35 research outputs found
Congruence schemes
A new category of algebro-geometric objects is defined. This construction is
a vast generalization of existing F1-theories, as it contains the the theory of
monoid schemes on the one hand and classical algebraic theory, e.g.
Grothendieck schemes, on the the other. It also gives a handy description of
Berkovich subdomains and thus contains Berkovich's approach to abstract
skeletons. Further it complements the theory of monoid schemes in view of
number theoretic applications as congruence schemes encode number theoretical
information as opposed to combinatorial data which are seen by monoid schemes
Analysis of Fourier transform valuation formulas and applications
The aim of this article is to provide a systematic analysis of the conditions
such that Fourier transform valuation formulas are valid in a general
framework; i.e. when the option has an arbitrary payoff function and depends on
the path of the asset price process. An interplay between the conditions on the
payoff function and the process arises naturally. We also extend these results
to the multi-dimensional case, and discuss the calculation of Greeks by Fourier
transform methods. As an application, we price options on the minimum of two
assets in L\'evy and stochastic volatility models.Comment: 26 pages, 3 figures, to appear in Appl. Math. Financ
On the Quantum Complexity of the Continuous Hidden Subgroup Problem
The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor's celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms.
The latest successful generalization (Eisentrager et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space Rm
, even for large dimensions m
. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices.
The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits.
This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Incidentally, our work clarifies certain claims from the extended abstract of Eisentrager et al
Decomposition of operator semigroups on W*-algebras
We consider semigroups of operators on a W-algebra and prove, under
appropriate assumptions, the existence of a Jacobs-DeLeeuw-Glicksberg type
decomposition. This decomposition splits the algebra into a "stable" and
"reversible" part with respect to the semigroup and yields, among others, a
structural approach to the Perron-Frobenius spectral theory for completely
positive operators on W-algebras.Comment: referee's comments incorporated. To appear in Semigroup Foru
On the Quantum Complexity of the Continuous Hidden Subgroup Problem
The Hidden Subgroup Problem (HSP) aims at capturing all problems that are susceptible to be solvable in quantum polynomial time following the blueprints of Shor’s celebrated algorithm. Successful solutions to this problems over various commutative groups allow to efficiently perform number-theoretic tasks such as factoring or finding discrete logarithms. The latest successful generalization (Eisenträger et al. STOC 2014) considers the problem of finding a full-rank lattice as the hidden subgroup of the continuous vector space Rm, even for large dimensions m. It unlocked new cryptanalytic algorithms (Biasse-Song SODA 2016, Cramer et al. EUROCRYPT 2016 and 2017), in particular to find mildly short vectors in ideal lattices. The cryptanalytic relevance of such a problem raises the question of a more refined and quantitative complexity analysis. In the light of the increasing physical difficulty of maintaining a large entanglement of qubits, the degree of concern may be different whether the above algorithm requires only linearly many qubits or a much larger polynomial amount of qubits. This is the question we start addressing with this work. We propose a detailed analysis of (a variation of) the aforementioned HSP algorithm, and conclude on its complexity as a function of all the relevant parameters. Our modular analysis is tailored to support the optimization of future specialization to cases of cryptanalytic interests. We suggest a few ideas in this direction