an upper bound on the aperiodic autocorrelation function for a maximal length sequence corresp the magnitude of the out of phase aperiodic autocorrelation function for a maximal length linear feedback shift register sequence of period n is at most 1 2 pi n 1 1 2 ln 4n pi previously the best upper bound was n 1 1 2 ln en

low correlation large linear span sequences from function fields a general method of generating families of binary sequences with low correlation as well as large linear span is presented the lower bound on the linear span is on the order of the square root of the period of each sequence within the family the design makes use of the theory of function fields two example applications of this method are presented in which the underlying function fields are the rational and elliptic function fields respectively

the algebra of complex 2 2 matrices and a general closed baker campbell hausdorff formula we derive a closed formula for the baker campbell hausdorff series expansion in the case of complex matrices for arbitrary matrices a and b and a matrix z such that our result expresses z as a linear combination of a and b their commutator and the identity matrix i the coefficients in this linear combination are functions of the traces and determinants of a and b and the trace of their product the derivation proceeds purely via algebraic manipulations of the given matrices and their products making use of relations developed here based on the cayley hamilton theorem as well as a characterization of the consequences of and or its determinant being zero or otherwise as a corollary of our main result we also derive a closed formula for the zassenhaus expansion we apply our results to several special cases most notably the parametrization of the product of two matrices and a verification of the recent result of van brunt and visser 2015 j phys a math theor 48 225207 for complex matrices in this latter case deriving also the related zassenhaus formula which turns out to be quite simple we then show that this simple formula should be valid for all matrices and operators

new results on periodic sequences with large k error linear complexity niederreiter showed that there is a class of periodic sequences which possess large linear complexity and large k error linear complexity simultaneously this result disproved the conjecture that there exists a trade off between the linear complexity and the k error linear complexity of a periodic sequence by ding by considering the orders of the divisors of xn 1 over bbf q we obtain three main results which hold for much larger k than those of niederreiter a sequences with maximal linear complexity and almost maximal k error linear complexity with general periods b sequences with maximal linear complexity and maximal k error linear complexity with special periods c sequences with maximal linear complexity and almost maximal k error linear complexity in the asymptotic case with composite periods besides we also construct some periodic sequences with low correlation and large k error linear complexity

an algorithm for the k error linear complexity of binary sequences with period 2 sup n certain applications require pseudo random sequences which are unpredictable in the sense that recovering more of the sequence from a short segment must be computationally infeasible it is shown that linear complexity is useful in determining such sequences a generalized linear complexity that has application to the security of stream ciphers is proposed and an efficient algorithm is given for the case in which the sequence is binary with period 2 sup n this algorithm generalizes an algorithm presented by r a games and a h chan 1983

on a limit formula for regular iterations abstract let f be an increasing homeomorphism of 0 onto itself with no nonzero fixed point such that d f 0 exists and 0 d 1 we investigate the limit f x lim n f n f n x x 0 0 and its properties we show that the schroder equation f x d x has a regularly varying solution if and only if for some a 0 the limit f a exists for in a dense set a in r and the map a f a is injective

the relationship between breaking the diffie hellman protocol and computing discrete logarithms both uniform and nonuniform results concerning the security of the diffie hellman key exchange protocol are proved first it is shown that in a cyclic group g of order g prod p i e i where all the multiple prime factors of g are polynomial in log g there exists an algorithm that reduces the computation of discrete logarithms in g to breaking the diffie hellman protocol in g and has complexity sqrt max nu p i cdot log g o 1 where nu p stands for the minimum of the set of largest prime factors of all the numbers d in the interval p 2 sqrt p 1 p 2 sqrt p 1 under the unproven but plausible assumption that nu p is polynomial in log p this reduction implies that the diffie hellman problem and the discrete logarithm problem are polynomial time equivalent in g second it is proved that the diffie hellman problem and the discrete logarithm problem are equivalent in a uniform sense for groups whose orders belong to certain classes there exists a polynomial time reduction algorithm that works for all those groups moreover it is shown that breaking the diffie hellman protocol for a small but nonnegligible fraction of the instances is equally difficult as breaking it for all instances finally efficient constructions of groups are described for which the algorithm reducing the discrete logarithm problem to the diffie hellman problem is efficiently constructible

