a remark concerning m divisibility and the discrete logarithm in the divisor class group of curves the aim of this paper is to show that the computation of the discrete logarithm in the m torsion part of the divisor class group of a curve x over a finite field ko with char ko prime to m or over a local field k with residue field ko can be reduced to the computation of the discrete logarithm in k0 4m for this purpose we use a variant of the tame tate pairing for abelian varieties over local fields in the same way the problem to determine all linear combinations of a finite set of elements in the divisor class group of a curve over k or ko which are divisible by m is reduced to the computation of the discrete logarithm in ko cm

theory of two dimensional multirate filter banks results are presented on 2 d fir two dimensional finite impulse response filter banks for multirate applications the theory is valid for all sampling lattices conditions for alias free and perfect signal reconstruction are derived synthesis structures for paraunitary and nonparaunitary polynomial matrices are derived which yield perfect reconstruction filter banks the degrees of freedom are given for these systems linear phase conditions are posed on the polyphase form of filter banks which is used to derive a design structure for the restricted but important case of linear phase filter banks

a filter bank for the directional decomposition of images theory and design the authors introduce a directionally oriented 2 d filter bank with the property that the individual channels may be critically sampled without loss of information the passband regions of the component filters are wedge shaped and thus provide directional information it is shown that these filter bank outputs may be maximally decimated to achieve a minimum sample representation in a way that permits the original signal to be exactly reconstructed the authors discuss the theory for directional decomposition and reconstruction in addition implementation issues are addressed where realizations based on both recursive and nonrecursive filters are considered

conjugacy classes of elements in the borovik group the borovik groupbis a lie primitive finite subgroup ofe8 c which containsalt5 sym6with index 2 and is not a direct product we determine its elements as members ofe8 c conjugacy classes and discuss itsalt5subgroups

boundary filter optimization for segmentation based subband coding this paper presents boundary optimization techniques for the nonexpansive decomposition of arbitrary length signals with multirate filterbanks both biorthogonal and paraunitary filterbanks are considered the paper shows how matching moments and orthonormality can be imposed as additional conditions during the boundary filter optimization process it provides direct solutions to the problem of finding good boundary filters for the following cases a biorthogonal boundary filters with exactly matching moments and b orthonormal boundary filters with almost matching moments with the proposed methods numerical optimization is only needed if orthonormality and exactly matching moments are demanded the proposed direct solutions are applicable to systems with a large number of subbands and or very long filter impulse responses design examples show that the methods allow the design of boundary filters with good frequency selectivity

nonseparable multidimensional perfect reconstruction filter banks and wavelet bases for r sup n new results on multidimensional filter banks and their connection to multidimensional nonseparable wavelets are presented among the topics discussed are sampling in multiple dimensions multidimensional perfect reconstruction filter banks the two channel case in multiple dimensions the synthesis of multidimensional filter banks and the design of compactly supported wavelets

asymptotically fast group operations on jacobians of general curves abramovich d 1996 int math res notices p1005 aho a v 1975 design anal computer becker t 1993 graduate texts math v141 brawley j 2004 density primitixe el burgisserr p 1997 fundamentals pricnip v315 cantor dg 1987 math comput v48 p95 doi 10 2307 2007876 chow wl 1954 am j math v76 p453 doi 10 2307 2372585 coppersmith d 1990 j symb comput v9 p251 doi 10 1016 s0747 7171 08 80013 2 decker wolfram 1999 algorithmic algebra p187 eberly w 2000 j symb comput v29 p441 doi 10 1006 jsco 1999 0308 greg w 2002 adv math v172 p169 griffiths p 1994 principles algebraic hartshorne robin 1977 algebraic geometry hess f 1999 thesis technische u hess f 2002 j symb comput v33 p425 doi 10 1006 jsco 2001 0513 huang md 1994 j symb comput v18 p519 doi 10 1006 jsco 1994 1063 kemper g 2002 j symb comput v34 p229 doi 10 1006 jsco 2002 0560 kemper g 2004 math comput v73 p245 khurimakdisi k 2004 asymptotically fast lazarsfeld r 1989 sampling vector bund p500 pari group bord pari gp stein w modular forms databa volcheck e 1994 lect notes computer v877 p221

