234 research outputs found
Counting points on hyperelliptic curves over finite fields
International audienceWe describe some algorithms for computing the cardinality of hyperelliptic curves and their Jacobians over finite fields. They include several methods for obtaining the result modulo small primes and prime powers, in particular an algorithm à la Schoof for genus 2 using Cantor's division polynomials. These are combined with a birthday paradox algorithm to calculate the cardinality. Our methods are practical and we give actual results computed using our current implementation. The Jacobian groups we handle are larger than those previously reported in the literature
Universal optimality of the and Leech lattices and interpolation formulas
We prove that the root lattice and the Leech lattice are universally
optimal among point configurations in Euclidean spaces of dimensions and
, respectively. In other words, they minimize energy for every potential
function that is a completely monotonic function of squared distance (for
example, inverse power laws or Gaussians), which is a strong form of robustness
not previously known for any configuration in more than one dimension. This
theorem implies their recently shown optimality as sphere packings, and broadly
generalizes it to allow for long-range interactions.
The proof uses sharp linear programming bounds for energy. To construct the
optimal auxiliary functions used to attain these bounds, we prove a new
interpolation theorem, which is of independent interest. It reconstructs a
radial Schwartz function from the values and radial derivatives of and
its Fourier transform at the radii for integers
in and in . To prove this
theorem, we construct an interpolation basis using integral transforms of
quasimodular forms, generalizing Viazovska's work on sphere packing and placing
it in the context of a more conceptual theory.Comment: 95 pages, 6 figure
Modular polynomials via isogeny volcanoes
We present a new algorithm to compute the classical modular polynomial Phi_n
in the rings Z[X,Y] and (Z/mZ)[X,Y], for a prime n and any positive integer m.
Our approach uses the graph of n-isogenies to efficiently compute Phi_n mod p
for many primes p of a suitable form, and then applies the Chinese Remainder
Theorem (CRT). Under the Generalized Riemann Hypothesis (GRH), we achieve an
expected running time of O(n^3 (log n)^3 log log n), and compute Phi_n mod m
using O(n^2 (log n)^2 + n^2 log m) space. We have used the new algorithm to
compute Phi_n with n over 5000, and Phi_n mod m with n over 20000. We also
consider several modular functions g for which Phi_n^g is smaller than Phi_n,
allowing us to handle n over 60000.Comment: corrected a typo in equation (14), 31 page
The Interpolating Random Spline Cryptosystem and the Chaotic-Map Public-Key Cryptosystem
The feasibility of implementing the interpolating cubic spline function as encryption and decryption transformations is presented. The encryption method can be viewed as computing a transposed polynomial. The main characteristic of the spline cryptosystem is that the domain and range of encryption are defined over real numbers, instead of the traditional integer numbers. Moreover, the spline cryptosystem can be implemented in terms of inexpensive multiplications and additions.
Using spline functions, a series of discontiguous spline segments can execute the modular arithmetic of the RSA system. The similarity of the RSA and spline functions within the integer domain is demonstrated. Furthermore, we observe that such a reformulation of RSA cryptosystem can be characterized as polynomials with random offsets between ciphertext values and plaintext values. This contrasts with the spline cryptosystems, so that a random spline system has been developed. The random spline cryptosystem is an advanced structure of spline cryptosystem. Its mathematical indeterminacy on computing keys with interpolants no more than 4 and numerical sensitivity to the random offset t( increases its utility.
This article also presents a chaotic public-key cryptosystem employing a one-dimensional difference equation as well as a quadratic difference equation. This system makes use of the El Gamal’s scheme to accomplish the encryption process. We note that breaking this system requires the identical work factor that is needed in solving discrete logarithm with the same size of moduli
Complex multiplication of abelian surfaces
The theory of complex multiplication makes it possible to construct certain class fields and abelian varieties. The main theme of this thesis is making these constructions explicit for the case where the abelian varieties have dimension 2. Chapter I is an introduction to complex multiplication, and shows that a general result of Shimura can be improved for degree-4 CM-fields. Chapter II gives an algorithm for computing class polynomials for quartic CM-fields, based on an algorithm of Spallek. We make the algorithm more explicit, and use Goren and Lauter___s recent bounds on the denominators of the coefficients, which yields the first running time bound and proof of correctness of an algorithm computing these polynomials. Chapter III studies and computes the irreducible components of the modular variety of abelian surfaces with CM by a given primitive quartic CM-field. We adapt the algorithm of Chapter II to compute these components. Chapters IV and V construct certain `Weil numbers'. They have properties that are number theoretic in nature and are motivated by cryptography. Chapter IV is joint work with David Freeman and Peter Stevenhagen. Chapter V is joint work with Laura Hitt O'Connor, Gary McGuire, and Michael Naehrig.UBL - phd migration 201
The full Quantum Spectral Curve for
The spectrum of planar N=6 superconformal Chern-Simons theory, dual to type
IIA superstring theory on , is accessible at finite coupling
using integrability. Starting from the results of [arXiv:1403.1859], we study
in depth the basic integrability structure underlying the spectral problem, the
Quantum Spectral Curve. The new results presented in this paper open the way to
the quantitative study of the spectrum for arbitrary operators at finite
coupling. Besides, we show that the Quantum Spectral Curve is embedded into a
novel kind of Q-system, which reflects the OSp(4|6) symmetry of the theory and
leads to exact Bethe Ansatz equations. The discovery of this algebraic
structure, more intricate than the one appearing in the case,
could be a first step towards the extension of the method to .Comment: 43 + 27 pages, 7 figures. v4: text improved, more details and App D
included. This is the same as the published version JHEP09(2017)140, with
small typos corrected in App
A boundary integral algorithm for the Laplace Dirichlet-Neumann mixed eigenvalue problem
We present a novel integral-equation algorithm for evaluation of Zaremba
eigenvalues and eigenfunctions}, that is, eigenvalues and eigenfunctions of the
Laplace operator with mixed Dirichlet-Neumann boundary conditions; of course,
(slight modifications of) our algorithms are also applicable to the pure
Dirichlet and Neumann eigenproblems. Expressing the eigenfunctions by means of
an ansatz based on the single layer boundary operator, the Zaremba eigenproblem
is transformed into a nonlinear equation for the eigenvalue . For smooth
domains the singular structure at Dirichlet-Neumann junctions is incorporated
as part of our corresponding numerical algorithm---which otherwise relies on
use of the cosine change of variables, trigonometric polynomials and, to avoid
the Gibbs phenomenon that would arise from the solution singularities, the
Fourier Continuation method (FC). The resulting numerical algorithm converges
with high order accuracy without recourse to use of meshes finer than those
resulting from the cosine transformation. For non-smooth (Lipschitz) domains,
in turn, an alternative algorithm is presented which achieves high-order
accuracy on the basis of graded meshes. In either case, smooth or Lipschitz
boundary, eigenvalues are evaluated by searching for zero minimal singular
values of a suitably stabilized discrete version of the single layer operator
mentioned above. (The stabilization technique is used to enable robust
non-local zero searches.) The resulting methods, which are fast and highly
accurate for high- and low-frequencies alike, can solve extremely challenging
two-dimensional Dirichlet, Neumann and Zaremba eigenproblems with high
accuracies in short computing times---enabling, in particular, evaluation of
thousands of eigenvalues and corresponding eigenfunctions for a given smooth or
non-smooth geometry with nearly full double-precision accurac
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