Uphill inflation

Primordial black holes (PBH) may form from large cosmological perturbations, produced during inflation when the inflaton's velocity is sufficiently slowed down. This usually requires very flat regions in the inflationary potential. In this paper we investigate another possibility, namely that the inflaton climbs up its potential. When it turns back, its velocity crosses zero, which triggers a short phase of ``uphill inflation'' during which cosmological perturbations grow at a very fast rate. This naturally occurs in double-well potentials if the width of the well is close to the Planck scale. We include the effect of quantum diffusion in this scenario, which plays a crucial role, by means of the stochastic-$\delta N$ formalism. We find that ultra-light black holes are produced with very high abundances, which do not depend on the energy scale at which uphill inflation occurs, and which suffer from substantially less fine tuning than in alternative PBH-production models. They are such that PBHs later drive a phase of PBH domination.Comment: 25 pages plus appendices (total 33 pages), 12 figure

An algebraic approach to the Rank Support Learning problem

Rank-metric code-based cryptography relies on the hardness of decoding a random linear code in the rank metric. The Rank Support Learning problem (RSL) is a variant where an attacker has access to N decoding instances whose errors have the same support and wants to solve one of them. This problem is for instance used in the Durandal signature scheme. In this paper, we propose an algebraic attack on RSL which clearly outperforms the previous attacks to solve this problem. We build upon Bardet et al., Asiacrypt 2020, where similar techniques are used to solve MinRank and RD. However, our analysis is simpler and overall our attack relies on very elementary assumptions compared to standard Gr{\"o}bner bases attacks. In particular, our results show that key recovery attacks on Durandal are more efficient than was previously thought

An explicit EtaEta rewrite rule

Projet EURECAIn this report, we extend \la-calculi of explicit substitutions by an $Eta$ rule. We do this in the framework of \la\ups, a \la-calculus of explicit substitutions introduced in~\citeLescannePOPL94 and thoroughly studied in~\citeLescanneR94. The main feature of such a calculus is that the classical $\beta$-contraction is expressed by a first-order term rewriting system. Our main result is the explicitation of the $\eta$-contraction by means of an unconditional, \it generic $Eta$ rewrite rule and of an extension of the substitution calculus, $\upsilon$. %Similar results Previous definitions of $Eta$ are due to~\citeHardin92 and~\citeRiosTHESE in the framework of \la\sigma-calculi. The principal difference between \la\upsilon and \la\sigma-calculi concerns confluence and strong normalization\,: \la\upsilon is ground confluent and its simply typed version is terminating, whenever \la\sigma_\Lift\ -calculus is confluent on open terms but non terminating on typed terms. In~\citeHardin92 and~\citeRiosTHESE, $Eta$ rule is presented as an extension of $\eta$-contraction. Hardin and R\'\ios present their extension as a conditional rewrite rule and do not stick fully to the philosophy of explicit substitutions. In our approach, the $Eta$-rule is a first order rewrite rule which uses an explicit substitution calculus. For that, one needs to introduce a new constant $\bot$ that denotes an unspecified term. Its behaviour is described by a rewrite rule. This report shows, in the one hand, how the $Eta$-rule associated with \la\ups and the rule for $\bot$ provides a correct implementation of the $\eta$-reduction and studies other properties of the \la\ups\eta-calculus namely confluenceon ground terms and strong normalisation on typed terms. On the other hand, this explicit $Eta$ leads to $\eta'$ a very general alternative to the classical $\eta$. Indeed, we prove that $\eta'$ allows confluent contractions which are not captured by classical $\eta$

Le pouvoir impérial romain et les cultes isiaques à Rome (IIIe-IVe siècles)

