416 research outputs found
Discrete Logarithms in Generalized Jacobians
D\'ech\`ene has proposed generalized Jacobians as a source of groups for
public-key cryptosystems based on the hardness of the Discrete Logarithm
Problem (DLP). Her specific proposal gives rise to a group isomorphic to the
semidirect product of an elliptic curve and a multiplicative group of a finite
field. We explain why her proposal has no advantages over simply taking the
direct product of groups. We then argue that generalized Jacobians offer poorer
security and efficiency than standard Jacobians
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(N^{1/12}) 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 F_{p^2) and trace zero varieties over F_{p^3}
with near-prime orders up to 372 bits in size. For p = 2^{61}-1, the average
time to find a group with 244-bit near-prime order is under an hour on a PC.Comment: 22 pages, to appear in Mathematics of Computatio
Gradient Flows in Filtering and Fisher-Rao Geometry
Uncertainty propagation and filtering can be interpreted as gradient flows
with respect to suitable metrics in the infinite dimensional manifold of
probability density functions. Such a viewpoint has been put forth in recent
literature, and a systematic way to formulate and solve the same for linear
Gaussian systems has appeared in our previous work where the gradient flows
were realized via proximal operators with respect to Wasserstein metric arising
in optimal mass transport. In this paper, we derive the evolution equations as
proximal operators with respect to Fisher-Rao metric arising in information
geometry. We develop the linear Gaussian case in detail and show that a
template two step optimization procedure proposed earlier by the authors still
applies. Our objective is to provide new geometric interpretations of known
equations in filtering, and to clarify the implication of different choices of
metric
Galois invariant smoothness basis
This text answers a question raised by Joux and the second author about the
computation of discrete logarithms in the multiplicative group of finite
fields. Given a finite residue field \bK, one looks for a smoothness basis
for \bK^* that is left invariant by automorphisms of \bK. For a broad class
of finite fields, we manage to construct models that allow such a smoothness
basis. This work aims at accelerating discrete logarithm computations in such
fields. We treat the cases of codimension one (the linear sieve) and
codimension two (the function field sieve)
A Computational Introduction to Elliptic and Hyperelliptic Curve Cryptography
At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic curves
Lie Algebraic Unscented Kalman Filter for Pose Estimation
An unscented Kalman filter for matrix Lie groups is proposed where the time
propagation of the state is formulated on the Lie algebra. This is done with
the kinematic differential equation of the logarithm, where the inverse of the
right Jacobian is used. The sigma points can then be expressed as logarithms in
vector form, and time propagation of the sigma points and the computation of
the mean and the covariance can be done on the Lie algebra. The resulting
formulation is to a large extent based on logarithms in vector form, and is
therefore closer to the UKF for systems in . This gives an
elegant and well-structured formulation which provides additional insight into
the problem, and which is computationally efficient. The proposed method is in
particular formulated and investigated on the matrix Lie group . A
discussion on right and left Jacobians is included, and a novel closed form
solution for the inverse of the right Jacobian on is derived, which
gives a compact representation involving fewer matrix operations. The proposed
method is validated in simulations
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