An algebraic property of Hecke operators and two indefinite theta series

We prove an algebraic property of the elements defining Hecke operators on period polynomials associated with modular forms, which implies that the pairing on period polynomials corresponding to the Petersson scalar product of modular forms is Hecke equivariant. As a consequence of this proof, we derive two indefinite theta series identities which can be seen as analogues of Jacobi's formula for the theta series associated with the sum of four squares.Comment: 11 pages. Published version. Forum Math., published online February 201

Elastic modulus of a colloidal suspension of rigid spheres at rest

International audienceBy modeling a colloidal suspension at rest as a solid, a new expression for the linear elastic modulus is obtained. This estimate is valid for a yield stress colloidal suspension submitted to a small strain. Interestingly, it is also possible to construct an hypothesis allowing one to recover the high-frequency modulus classically found by means of a classical ‘fluid approach'. However, in most of the situations, the moduli obtained by the two approaches are different

Bounded Fully Homomorphic Encryption from Monoid Algebras

We present a new method that produces bounded FHE schemes (see Definition 3), starting with encryption schemes that support one algebraic operation. We use this technique to construct examples of encryption schemes that, theoretically can handle any algebraic function on encrypted data

Ring Homomorphic Encryption Schemes

We analyze the structure of commutative ring homomorphic encryption schemes and show that they are not quantum IND-CCA secure

Computing Primitive Idempotents in Finite Commutative Rings and Applications

In this paper, we compute an algebraic decomposition of blackbox rings in the generic ring model. More precisely, we explicitly decompose a black-box ring as a direct product of a nilpotent black-box ring and local Artinian black-box rings, by computing all its primitive idempotents. The algorithm presented in this paper uses quantum subroutines for the computation of the p-power parts of a black-box ring and then classical algorithms for the computation of the corresponding primitive idempotents. As a by-product, we get that the reduction of a black-box ring is also a black-box ring. The first application of this decomposition is an extension of the work of Maurer and Raub [26] on representation problem in black-box finite fields to the case of reduced p-power black-box rings. Another important application is an IND-CCA1 attack for any ring homomorphic encryption scheme in the generic ring model. Moreover, when the plaintext space is a nite reduced black-box ring, we present a plaintext-recovery attack based on representation problem in black-box prime fields. In particular, if the ciphertext space has smooth characteristic, the plaintext-recovery attack is effectively computable in the generic ring model

Ecoulement de fluide visqueux autour d'une sphère proche d'une paroi avec condition de glissement.

On considère une particule solide et sphérique dans un écoulement de fluide visqueux à petit nombre de Reynolds parallèle à une paroi plane sur laquelle s'applique une condition de glissement. Par linéarité des équations de Stokes, la solution générale pour une sphère en translation et rotation dans un écoulement de cisaillement s'obtient par superposition de trois problèmes : sphère au repos dans un écoulement de cisaillement, sphère en rotation ou translation dans un fluide au repos. Les solutions de ces trois problèmes sont obtenues par la méthode des coordonnées bisphériques. Les résultats pour la force et le couple sont calculés avec une précision de 10^ (-7), même pour de très petites distances particule-paroi de l'ordre de 10^(-3). Les vitesses de translation et de rotation d'une sphère libre dans un écoulement de cisaillement sont alors obtenues avec une précision de 10^(-7)