analysis synthesis techniques for subband image coding analysis synthesis systems designed for low bit rate image coding their impact on overall system quality and their computational complexity are discussed the investigation focuses on the design of analysis synthesis systems for image coding and the perceptual impact of these systems at low bit rates two objectives are emphasized in developing these systems confining the total size of the subband images to be equal to the original image size and designing the filters so that perceptual distortion is not introduced by the analysis synthesis system methods based on circular convolution and symmetric extensions are developed and discussed in detail the theory design and implementation of both recursive and nonrecursive filtering systems are discussed methods are introduced which display advantages over conventional quadrature mirror filter based approaches

a rapid method of evaluating the regulator and class number of a pure cubic field let 2 0 be the algebraic number field formed by adjoining 9 to the rationals 2 let r and h be respectively the regulator and class number of shanks has described a method of evaluating r for 2 vd where d is a positive integer his technique improved the speed of the usual continued fraction algorithm for finding r by allowing one to proceed almost directly from the nth to the mth step where m is approximately in in the continued fraction expansion of id this paper shows how shanks idea can be extended to the voronoi algorithm which is used to find r in cubic fields of negative discriminant it also discusses at length an algorithm for finding r and h for pure cubic fields 1 fd d an integer under a certain generalized riemann hypothesis the ideas developed here will provide a new method which will find r and h in 0 02 5 e operations when h is small this is an improvement over the 0 d h operations required by voronoi s algorithm to find r for example with d 200171999 it required only 5 minutes for an amdahl 470 vi computer to find that r 518594546 969083 and h 1 this same calculation would require about 8 days of computer time if it used only the standard voronoi algorithm a 0 and gcd w m2 m3 n2 n3 a 1 also if e m2n3 m3n2 then e a if u e we denote its conjugates by w and we also write the norm of as nia and define it as nico oco w from the simple fact that

computing discrete logarithms in high genus hyperelliptic jacobians in provably subexponential time we provide a subexponential algorithm for solving the discrete logarithm problem in jacobians of high genus hyperelliptic curves over finite fields its expected running time for instances with genus g and underlying finite field fq satisfying g log q for a positive constant is given by o e f 1 3 2 3 2 o 1 g log q log g log q the algorithm works over any finite field and its running time does not rely on any unproven assumptions

computing in the jacobian of a hyperelliptic curve in this paper we present algorithms suitable for computer use for computation in the jacobian of a hyperelliptic curve we present a reduction algorithm which is asymptotically faster than that of gauss when the genus g is very large

linear algebra algorithms for divisors on an algebraic curve we use an embedding of the symmetric dth power of any algebraic curve c of genus g into a grassmannian space to give algorithms for working with divisors on c using only linear algebra in vector spaces of dimension o g and matrices of size o g2 o g when the base field k is finite or if c has a rational point over k these give algorithms for working on the jacobian of c that require o g4 field operations arising from the gaussian elimination our point of view is strongly geometric and our representation of points on the jacobian is fairly simple to deal with in particular none of our algorithms involves arithmetic with polynomials we note that our algorithms have the same asymptotic complexity for general curves as the more algebraic algorithms in florian hess 1999 ph d thesis which works with function fields as extensions of k x however for special classes of curves hess algorithms are asymptotically more efficient than ours generalizing other known efficient algorithms for special classes of curves such as hyperelliptic curves cantor 1987 superelliptic curves galbraith paulus and smart 2002 and cab curves harasawa and suzuki 2000 in all those cases one can attain a complexity of o g2

a development of symmetric extension method for subband image coding describes the development of a technique for subband image coding called the symmetric extension method which utilizes the nature of a symmetrically extended image to achieve high quality coding there are however some restrictions on the usable subband coding system the development is done based on the property of a symmetrically extended signal in the analysis process to remove the restriction the behavior of the sequence after two processes filtering and decimation is examined and formulated the development extends the application area of the technique

