Showing papers on "Divisor published in 2006"

Security analysis of the strong diffie-hellman problem

Abstract: Let g be an element of prime order p in an abelian group and $\alpha\in {{\mathbb Z}}_p$. We show that if g, gα, and $g^{\alpha^d}$ are given for a positive divisor d of p–1, we can compute the secret α in $O(\log p \cdot (\sqrt{p/d}+\sqrt d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. If $g^{\alpha^i}$ (i=0,1,2,..., d) are provided for a positive divisor d of p+1, α can be computed in $O(\log p \cdot (\sqrt{p/d}+d))$ group operations using $O(\max\{\sqrt{p/d},\sqrt d\})$ memory. This implies that the strong Diffie-Hellman problem and its related problems have computational complexity reduced by $O(\sqrt d)$ from that of the discrete logarithm problem for such primes. Further we apply this algorithm to the schemes based on the Diffie-Hellman problem on an abelian group of prime order p. As a result, we reduce the complexity of recovering the secret key from $O(\sqrt p)$ to $O(\sqrt{p/d})$ for Boldyreva's blind signature and the original ElGamal scheme when p–1 (resp. p+1) has a divisor d ≤p1/2 (resp. d ≤p1/3) and d signature or decryption queries are allowed.

Boundedness of pluricanonical maps of varieties of general type

Abstract: Using the techniques of [20] and [10], we prove that certain log forms may be lifted from a divisor to the ambient variety. As a consequence of this result, following [22], we show that: For any positive integer n there exists an integer r n such that if X is a smooth projective variety of general type and dimension n, then $\phi_{rK_X}\colon X\dasharrow\mathbb{P}(H^0(\mathcal{O}_{X}(rK_X)))$ is birational for all r≥r n .

Security analysis of the strong diffie-hellman problem

Abstract: Let g be an element of prime order p in an abelian group and a ∈ Zp. We show that if g,gα, and g αd are given for a positive divisor d of p - 1, we can compute the secret a in O(log p. (√p/d + √d)) group operations using O(max{√p/d, √d}) memory. If g αi (i = 0,1, 2,..., d) are provided for a positive divisor d of p + 1, a can be computed in O(log p . (√p/d + d)) group operations using O(max{√p/d, √d}) memory. This implies that the strong Diffie-Hellman problem and its related problems have computational complexity reduced by O(√2) from that of the discrete logarithm problem for such primes. Further we apply this algorithm to the schemes based on the Diffie-Hellman problem on an abelian group of prime order p. As a result, we reduce the complexity of recovering the secret key from O(√p) to O( √p/d) for Boldyreva's blind signature and the original ElGamal scheme when p - 1 (resp. p + 1) has a divisor d < p 1/2 (resp. d < p 1/3 ) and d signature or decryption queries are allowed.

Koszul divisors on moduli spaces of curves

Abstract: Given a moduli problem with a nice coarse moduli space, which is the most intrinsic and natural divisor on this moduli space? We describe a general method of constructing effective divisors on a large class of moduli spaces using the syzygies of the parametrized objects. In this paper we treat the case of the moduli space of curves and we compute the slope of a doubly infinite sequence of Koszul divisors on M_g. We present a single formula which apart from giving myriads of new counterexamples to the Harris-Morrison Slope Conjecture it also encodes virtually all interesting divisor class calculations on M_g. In particular our formula specializes to earlier results due to Harris-Mumford (the case of the Brill-Noether divisors), Eisenbud-Harris (Giseker-Petri divisors), Khosla as well as to our earlier counterexamples to the Slope Conjecture obtained using K3 surfaces and Hurwitz schemes. We also describe five ways of constructing Koszul divisors on moduli spaces of pointed curves. As an application, we improve most of Logan's results on the Kodaira type of M_{g,n}.

Multidimensional continued fractions, dynamical renormalization and KAM theory

Abstract: The disadvantage of ‘traditional’ multidimensional continued fraction algorithms is that it is not known whether they provide simultaneous rational approximations for generic vectors Following ideas of Dani, Lagarias and Kleinbock-Margulis we describe a simple algorithm based on the dynamics of flows on the homogeneous space $$SL(d, \mathbb{Z}) \backslash SL(d, \mathbb{R})$$ (the space of lattices of covolume one) that indeed yields best possible approximations to any irrational vector The algorithm is ideally suited for a number of dynamical applications that involve small divisor problems As an example, we explicitly construct a renormalization scheme for the linearization of vector fields on tori of arbitrary dimension

