schrijver siam j optim 1 1991 pp 166 190 showed how to formulate increasingly tight approximations of the stable set polytope of a graph by solving semidefinite programs sdps of increasing size lift and project method in this paper we present a similar idea we show how the stability number can be computed as the solution of a conic linear program lp over the cone of copositive matrices subsequently we show how to approximate the copositive cone ever more closely via a hierarchy of linear or semidefinite programs of increasing size liftings the latter idea is based on recent work by parrilo structured semidefinite programs and semi algebraic geometry methods in robustness and optimization ph d thesis california institute of technology pasadena ca 2000 in this way we can compute the stability number alpha g of any graph g v e after at most alpha g 2 successive liftings for the lp based approximations one can compare this to the n alpha g 1 bound for the lp based lift and project scheme of lovasz and schrijver our approach therefore requires fewer liftings for families of graphs where alpha g o sqrt n we show that the first sdp based approximation for alpha g in our series of increasingly tight approximations coincides with the vartheta function of schrijver ieee trans inform theory 25 1979 pp 425 429 we further show that the second approximation is tight for complements of triangle free graphs and for odd cycles

elliptic curve cryptosystems we discuss analogs based on elliptic curves over finite fields of public key cryptosystems which use the multiplicative group of a finite field these elliptic curve cryptosystems may be more secure because the analog of the discrete logarithm problem on elliptic curves is likely to be harder than the classical discrete logarithm problem especially over gf 2 we discuss the question of primitive points on an elliptic curve modulo p and give a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point

on the group matrices for a generalized dihedral group abstract for the generalized dihedral group an effective method is given to construct a matrix which transforms group matrices to a convenient block diagonal form

a variational approach to copositive matrices this work surveys essential properties of the so called copositive matrices the study of which has been spread over more than fifty five years special emphasis is given to variational aspects related to the concept of copositivity in addition some new results on the geometry of the cone of copositive matrices are presented here for the first time

the operator psi for the chromatic number of a graph we investigate hierarchies of semidefinite approximations for the chromatic number chi g of a graph g we introduce an operator psi mapping any graph parameter beta g nested between the stability number alpha g and chi overline g to a new graph parameter psi beta g nested between alpha overline g and chi g psi beta g is polynomial time computable if beta g is as an application there is no polynomial time computable graph parameter nested between the fractional chromatic number chi cdot and chi cdot unless text p text np moreover based on the motzkin straus formulation for alpha g we give quadratically constrained quadratic and copositive programming formulations for chi g under some mild assumptions n beta g le psi beta g but while n beta g remains below chi g psi beta g can reach chi g e g for beta cdot alpha cdot we also define new polynomial time computable lower bounds for chi g improving the classic lovasz theta number and its strengthenings obtained by adding nonnegativity and triangle inequalities experimental results on hamming graphs kneser graphs and dimacs benchmark graphs will be given in the follow up paper n gvozdenovic and m laurent siam j optim 19 2008 pp 592 615

on the copositive representation of binary and continuous nonconvex quadratic programs in this paper we model any nonconvex quadratic program having a mix of binary and continuous variables as a linear program over the dual of the cone of copositive matrices this result can be viewed as an extension of earlier separate results which have established the copositive representation of a small collection of np hard problems a simplification which reduces the dimension of the linear conic program and an extension to complementarity constraints are established and computational issues are discussed

reducing elliptic curve logarithms to logarithms in a finite field elliptic curve cryptosystems have the potential to provide relatively small block size high security public key schemes that can be efficiently implemented as with other known public key schemes such as rsa and discrete exponentiation in a finite field some care must be exercised when selecting the parameters involved in this case the elliptic curve and the underlying field specific classes of curves that give little or no advantage over previously known schemes are discussed the main result of the paper is to demonstrate the reduction of the elliptic curve logarithm problem to the logarithm problem in the multiplicative group of an extension of the underlying finite field for the class of supersingular elliptic curves the reduction takes probabilistic polynomial time thus providing a probabilistic subexponential time algorithm for the former problem

on random walks for pollard s rho method we consider pollard s rho method for discrete logarithm computation usually in the analysis of its running time the assumption is made that a random walk in the underlying group is simulated we show that this assumption does not hold for the walk originally suggested by pollard its performance is worse than in the random case we study alternative walks that can be efficiently applied to compute discrete logarithms we introduce a class of walks that lead to the same performance as expected in the random case we show that this holds for arbitrarily large prime group orders thus making pollard s rho method for prime group orders about 20 faster than before

on cones of nonnegative quadratic functions we derive linear matrix inequality lmi characterizations and dual decomposition algorithms for certain matrix cones which are generated by a given set using generalized co positivity these matrix cones are in fact cones of nonconvex quadratic functions that are nonnegative on a certain domain as a domain we consider for instance the intersection of a upper level set of a quadratic function and a half plane consequently we arrive at a generalization of yakubovich s s procedure result although the primary concern of this paper is to characterize the matrix cones by lmis we show as an application of our results that optimizing a general quadratic function over the intersection of an ellipsoid and a half plane can be formulated as semidefinite programming sdp thus proving the polynomiality of this class of optimization problems which arise e g from the application of the trust region method for nonlinear programming other applications are in control theory and robust optimization