embeddability of homeomorphisms of the circle in set valued iteration groups abstract let f s 1 s 1 be a homeomorphism without periodic points it is known that f is embeddable in a continuous iteration group if and only if f is minimal we deal with f which is not minimal in this case f satisfying some additional assumptions can be embedded but only in a nonmeasurable iteration groups there are infinitely many such nonmeasurable groups we propose here a new approach to the problem of embeddability for a given homeomorphism f without periodic points we construct some substitute of an iteration group namely the unique special set valued iteration group f t s 1 cc s 1 t r which is regular in a sense and in which f can be embedded i e f x f 1 x we also determine a maximal countable and dense subgroup t r such that f t s 1 cc s 1 t t has a continuous selection f t s 1 s 1 t t being the best regular embedding of f if there exists a nonmeasurable embedding f t s 1 s 1 t r of f then there exists an additive function r t such that f t z f t z t r we determine a unique maximal subgroup t with this property

lattices from hermitian function fields abstract we consider the well known rosenbloom tsfasman function field lattices in the special case of hermitian function fields we show that in this case the resulting lattices are generated by their minimal vectors provide an estimate on the total number of minimal vectors and derive properties of the automorphism groups of these lattices our study continues previous investigations of lattices coming from elliptic curves and finite abelian groups the lattices we are faced with here are more subtle than those considered previously and the proofs of the main results require the replacement of the existing linear algebra approaches by deep results of gerhard hiss on the factorization of functions with particular divisor support into lines and their inverses

realization extension and classification of certain physically important groups and algebras an associative algebra of differential forms with division has been constructed the algebra of forms in each different space provides a practical realization of the universal clifford algebra of that space a classification of all such algebras is given in terms of two distinct types of algebras nk and sk the former include the dihedral quaternion and majorana algebras the latter include the complex spinor and dirac algebras the associative product expresses hodge duality as multiplication by a basis element this makes possible the realization of higher order algebras in a calculationally useful algebraic setting the fact that the associative algebras as well as the enveloped lie algebras are precisely those arising in physics suggests that this formalism may be a convenient setting for the formulation of basic physical laws

pseudorandom bits from points on elliptic curves let e be an elliptic curve over a finite field fq of q elements with gcd q 6 1 given by an affine weierstras equation we use x p to denote the x component of a point p x p y p e we estimate character sums of the form n 1n x np x nq and n1 nk 1n j 1k cjx i 1j ni r on average over all fq rational points p q and r on e where is a quadratic character is a nontrivial additive character in fq and c1 ck fqk is a nonzero vector these bounds confirm several recent conjectures of jao jetchev and venkatesan related to extracting random bits from various sequences of points on the elliptic curves

on the single exponential closed form of the product of two exponential operators in the frame of a vectorial pauli algebraic approach it is shown that the product of two exponentials of any two by two linear operators can be put in a single exponential closed form as a first application a compact vectorial expression for the characteristics angle and axis of the product of two r3 rotations is established the same mathematics can be used in a large diversity of problems of the whole class of two states physical systems an exemplification in the field of polarization optics is given

determining the 2 sylow subgroup of an elliptic curve over a finite field in this paper we describe an algorithm that outputs the order and the structure including generators of the 2 sylow subgroup of an elliptic curve over a finite field to do this we do not assume any knowledge of the group order the results that lead to the design of this algorithm are of inductive type then a right choice of points allows us to reach the end within a linear number of successive halvings the algorithm works with abscissas so that halving of rational points in the elliptic curve becomes computing of square roots in the finite field efficient methods for this computation determine the efficiency of our algorithm

the exponential mapping in clifford algebras the exponential mapping takes the lie algebra of the lorentz group into the lorentz group each element of the group is defined as a formal power series while the product of two exponential elements usually involves an infinite sum of commutator terms such as in the baker campbell hausdorff formula because of the special arithmetic in clifford algebras many baker campbell hausdorff like formulas and identities can be calculated or summed exactly when they involve only elements from the algebra we calculate exact identities for the baker campbell hausdorff formula and related formulas in the quaternion and dihedral algebras these are useful in treatments of the lorentz group and make possible a truly finite as opposed to infinitesimal description of a transformation group in physics