computing discrete logarithms in the jacobian of high genus hyperelliptic curves over even characteristic finite fields we describe improved versions of index calculus algorithms for solving discrete logarithm problems in jacobians of high genus hyperelliptic curves de ned over even characteristic elds our rst improvement is to incorporate several ideas for the low genus case by gaudry and theriault including the large prime variant and using a smaller factor base into the large genus algorithm of enge and gaudry we extend the analysis in 24 to our new algorithm allowing us to predict accurately the number of random walk steps required to nd all relations and to select optimal degree bounds for the factor base our second improvement is the adaptation of sieving techniques from flassenberg and paulus and jacobson to our setting the new algorithms are applied to concrete problem instances arising from the weil descent attack methodology for solving the elliptic curve discrete logarithm problem demonstrating signi cant improvements in practice

approach to pairing inversions without solving miller inversion in the present paper we show that the pairing inversion problem of atei pairing can be solved under the assumption that we have a generic algorithm for solving exponentiation inversion problem with such an algorithm the inversion problem of atei pairing can be solved without solving the miller inversion thus the pairing inversion problem of atei pairing is reduced to the exponentiation inversion problem

cyclic lti systems in digital signal processing cyclic signal processing refers to situations where all the time indices are interpreted modulo some integer l in such cases the frequency domain is defined as a uniform discrete grid as in l point dft this offers more freedom in theoretical as well as design aspects while circular convolution has been the centerpiece of many algorithms in signal processing for decades such freedom especially from the viewpoint of linear system theory has not been studied in the past in this paper we introduce the fundamentals of cyclic multirate systems and filter banks presenting several important differences between the cyclic and noncyclic cases cyclic systems with allpass and paraunitary properties are studied the paraunitary interpolation problem is introduced and it is shown that the interpolation does not always succeed state space descriptions of cyclic lti systems are introduced and the notions of reachability and observability of state equations are revisited it is shown that unlike in traditional linear systems these two notions are not related to the system minimality in a simple way throughout the paper a number of open problems are pointed out from the perspective of the signal processor as well as the system theorist

a filter based bit allocation scheme for subband compression of hdtv the authors compare the subband compression capabilities of eight filter sets consisting of linear phase quadrature mirror filters qmfs perfect reconstruction filters and nonlinear phase wavelets at different bit rates using a filter based bit allocation procedure using dpcm and pcm in hdtv subband coding it is found that qmfs have an edge over the rest

generalized symmetric extension for size limited multirate filter banks a multirate mr filter bank is called size limited if the total number of output samples equals the number of input samples a method called symmetric extension improved performance in subband image compression systems compared to the earlier method of circular convolution however the symmetric extension method was developed only for two band uniform filter banks and required even length linear phase analysis filters the authors generalize the symmetric extension method to the m band possibly nonuniform filter banks where m spl ges 2 the length restriction on the analysis filters is relaxed

subband coding of images subband coding has become quite popular for the source encoding of speech this paper presents a simple yet efficient extension of this concept to the source coding of images we specify the constraints for a set of two dimensional quadrature mirror filters qmf s for a particular frequency domain partition and show that these constraints are satisfied by a separable combination of one dimensional qmf s bits are then optimally allocated among the subbands to minimize the mean squared error for dpcm coding of the subbands also an adaptive technique is developed to allocate the bits within each subband by means of a local variance mask optimum quantization is employed with quantizers matched to the laplacian distribution subband coded images are presented along with their signal to noise ratios snr s the snr performance of the subband coder is compared to that of the adaptive discrete cosine transform dct vector quantization and differential vector quantization for bit rates of 0 67 1 0 and 2 0 bits per pixel for 256 256 monochrome images the adaptive subband coder has the best snr performance

speeding the pollard and elliptic curve methods of factorization since 1974 several algorithms have been developed that attempt to factor a large number n by doing extensive computations module n and occasionally taking gcds with n these began with pollard s p 1 and monte carlo methods more recently williams published a p 1 method and lenstra discovered an elliptic curve method ecm we present ways to speed all of these one improvement uses two tables during the second phases of p 1 and ecm looking for a match polynomial preconditioning lets us search a fixed table of size n with n 2 o n multiplications a parametrization of elliptic curves lets step 1 of ecm compute the x coordinate of np from that of p in about 9 3 1og2 n multiplications for arbitrary p