multicore curve based cryptoprocessor with reconfigurable modular arithmetic logic units over gf 2 n this paper presents a reconfigurable curve based cryptoprocessor that accelerates scalar multiplication of elliptic curve cryptography ecc and hyperelliptic curve cryptography hecc of genus 2 over gf 2n by allocating a copies of processing cores that embed reconfigurable modular arithmetic logic units malus over gf 2n the scalar multiplication of ecc hecc can be accelerated by exploiting instruction level parallelism ilp the supported field size can be arbitrary up to a n 1 1 the superscaling feature is facilitated by defining a single instruction that can be used for all field operations and point divisor operations in addition the cryptoprocessor is fully programmable and it can handle various curve parameters and arbitrary irreducible polynomials the cost performance and security trade offs are thoroughly discussed for different hardware configurations and software programs the synthesis results with a 0 13 mum cmos technology show that the proposed reconfigurable cryptoprocessor runs at 292 mhz whereas the field sizes can be supported up to 587 bits the compact and fastest configuration of our design is also synthesized with a fixed field size and irreducible polynomial the results show that the scalar multiplication of ecc over gf 2163 and hecc over gf 283 can be performed in 29 and 63 mus respectively

improved computation of square roots in specific finite fields in this paper we study exponentiation in the specific finite fields f with very special exponents such as those that occur in algorithms for computing square roots here q is a prime power q pk where k 1 and k is odd our algorithmic approach improves the corresponding exponentiation resulted from the better rewritten exponent to the best of our knowledge it is the first major improvement to the tonelli shanks algorithm for example the number of multiplications can be reduced to at least 60 percent on the average when p 1 mod 16 several numerical examples are given that show the speedup of the proposed methods

some np complete problems in quadratic and nonlinear programming in continuous variable smooth nonconvex nonlinear programming we analyze the complexity of checking whether a a given feasible solution is not a local minimum and b the objective function is not bounded below on the set of feasible solutions we construct a special class of indefinite quadratic programs with simple constraints and integer data and show that checking a or b on this class is np complete as a corollary we show that checking whether a given integer square matrix is not copositive is np complete

detecting copositivity of a symmetric matrix by an adaptive ellipsoid based approximation scheme abstract it is co np complete to decide whether a given matrix is copositive or not in this paper this decision problem is transformed into a quadratic programming problem which can be approximated by solving a sequence of linear conic programming problems defined on the dual cone of the cone of nonnegative quadratic functions over the union of a collection of ellipsoids using linear matrix inequalities lmi representations each corresponding problem in the sequence can be solved via semidefinite programming in order to speed up the convergence of the approximation sequence and to relieve the computational effort of solving linear conic programming problems an adaptive approximation scheme is adopted to refine the union of ellipsoids the lower and upper bounds of the transformed quadratic programming problem are used to determine the copositivity of the given matrix

elliptic curves over finite fields and the computation of square roots mod p in this paper we present a deterministic algorithm to compute the number of f points of an elliptic curve that is defined over a finite field fv and which is given by a weierstrass equation the algorithm takes 0 log9 q elementary operations as an application wc give an algorithm to compute square roots mod p for fixed i e z it takes 0 log9p elementary operations to compute fx mod p 1 introduction in this paper we present an algorithm to compute the number of f points of an elliptic curve defined over a finite field f which is given by a weierstrass equation we restrict ourselves to the case where the characteristic of f is not 2 or 3 the algorithm is deterministic does not depend on any unproved hypotheses and takes 0 log9 0 if one applies fast multiplication techniques the algorithm will take 0 x 1 2log p 6 f elementary operations for any e 0 let be an elliptic curve defined over the prime field fp and let an affine model of it be given by a weierstrass equation y2 x3 ax b a befp an explicit formula for the number of f points on is given by

on some computational problems in finite abelian groups we present new algorithms for computing orders of elements discrete logarithms and structures of finite abelian groups we estimate the computational complexity and storage requirements and we explicitly determine the o constants and constants we implemented the algorithms for class groups of imaginary quadratic orders and present a selection of our experimental results our algorithms are based on a modification of shanks baby step giant step strategy and have the advantage that their computational complexity and storage requirements are relative to the actual order discrete logarithm or size of the group rather than relative to an upper bound on the group order

a w nnaf method for the efficient computation of scalar multiplication in elliptic curve cryptography based on the non adjacent form naf expression here we present a new method to express a large positive integer k with the base of any positive integer w other than 2 we call this method the w nnaf method as the expression is near to the naf one and with base w this expression leads to the minimal hamming weight of all the base w expressions based on the proposed expression and some existing methods an algorithm is developed to compute kp efficiently