On surfaces of class VII + 0 with numerically anticanonical divisor

Abstract: We consider minimal compact complex surfaces S with Betti numbers b\ = 1 and n = bi > 0. A theorem of Donaldson gives n exceptional line bundles. We prove that if in a deformation, these line bundles have sections, S is a degeneration of blown-up Hopf surfaces. Besides, if there exists an integer m > 1 and a flat line bundle F such that H®(S,-mK 0; these surfaces admit no nonconstant mero-morphic functions. The major problem in classification of non-kahlerian surfaces is to achieve the classification of surfaces S of class VIIJ. All known surfaces of this class contain Global Spherical Shells (GSS), i.e., admit a biholomorphic map ip\ U-> V from a neighbourhood U C C2\ {0} of the sphere 53 = dB2 onto an open set V such that I = (f(S3) does not disconnect S. Are there other surfaces ? In first section we investigate the general situation: A theorem of Donaldson [13] gives a Z-base (£,) of //2(S,Z), such that £/£, =- B of S these line bundles form families £/. We propose the following conjecture which can be easily checked for surfaces with GSS:

On Q-conic bundles

Abstract: A $\mathbb Q$-conic bundle is a proper morphism from a threefold with only terminal singularities to a normal surface such that fibers are connected and the anti-canonical divisor is relatively ample. We study the structure of $\mathbb Q$-conic bundles near their singular fibers. One corollary to our main results is that the base surface of every $\mathbb Q$-conic bundle has only Du Val singularities of type A (a positive solution of a conjecture by Iskovskikh). We obtain the complete classification of $\mathbb Q$-conic bundles under the additional assumption that the singular fiber is irreducible and the base surface is singular.

Algorithmic Aspects of Divisor-Based Biproportional Rounding

Abstract: Biproportional rounding of a matrix is the problem of assigning values to the elements of a matrix that are proportional to a given input matrix. The assignment should be integral and fulfill a set of rowand column-sum requirements. In a divisor-based method the problem is solved by computing appropriate rowand column-divisors, and by rounding the quotients. The only known divisor-based method that provably solves the problem is the tie-and-transfer algorithm by Balinski, Demange and Rachev. We analyze the complexity of this algorithm, and show that it is pseudo-polynomial. Two different approaches for reducing the complexity to (weakly) polynomial are presented. Finally, we give efficient algorithms for identifying ties, quotient intervals and divisor intervals.

Multi-bit programmable frequency divider

Abstract: A multi-bit, programmable, modular digital frequency divider divides an input frequency by an m-bit integer divisor to produce an output frequency. The integer divisor re-initializes m-number of flip-flop stages with the divisor input at the end of every output clock. Each divisor bit is gated to a D-input through a respective data multiplexer controlled by a clock output. A run/initialize mode controller receives the input frequency and produces the divided output frequency and controls the timing of the re-initialization.

Weighted Divisor Sums and Bessel Function Series

Abstract: On page 335 in his lost notebook, Ramanujan records without proof an identity involving a finite trigonometric sum and a doubly infinite series of ordinary Bessel functions. We provide the first published proof of this result. The identity yields as corollaries representations of weighted divisor sums, in particular, the summatory function for r 2(n), the number of representations of the positive integer n as a sum of two squares.

Stability of tri-canonical curves of genus two

Abstract: We completely classify tri-canonically embedded curves of genus two that are Chow semistable, and identify the moduli space of them with the compact moduli space of binary sextics. This moduli space is the log canonical model for the pair \(\big(\overline{M}_2,\alpha\Delta_0+\frac{1+\alpha}{2}\Delta_1+\frac{1}{2}\Xi\big)\) for 7/10 \(< \alpha \le\) 9/11 whose log canonical divisor pulls back to \(K_{\overline{M}_2}+\alpha\delta\) on the moduli stack

Families of higher dimensional germs with bijective Nash map

Abstract: Let (X,0) be a germ of complex analytic normal variety, non-singular outside 0. An essential divisor over (X,0) is a divisorial valuation of the field of meromorphic functions on (X,0), whose center on any resolution of the germ is an irreducible component of the exceptional locus. The Nash map associates to each irreducible component of the space of arcs through 0 on X the unique essential divisor intersected by the strict transform of the generic arc in the component. Nash proved its injectivity and asked if it was bijective. We prove that this is the case if there exists a divisorial resolution \pi of (X,0) such that its reduced exceptional divisor carries sufficiently many \pi-ample divisors (in a sense we define). Then we apply this criterion to construct an infinite number of families of 3-dimensional examples, which are not analytically isomorphic to germs of toric 3-folds (the only class of normal 3-fold germs with bijective Nash map known before).