incomplete exponential sums over galois rings with applications to some binary sequences derived from z sub 2 sup l an upper bound for the incomplete exponential sums over galois rings is derived explicitly based on the incomplete exponential sums we analyze the partial period properties of some binary sequences derived from z sub 2 sup l in detail such as the kerdock code binary sequences and the highest level sequences of primitive sequences over z sub 2 sup l the results show that the partial period distributions and the partial period independent r pattern distributions of these binary sequences are asymptotically uniform nontrivial upper bounds for the aperiodic autocorrelation of these sequences are also given

on the semigroup property for some structured iterations abstract nonlinear matrix equations play a crucial role in science and engineering problems however solutions of nonlinear matrix equations cannot in general be given analytically one standard way of solving nonlinear matrix equations is to apply the fixed point iteration with usually only the linear convergence rate to advance the existing methods we exploit in this work one type of semigroup property and use this property to propose a technique for solving the equations with the speed of convergence of any desired order we realize our way by starting with examples of solving the scalar equations and also connect this method with some well known equations including but not limited to the stein matrix equation the generalized eigenvalue problem the generalized nonlinear matrix equation the discrete time algebraic riccati equations to express the capacity of this method

multisequences with large linear and k error linear complexity from hermitian function fields in the present paper by making use of some special properties of the hermitian function fields we construct multisequences with both large linear complexity and k error linear complexity moreover these sequences can be explicitly constructed

a note on elliptic curves over finite fields let e be an elliptic curve over a finite field k and let e k be the group of k rational points on e we evaluate all the possible groups e k where e runs through all the elliptic curves over a given fixed finite field k

on the main conjecture on geometric mds codes a new way to attack the main conjecture on mds codes for geometric codes is proposed in particular the conjecture for codes arising from curves of genus one or two when the cardinal of the ground field is large enough is proven

constructing linear unequal error protection codes from algebraic curves we show that the concept of generalized algebraic geometry codes which was introduced by xing niederreiter and lam see ibid vol 45 p 2498 2501 nov 1999 gives a natural framework for constructing linear unequal error protection codes

on set valued iteration groups generated by commuting functions abstract let i be an open interval and f g i i be commuting homeomorphisms such that f i d g i d and inf n m n m n f n g m s q define f t sup f n g m n m n n s m t f t inf f n g m n m n n s m t and f t x f t x f t x x i we show that the family f t i c c i t r is an iteration group in the sense of set valued functions such that f x f 1 x and g x f s x if f and g are embeddable in an iteration group of continuous functions f t t r then there exists an additive function such that f t x f t x t r several other properties of set valued iteration groups f t t r as well as iteration groups f t t r and f t t r are given

on the expected value of the linear complexity and the k error linear complexity of periodic sequences rueppel 1986 conjectured that periodic binary sequences have expected linear complexity close to the period length n in this paper we determine the expected value of the linear complexity of n periodic sequences explicitly and confirm rueppel s conjecture for arbitrary finite fields cryptographically strong sequences should not only have a large linear complexity but also the change of a few terms should not cause a significant decrease of the linear complexity this requirement leads to the concept of the k error linear complexity of n periodic sequences we present a method to establish a lower bound on the expected k error linear complexity of n periodic sequences based on the knowledge of the counting function spl nscr sub n sub 0 c i e the number of n periodic sequences with given linear complexity c for some cases we give explicit formulas for that lower bound and we also determine spl nscr sub n 0 c

bi iterative limits used to the theory of the schroder equation abstract let f be a fixed point free increasing homeomorphism of r onto itself such that the limit d lim n f n 1 x f n x exits belongs to the interval 0 1 and is independent of x r for each r we define the bi iterative limits f and f of f to be the lower and the upper limits of the sequence f n f n x n n x r respectively we show that the following statements are equivalent a the schroder equation f x d x has a continuous regularly varying solution b the set consisting of the differentials at zero of the bi iterative limits of f is dense in the multiplicative group r c there exists r such that lim x 0 f x x exists belongs to r and log log d is irrational

algebras with three anticommuting elements i spinors and quaternions a general construction of alternative algebras with three anticommuting elements and a unit is given as an exhaustive result over the real and complex fields we obtain the clifford algebras h quaternions n1 dihedral clifford algebra which is related to real 2 spinors and s1 algebra of pauli matrices which is related to complex 2 spinors what is important is that the algebras n1 and s1 possess inverses everywhere except on a region akin to the light cone of the minkowski space while the quaternion algebra h has inverses everywhere except at the zero element we discuss the reasons why the three algebras n1 h and s1 are so difficult to distinguish in the representation space of 2 2 complex matrices