computing hilbert class polynomials with the chinese remainder theorem we present a space efficient algorithm to compute the hilbert class polynomial h d x modulo a positive integer p based on an explicit form of the chinese remainder theorem under the generalized riemann hypothesis the algorithm uses o d 1 2 e log p space and has an expected running time of o d 1 e we describe practical optimizations that allow us to handle larger discriminants than other methods with d as large as 10 13 and h d up to 10 6 we apply these results to construct pairing friendly elliptic curves of prime order using the cm method

new results on quadratic minimization in this paper we present several new results on minimizing an indefinite quadratic function under quadratic linear constraints the emphasis is placed on the case in which the constraints are two quadratic inequalities this formulation is termed the extended trust region subproblem in this paper to distinguish it from the ordinary trust region subproblem in which the constraint is a single ellipsoid the computational complexity of the extended trust region subproblem in general is still unknown in this paper we consider several interesting cases related to this problem and show that for those cases the corresponding semidefinite programming relaxation admits no gap with the true optimal value and consequently we obtain polynomial time procedures for solving those special cases of quadratic optimization for the extended trust region subproblem itself we introduce a parameterized problem and prove the existence of a trajectory that will lead to an optimal solution combining this with a result obtained in the first part of the paper we propose a polynomial time solution procedure for the extended trust region subproblem arising from solving nonlinear programs with a single equality constraint

an application of lemke s method to a class of markov decision problems this paper presents an application of lemke s method to a class of markov decision problems appearing in the optimal stopping problems and other well known optimization problems we consider a special case of the markov decision problems with finitely many states where the agent can choose one of the alternatives getting a fixed reward immediately or paying the penalty for one term we show that the problem can be reduced to a linear complementarity problem that can be solved by lemke s method with the number of iterations less than the number of states the reduced linear complementarity problem does not necessarily satisfy the copositive plus condition nevertheless we show that the lemke s method succeeds in solving the problem by proving that the problem satisfies a necessary and sufficient condition for the extended lemke s method to compute a solution in the piecewise linear complementarity problem

copositive optimization recent developments and applications due to its versatility copositive optimization receives increasing interest in the operational research community and is a rapidly expanding and fertile field of research it is a special case of conic optimization which consists of minimizing a linear function over a cone subject to linear constraints the diversity of copositive formulations in different domains of optimization is impressive since problem classes both in the continuous and discrete world as well as both deterministic and stochastic models are covered copositivity appears in local and global optimality conditions for quadratic optimization but can also yield tighter bounds for np hard combinatorial optimization problems here some of the recent success stories are told along with principles algorithms and applications

a space efficient algorithm for group structure computation we present a new algorithm for computing the structure of a finite abelian group which has to store only a fixed small number of group elements independent of the group order we estimate the computational complexity by counting the group operations such as multiplications and equality checks under some plausible assumptions we prove that the expected run time is o n with n denoting the group order and we explicitly determine the o constants we implemented our algorithm for ideal class groups of imaginary quadratic orders and present experimental results

kkt solution and conic relaxation for solving quadratically constrained quadratic programming problems to find a global optimal solution to the quadratically constrained quadratic programming problem we explore the relationship between its lagrangian multipliers and related linear conic programming problems this study leads to a global optimality condition that is more general than the known positive semidefiniteness condition in the literature moreover we propose a computational scheme that provides clues of designing effective algorithms for more solvable quadratically constrained quadratic programming problems

an efficient undeniable group oriented signature scheme three main methods with favorable security and efficiency can be considered in the design of cryptosystems these methods are integer factorization systems of which rsa is the best known example discrete logarithm systems such as dsa and elliptic curve cryptosystems ecc math comput 48 1987 203 advances in cryptology crypto 85 springer verlag 1986 p 417 the elliptic curve cryptosystem has been applied in this study because of its low computational cost and short key size which reduces the overheads in a limited hardware environment thus the elliptic curve constitutes a suitable environment for the cryptosystems this study improves the undeniable signature scheme proceedings of the eleventh national conference on information security 2001 p 331 by reducing its time complexity such an improvement makes the proposed scheme more efficient and secure

security analysis of the public key algorithm based on chebyshev polynomials over the integer ring zn recently kocarev and tasev 20 proposed to use chebyshev polynomials over real numbers to design a public key algorithm by employing the semigroup property bergamo et al 4 pointed out that the public key algorithm based on chebyshev polynomials working on real numbers is not secure and devised an attack which permits to recover the corresponding plaintext from a given ciphertext later kocarev et al 19 generalized the chebyshev polynomials from real number fields to finite fields and finite rings to make the public key algorithm more secure and practical however we analyzed the period distribution of the sequences generated by the chebyshev polynomials over finite fields 21 when the modulus n is prime we found this algorithm was also not secure and proposed an attack on this algorithm over finite fields we then proposed some schemes to improve the security in this paper we further analyze in detail the period distribution of the sequences generated by chebyshev polynomials over the integer ring z n when n is composite it turns out that the period distribution is poor if n is not chosen properly and there are many small periods which are not secure in the sense of cryptology based on these findings we devise an attack on the public key algorithm based on chebyshev polynomials over the integer ring z n we also propose some suggestions to avoid this attack