On certain arithmetic functions involving exponential divisors

Abstract: The integer $d$ is called an exponential divisor of $n=\prod_{i=1}^r p_i^{a_i}>1$ if $d=\prod_{i=1}^r p_i^{c_i}$, where $c_i \mid a_i$ for every $1\le i \le r$. The integers $n=\prod_{i=1}^r p_i^{a_i}, m=\prod_{i=1}^r p_i^{b_i}>1$ having the same prime factors are called exponentially coprime if $(a_i,b_i)=1$ for every $1\le i\le r$. In the paper we investigate asymptotic properties of certain arithmetic functions involving exponential divisors and exponentially coprime integers.

On the mean square of the zeta-function and the divisor problem

Abstract: Let $\Delta(x)$ denote the error term in the Dirichlet divisor problem, and $E(T)$ the error term in the asymptotic formula for the mean square of $|\zeta(1/2+it)|$. If $E^*(t) = E(t) - 2\pi\Delta^*(t/2\pi)$ with $\Delta^*(x) = -\Delta(x) + 2\Delta(2x) - {1\over2}\Delta(4x)$, then we obtain the asymptotic formula $$ \int_0^T (E^*(t))^2 {\rm d} t = T^{4/3}P_3(\log T) + O_\epsilon(T^{7/6+\epsilon}), $$ where $P_3$ is a polynomial of degree three in $\log T$ with positive leading coefficient. The exponent 7/6 in the error term is the limit of the method.

Divisors on the space of maps to Grassmannians

Abstract: In this note we study the divisor theory of the Kontsevich moduli spacesM0,0(G(k, n), d) of genus-zero stable maps to the Grassmannians. We calculate the classes of several geometrically significant divisors. We prove that the cone of effective divisors stabilizes as n increases and we determine the stable effective cone. We also characterize the ample cone.

Fourier–Mukai transforms and canonical divisors

Abstract: Let $X$ be a smooth projective variety. We study a relationship between the derived category of $X$ and that of a canonical divisor. As an application, we study Fourier–Mukai transforms when $\kappa (X)=\dim X -1$ .

A fansy divisor on M_{0,n}

Abstract: We study the relation between projective T-varieties and their affine cones in the language of the so-called divisorial fans and polyhedral divisors. As an application, we present the Grassmannian Grass(2,n) as a ``fansy divisor'' on the moduli space of stable, n-pointed, rational curves.

Fast bounds on the distribution of smooth numbers

Abstract: Let P(n) denote the largest prime divisor of n, and let Ψ(x,y) be the number of integers n≤x with P(n)≤y. In this paper we present improvements to Bernstein’s algorithm, which finds rigorous upper and lower bounds for Ψ(x,y). Bernstein’s original algorithm runs in time roughly linear in y. Our first, easy improvement runs in time roughly y2/3. Then, assuming the Riemann Hypothesis, we show how to drastically improve this. In particular, if logy is a fractional power of logx, which is true in applications to factoring and cryptography, then our new algorithm has a running time that is polynomial in logy, and gives bounds as tight as, and often tighter than, Bernstein’s algorithm.

The divisor problem for d 4 ( n ) in short intervals

Abstract: In this paper we establish an asymptotic formula for the sum $$ {\sum\limits_{x < n \leqq x + y} {d_{4} (n)} } $$ when y is large compared to x 1/2 log x.

On the $D$-dimension of a certain type of threefolds

Abstract: Let $Y$ be an algebraic manifold of dimension 3 with $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and $h^0(Y, {\mathcal{O}}_Y) > 1$. Let $X$ be a smooth completion of $Y$ such that the boundary $X-Y$ is the support of an effective divisor $D$ on $X$ with simple normal crossings. We prove that the $D$-dimension of $X$ cannot be 2, i.e., either any two nonconstant regular functions are algebraically dependent or there are three algebraically independent nonconstant regular functions on $Y$. Secondly, if the $D$-dimension of $X$ is greater than 1, then the associated scheme of $Y$ is isomorphic to Spec$\Gamma(Y, {\mathcal{O}}_Y)$. Furthermore, we prove that an algebraic manifold $Y$ of any dimension $d\geq 1$ is affine if and only if $H^i(Y, \Omega^j_Y)=0$ for all $j\geq 0$, $i>0$ and it is regularly separable, i.e., for any two distinct points $y_1$, $y_2$ on $Y$, there is a regular function $f$ on $Y$ such that $f(y_1) eq f(y_2)$.