the parallelized pollard kangaroo method in real quadratic function fields we show how to use the parallelized kangaroo method for computing invariants in real quadratic function fields specifically we show how to apply the kangaroo method to the infrastructure in these fields we also show how to speed up the computation by using heuristics on the distribution of the divisor class number and by using the relatively inexpensive baby steps in the real quadratic model of a hyperelliptic function field furthermore we provide examples for regulators and class numbers of hyperelliptic function fields of genus 3 that are larger than those ever reported before

modular equations for hyperelliptic curves we dene modular equations describing the torsion subgroups of the jacobian of a hyperelliptic curve over a nite base eld we prove factorization properties that extend the well known results used in atkin s improvement of schoof s genus 1 point counting algorithm

the arithmetic of certain cubic function fields in this paper we discuss the properties of curves of the form y3 f x over a given field k of characteristic different from 3 if f x satisfies certain properties then the jacobian of such a curve is isomorphic to the ideal class group of the maximal order in the corresponding function field we seek to make this connection concrete and then use it to develop an explicit arithmetic for the jacobian of such curves from a purely mathematical perspective this provides explicit and efficient techniques for performing arithmetic in certain ideal class groups which are of fundamental interest in algebraic number theory at the same time it provides another source of groups which are suitable for diffie hellman type protocols in cryptographic applications

on the modeling of dct and subband image data for compression image subband and discrete cosine transform coefficients are modeled for efficient quantization and noiseless coding quantizers and codes are selected based on laplacian fixed generalized gaussian and adaptive generalized gaussian models the quantizers and codes based on the adaptive generalized gaussian models are always superior in mean squared error distortion performance but generally by no more than 0 08 bit pixel compared with the much simpler laplacian model based quantizers and noiseless codes this provides strong motivation for the selection of pyramid codes for transform and subband image coding

faster computation of self pairings self pairings have found interesting applications in cryptographic schemes in this paper we present a novel method for constructing a self pairing on supersingular elliptic curves with even embedding degrees which we call the ateil pairing this new pairing improves the efficiency of the self pairing computation on supersingular curves over finite fields with large characteristic based on the t pairing we propose a generalization of the ateil pairing which we call the ateili pairing the optimal ateili pairing which has the shortest miller loop is faster than previously known self pairings on supersingular elliptic curves over finite fields with small characteristic we also present a new self pairing based on the weil pairing which is faster than the self pairing based on the tate pairing on ordinary elliptic curves with embedding degree one

a probabilistic factorization algorithm with quadratic forms of negative discriminant we propose a probabilistic algorithm for factorization of an integer n with run time expvlog n log log n v54 1 asymptotically our algorithm will be as fast as the wellknown factorization algorithm of morrison and brillhart the latter algorithm will fail in several cases and heuristic assumptions are needed for its run time analysis our new algorithm will be analyzed under the assumption of the extended riemann hypothesis and it will be of las vegas type on input n the new algorithm will factor n with probability 12 in case of prime n the algorithm will prove the primality of n with probability 2 introduction until the last decade the centuries old problem of factoring integers was mainly a problem for specialists worldwide interest in factoring integers increased dramatically in 1978 when rivest shamir and adleman 32 published their public key cryptosystem whose security relies on the fact that some large integers are hard to factor gauss 9 already discovered a close connection between the factorization of a natural number n and the theory of quadratic forms of discriminant 4n now quadratic forms are one of the most important tools for factoring integers examples for efficient factorization algorithms are among others the algorithms of morrison and brillhart 26 and lenstra and schnorr 22 the former works with the continued fraction expansion of fn which is closely related to the theory of quadratic forms of discriminant 4n see 20 while the latter works with quadratic forms of discriminant 4n at present the most efficient factorization algorithm is the quadratic sieve algorithm see 29 which can also be expressed in terms of quadratic forms for an overview of modem factorization algorithms we refer to the papers of guy 10 monier 25 and pomerance 29 a deeper understanding of the theory of quadratic forms is of great importance for the analysis of modem factorization algorithms in this paper we shall only deal with the theory of quadratic forms of negative discriminant which is considerably more simple than the theory of forms of positive discriminants using this theory we obtain a probabilistic factorization algorithm with run time exp vlog n log log n v4 in this paper we denote by log x the natural logarithm of x we have 5 4 1 118 received may 17 1985 revised june 12 1986 1980 mathematics subject classification primary 10a30 68c25 1987 american mathematical society 0025 5718 87 1 00 25 per page

