109 research outputs found
Minors solve the elliptic curve discrete logarithm problem
The elliptic curve discrete logarithm problem is of fundamental importance in
public-key cryptography. It is in use for a long time. Moreover, it is an
interesting challenge in computational mathematics. Its solution is supposed to
provide interesting research directions.
In this paper, we explore ways to solve the elliptic curve discrete logarithm
problem. Our results are mostly computational. However, it seems, the methods
that we develop and directions that we pursue can provide a potent attack on
this problem. This work follows our earlier work, where we tried to solve this
problem by finding a zero minor in a matrix over the same finite field on which
the elliptic curve is defined. This paper is self-contained
On the complexity of computing Gr\"obner bases for weighted homogeneous systems
Solving polynomial systems arising from applications is frequently made
easier by the structure of the systems. Weighted homogeneity (or
quasi-homogeneity) is one example of such a structure: given a system of
weights , -homogeneous polynomials are polynomials
which are homogeneous w.r.t the weighted degree
. Gr\"obner bases for weighted homogeneous systems can be
computed by adapting existing algorithms for homogeneous systems to the
weighted homogeneous case. We show that in this case, the complexity estimate
for Algorithm~\F5 \left(\binom{n+\dmax-1}{\dmax}^{\omega}\right) can be
divided by a factor . For zero-dimensional
systems, the complexity of Algorithm~\FGLM (where is the
number of solutions of the system) can be divided by the same factor
. Under genericity assumptions, for
zero-dimensional weighted homogeneous systems of -degree
, these complexity estimates are polynomial in the
weighted B\'ezout bound .
Furthermore, the maximum degree reached in a run of Algorithm \F5 is bounded by
the weighted Macaulay bound , and this bound is
sharp if we can order the weights so that . For overdetermined
semi-regular systems, estimates from the homogeneous case can be adapted to the
weighted case. We provide some experimental results based on systems arising
from a cryptography problem and from polynomial inversion problems. They show
that taking advantage of the weighted homogeneous structure yields substantial
speed-ups, and allows us to solve systems which were otherwise out of reach
Where Quantum Complexity Helps Classical Complexity
Scientists have demonstrated that quantum computing has presented novel
approaches to address computational challenges, each varying in complexity.
Adapting problem-solving strategies is crucial to harness the full potential of
quantum computing. Nonetheless, there are defined boundaries to the
capabilities of quantum computing. This paper concentrates on aggregating prior
research efforts dedicated to solving intricate classical computational
problems through quantum computing. The objective is to systematically compile
an exhaustive inventory of these solutions and categorize a collection of
demanding problems that await further exploration
A heuristic for boundedness of ranks of elliptic curves
We present a heuristic that suggests that ranks of elliptic curves over the
rationals are bounded. In fact, it suggests that there are only finitely many
elliptic curves of rank greater than 21. Our heuristic is based on modeling the
ranks and Shafarevich-Tate groups of elliptic curves simultaneously, and relies
on a theorem counting alternating integer matrices of specified rank. We also
discuss analogues for elliptic curves over other global fields.Comment: 41 pages, typos fixed in torsion table in section
Recommended from our members
Geometric and Algebraic Combinatorics
The 2015 Oberwolfach meeting âGeometric and Algebraic Combinatoricsâ was organized by Gil Kalai (Jerusalem), Isabella Novik (Seattle), Francisco Santos (Santander), and Volkmar Welker (Marburg). It covered a wide variety of aspects of Discrete Geometry, Algebraic Combinatorics with geometric flavor, and Topological Combinatorics. Some of the highlights of the conference included (1) counterexamples to the topological Tverberg conjecture, and (2) the latest results around the Heron-Rota-Welsh conjecture
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