a relationship between linear complexity and k error linear complexity linear complexity is an important cryptographic criterion of stream ciphers the k error linear complexity of a periodic sequence of period n is defined as the smallest linear complexity that can be obtained by changing k or fewer bits of the sequence per period this article shows a relationship between the linear complexity and the minimum value k for which the k error linear complexity is strictly less than the linear complexity

practical tests for the schroder equation to have a regularly varying solution abstract let f r r be an increasing continuous function having no fixed point such that the differential d lim x 0 f x x exists and 0 d 1 in this paper we present practical tests to check if the schroder equation f x d x has a regularly varying solution on r to do this we associate to f a sequence u 0 u 1 of continuous functions of 0 1 onto itself in such a way that the desired solution exists if and only if the iterative sequence u n u n u n 1 u 0 converges uniformly concerning this we present three tests called the norm test the root test of iterations the alternating iterative sequence test

computing the l power torsion of an elliptic curve over a finite field the algorithm we develop outputs the order and the structure including generators of the l sylow subgroup of the group of rational points of an elliptic curve defined over a finite field to do this we do not assume any knowledge of the group order we are able to choose points in such a way that a linear number of successive l divisions leads to generators of the subgroup under consideration after the computation of a couple of polynomials each division step relies on finding rational roots of polynomials of degree l we specify in complete detail the case l 3 when the complexity of each trisection is given by the computation of cubic roots in finite fields

the lorentz group and the thomas precession ii exact results for the product of two boosts the product of two lorentz boosts in different directions is equal to the product of a pure boost and a spatial rotation to second order the resulting boost is simply the sum of the individual boosts and the rotation is responsible for the thomas precession here the resulting boost and the rotation correction are calculated exactly the calculation of higher order corrections to the sum of two finite lorentz boosts is not usually done and here previous exact results are critically compared the complete expression is to the best of the author s knowledge a new result

exponentiation of the spinor representation of the lorentz group an exact finite expression for the exponentiation of the spin 12 spinor representation of the lorentz group is obtained from this expression an exact finite expression for the exponentiation of the vector representation of the lorentz group is derived the two expressions are compared with the literature in the special cases of either spatial rotations or boosts only

on the aperiodic and odd correlations of the binary shanbhag kumar helleseth sequences a previously derived upper bound on hybrid sums over galois rings is combined with a method due to vinogradov 1954 to obtain an upper bound for the maximum aperiodic and odd correlations of the binary sequences constructed by shanbhag kumar and helleseth see proc 1995 ieee int symp on information theory 1995

upper bound for a hybrid sum over galois rings with applications to aperiodic correlation of some q ary sequences an upper bound for a hybrid exponential sum over galois rings is derived this bound is then used to obtain an upper bound for the maximum aperiodic correlation of some sequence families over galois rings the bound is of the order of spl radic qlnq where q 1 is the period of the sequences

on a class of pseudorandom sequences from elliptic curves over finite fields following the idea of xing et al we investigate a general method for constructing families of pseudorandom sequences with low correlation and large linear complexity from elliptic curves over finite fields in this correspondence with the help of the tool of exponential sums on elliptic curves we study their periods linear complexities linear complexity profiles distributions of r patterns periodic correlation partial period distributions and aperiodic correlation in detail the results show that they have nice randomness

recursion formulas for the lie integral abstract an explicit definition is given of a non linear integral which is termed the lie integral this integral is defined for lie algebra valued functions in terms of the representations of the algebra various properties of the integral are investigated and applications are given to differential equations applications to differential geometry will be given elsewhere

sequences with almost perfect linear complexity profiles and curves over finite fields for stream ciphers we need to generate pseudorandom sequences which are of properties of unpredictability and randomness a important measure of unpredictability and randomness is the linear complexity profile l c p l sub a n of a sequence a a sequence a is called almost perfect if the l c p is l sub a n n 2 o 1 based on curves over finite fields we present a method to construct almost perfect sequences we also illustrate our construction by explicit examples from the projective line and elliptic curves over the binary field

periodic sequences with large k error linear complexity we establish the existence of periodic sequences over a finite field which simultaneously achieve the maximum value for the given period length of the linear complexity and of the k error linear complexity for small values of k this disproves a conjecture of ding xiao and shan 1991 the result is of relevance for the theory of stream ciphers