trellis coded quantization of memoryless and gauss markov sources trellis coded quantization tcq is developed and applied to the encoding of memoryless and gauss markov sources the theoretical justification for the approach is alphabet constrained rate distortion theory which is a dual to the channel capacity argument that motivates trellis coded modulation tcm the authors adopt the notions of signal set expansion set partitioning and branch labeling of tcm but modify the techniques to account for the source distribution to design tcq coders of low complexity with excellent mean squared error mse performance for a memoryless uniform source tcq provides an mse within 0 21 db of the distortion rate bound at all positive integral rates the performance is superior to that promised by the coefficient of quantization for all of the best lattices known in dimensions 24 or less for a memoryless gaussian source the tcq performance at rates of 0 5 1 and 2 b sample is superior to all previous results the authors found in the literature the encoding complexity of tcq is very modest tcq is incorporated into a predictive coding structure for the encoding of gauss markov sources simulation results for first second and third order gauss markov sources are presented

real and imaginary quadratic representations of hyperelliptic function fields a hyperelliptic function field can be always be represented as a real quadratic extension of the rational function field if at least one of the rational prime divisors is rational over the field of constants then it also can be represented as an imaginary quadratic extension of the rational function field the arithmetic in the divisor class group can be realized in the second case by cantor s algorithm we show that in the first case one can compute in the divisor class group of the function field using reduced ideals and distances of ideals in the orders involved furthermore we show how the two representations are connected and compare the computational complexity

embeddings of altn and its perfect covers for 6 in exceptional complex lie groups abstract we classify alt n and its perfect covers for n 6 up to conjugacy as subgroups of the exceptional complex lie groups with some exceptions where we give lower bounds and discuss the possibilities of lie primitive subgroups

superintegrable systems and riemann roch theorem in algebraic geometry there is a reduction algorithm that transforms the unreduced divisor into a unique reduced divisor which existence is guaranteed by the riemann roch theorem we discuss application of this algorithm to construction of finite dimensional superintegrable systems with n degrees of freedom identifying coordinates of the reduced divisor with integrals of motion

aspects of pairing inversion in this paper we discuss some applications of the pairing inversion problem and outline some potential approaches for solving it our analysis of these approaches gives further evidence that pairing inversion is a difficult problem

the eta pairing revisited in this paper we simplify and extend the eta pairing originally discovered in the setting of supersingular curves by barreto to ordinary curves furthermore we show that by swapping the arguments of the eta pairing one obtains a very efficient algorithm resulting in a speed up of a factor of around six over the usual tate pairing in the case of curves that have large security parameters complex multiplication by an order of qopf radic 3 and when the trace of frobenius is chosen to be suitably small other more minor savings are obtained for more general curves

a double large prime variation for small genus hyperelliptic index calculus in this article we examine how the index calculus approach for computing discrete logarithms in small genus hyperelliptic curves can be improved by introducing a double large prime variation two algorithms are presented the first algorithm is a rather natural adaptation of the double large prime variation to the intended context on heuristic and experimental grounds it seems to perform quite well but lacks a complete and precise analysis our second algorithm is a considerably simplified variant which can be analyzed easily the resulting complexity improves on the fastest known algorithms computer experiments show that for hyperelliptic curves of genus three our first algorithm surpasses pollard s rho method even for rather small field sizes

size limited filter banks for subband image compression a size limited filter bank slfb is a maximally decimated filter bank operating on a finite length duration input signal resulting in subband components whose total number of independent samples is equal to the number of input samples a theoretical framework for the design analysis and implementation of such filter banks resulting from applying fir separable filter banks to finite length signals is presented the concept of maximizing theoretical coding gain tcg using optimal bit allocation is generalized for this special case using tcg the relative merits of various different slfbs are addressed