The number of rational curves on K3 surfaces

Abstract: Let X be a K3 surface with a primitive ample divisor H, and let $\beta=2[H]\in H_2(X, \mathbf Z)$. We calculate the Gromov-Witten type invariants $n_{\beta}$ by virtue of Euler numbers of some moduli spaces of stable sheaves. Eventually, it verifies Yau-Zaslow formula in the non primitive class $\beta$.

Apparatus for multiple-divisor prescaler

Abstract: Disclosed is an apparatus for multiple-divisor prescaler, which includes an odd/even core divider, a divisor control logic unit, an odd number inserted mechanism, and an n-order divided-by-2 divider with changeable trigger edges. This invention uses a clock toggle mechanism to vary the trigger edges of each divided-by-2 divider in the n-order divider, and associates the odd/even core divider to realize the multiple-divisor prescaler apparatus. Thereby, it achieves the purpose of being divided by 30/31. In addition, it increases the divisor range up to 2 n−1 +2 and 2 n +1 through the use of the clock toggle mechanism.

Efficient computation of the modulo operation based on divisor (2n-1)

Abstract: A system and method for computing A mod (2 n −1), where A is an m bit quantity, where n is a positive integer, where m is greater than or equal to n. The quantity A may be partitioned into a plurality of sections, each being at most n bits long. The value A mod (2 n −1) may be computed by adding the sections in mod(2 n −1) fashion. This addition of the sections of A may be performed in a single clock cycle using an adder tree, or, sequentially in multiple clock cycles using a two-input adder circuit provided the output of the adder circuit is coupled to one of the two inputs. The computation A mod (2 n −1) may be performed as a part of an interleaving/deinterleaving operation, or, as part of an encryption/decryption operation.

On the Order of Magnitude of the Divisor Function

Abstract: Let \({\fancyscript D}\) be an increasing sequence of positive integers, and consider the divisor functions: $$ \begin{array}{*{20}c} {{d{\left( {n,{\fancyscript D}} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{d\left| n \right.}} \\ {{d \in {\fancyscript D},d \leqslant {\sqrt n }}} \\ \end{array} } {1,} }}} & {{d_{2} {\left( {n,{\fancyscript D}} \right)} = {\sum\limits_{\begin{array}{*{20}c} {{{\left[ {d,\delta } \right]}\left| n \right.}} \\ {{d,\delta \in {\fancyscript D},{\left[ {d,\delta } \right]} \leqslant {\sqrt n }}} \\ \end{array} } {1,} }}} \\ \end{array} $$ where [d, δ] = l.c.m.(d, δ). A probabilistic argument is introduced to evaluate the series \( {\sum olimits_{n = 1}^\infty {\alpha _{n} d{\left( {n,{\fancyscript D}} \right)}} } \) and \( {\sum olimits_{n = 1}^\infty {\alpha _{n} d_{2} {\left( {n,{\fancyscript D}} \right)}} } \).

Digital computation method involving euclidean division

Abstract: A computational method for implementation in an electronic digital processing system performs integer division upon very large (multi-word) operands. An approximated reciprocal of the divisor is obtained by extracting the two most significant words of the divisor, adding one to the extracted value and dividing from a power of two out to two significant words. Multiplying this reciprocal value by a remainder (initialized as the dividend) obtains a quotient value, which is then decremented by a random value. The randomized quotient is multiplied by the actual divisor, and decremented from the remainder. The quotient value is accumulated to obtain updated quotient values. This process is repeated over a fixed number of rounds related to the relative sizes in words of the dividend and divisor. Each round corrects approximation and randomization errors from a preceding round.

On the k-free divisor problem

Abstract: Let $\Delta^{(k)}(x)$ denote the error term of the $k$-free divisor problem for $k\geq 2$. In this paper we establish an asymptotic formula of the integral $\int_1^T|\Delta^{(k)}(x)|^2dx$ for each $k\geq 4.$

A note of perfect nonlinear functions

Abstract: Perfect nonlinear functions are of importance in cryptography. By using Galois rings and investigating the character values of corresponding relative difference sets, we construct a perfect nonlinear function from $\mathbb{Z}^{n}_{p_{2}}$ to $\mathbb{Z}^{m}_{p_{2}}$ where 2m is possibly larger than the largest divisor of n. Meanwhile we prove that there exists a perfect nonlinear function from $\mathbb{Z}^{2}_{2_{p}}$ to $\mathbb{Z}_{2_{p}}$ if and only if p=2, and that there doesn't exist a perfect nonlinear function from $\mathbb{Z}^{2n}_{2k_{l}}$ to $\mathbb{Z}^{m}_{2k_{l}}$ if m>n and l(l is odd) is self-conjugate modulo 2k(k≥1) .