Les cultes isiaques se répandent autour du bassin méditerranéen entre le IVe s. av. et la fin du IVe s. apr. J.-C., arrivant à Rome au Ier siècle av. notre ère et y disparaissant conjointement aux cultes païens traditionnels avec ou peu après le coup porté par Théodose. Leur diffusion romaine s’étale donc sur une grande partie de l’histoire de l’Empire d’Occident, et ils vont ainsi se retrouver face à l’homme désormais considéré comme primus inter pares, grâce à l’accumulation de ses pouvoirs. L’empereur est pontifex maximus, c’est-à-dire maitre des cultes publics et du droit religieux, et il est ainsi l’agent du pouvoir qui a le potentiel religieux et législatif pour avoir un discours varié sur les autres traditions religieuses. Il semble donc inévitable que la sphère cultuelle isiaque rencontre et interagisse avec la sphère religieuse traditionnelle romaine, et que certains empereurs interfèrent, par renforcement ou au contraire par opposition, avec les cultes isiaques arrivés peu de temps avant l’avènement du pouvoir impérial. Cette thèse se propose non seulement d’étudier la dialectique entre institué romain (la force d’inertie) et instituant isiaque (la force de changements) dans la direction d’une potentielle altérité incluse (soit le dialogue évolutif du Nous par rapport à l’Autre), mais notamment par les liens publics romains et évolutifs entre l’empereur et les divinités isiaques. Nous effectuons cette recherche grâce à quatre types de sources antiques : la littérature pour le point de vue de l’élite littéraire gréco-romaine ; les inscriptions isiaques pour une définition surtout populaire de l’identité évolutive de l’instituant isiaque ; les monnaies et les oeuvres monumentales pour le point de vue public (et parfois plus personnel) impérial. Nous concentrons notre étude à partir de la dynastie sévérienne, qui laisse supposer que les cultes romano-orientaux profitent de la nouvelle configuration impériale avec des empereurs originaires d’Afrique et d’Orient, dans un phénomène d’« impérialisation » isiaque intensifié par un engagement tant impérial que populaire. En outre, nous nuancerons les conséquences sur la diffusion isiaque des troubles qui surviennent dans la deuxième moitié du IIIe siècle. Enfin, le IVe siècle, avec la christianisation de l’Empire et donc un christianisme devenant institué, ouvre sur une analyse des débats entre les défenseurs actifs du paganisme et les auteurs chrétiens, et de là, vers des interrogations sur l’intervention de la sphère isiaque dans ce face-à-face.Isiac worship spread around the mediterranean basin between IVth c. BC and late IVth century AD, arriving to Rome at Ist century BC and disappearing there at the same time as traditional pagan worship, with or soon after Theodosius’policy. So, their Italian distribution spreads during a large part of the history of Western Empire, and thus, they’re going to meet the man henceforth considered as primus inter pares, thanks to the accumulation of his powers. The emperor is pontifex maximus that is master of the public worship and the religious right, thus he is the powerful agent who possesses the religious and legislative potential to have a varied discourse on other religious traditions. It seems inevitable that the sphere of isiac worship meets and interacts with Roman traditional religious sphere, and that some emperors interfere, by reinforcement or opposition, with the isiac cults arrived not enough time before the advent of the imperial power. This thesis studies non only the dialectic between Roman institué (inertial force) and isiac instituant (force of changes) in the direction of a potential included otherness (that to say the evolving dialogue of Us in relation to the Other), but especially by Roman publics links (and processing) between the isiac worship and the emperors. We do this research with four types of antique sources : literature for the views of the Greco-Roman literary elite ; the isiac inscriptions for (popular) definition of the identity of the isiac instituant ; the coins and the monumental works for the public (and sometimes more personal) imperial point of view. We focus our study from the Severan dynasty, what can let suppose that these romano-oriental worship take advantage of this new imperial configuration with emperors from Africa and the East, in a phenomenon of isiac " imperialization " intensified by both imperial and popular involvement. Moreover, we soften the consequences on isiac spread of disorders that arise in the second half of the IIIth century. Finally, the IVth century, with the christianization of the Empire and so christianity becoming institué, brings to an analysis of the debates between the active defenders of the paganism and the christian writers, and from there, on questions over the intervention of the isiac group in this face to face

A Note on Anemoi Gröbner Bases

This paper focuses on algebraic attacks on the $\mathsf{Anemoi}$ family of arithmetization-oriented permutations [Crypto \u2723]. We consider a slight variation of the naive modeling of the $\mathsf{CICO}$ problem associated to the primitive, for which we can very easily obtain a Gröbner basis and prove the degree of the associated ideal. For inputs in $\mathbb{F}_{q}^2$ when $q$ is an odd prime, we recover the same degree as conjectured for alternative polynomial systems used in other recent works [eprint/2024/250][eprint/2024/347]. Furthermore, our approach can be adapted to cases which have not been studied there, i.e., even characteristic fields and inputs in $\mathbb{F}_q^{2\ell}$ for $\ell > 1$