Keywords: Acetylcholine (ACh), Acetylcholinesterase (AChE), Alzheimer

Introduction

Dementia is a debilitating condition frequent in ageing populations, and Alzheimer's Disease (AD) accounts for 70% of all dementia cases. AD is characterized by neuropathological hallmarks consisting of an accumulation of Amyloid ß peptide (Aß) in extracellular plaques, intracellular deposits of tau protein, neuronal loss and, more recently, a prominent synaptic loss was identified [START_REF] Braak | Neuropathological stageing of Alzheimer-related changes[END_REF][START_REF] Masliah | Altered expression of synaptic proteins occurs early during progression of Alzheimer's disease[END_REF][START_REF] Selkoe | The molecular pathology of Alzheimer's disease[END_REF][START_REF] Spires-Jones | The Intersection of Amyloid Beta and Tau at Synapses in Alzheimer's Disease[END_REF]. In addition, anatomical studies in AD patients showed a massive loss of brain white matter and a specific reduction of cholinergic neurons of the basal forebrain [START_REF] Auld | Alzheimer's disease and the basal forebrain cholinergic system: relations to β-amyloid peptides, cognition, and treatment strategies[END_REF][START_REF] Bowen | Neurotransmitter-related enzymes and indices of hypoxia in senile dementia and other abiotrophies[END_REF][START_REF] Coyle | Alzheimer's disease: a disorder of cortical cholinergic innervation[END_REF][START_REF] Kim | Differential Cholinergic Pathway Involvement in Alzheimer's Disease and Subcortical Ischemic Vascular Dementia[END_REF][START_REF] Whitehouse | Alzheimer disease: Evidence for selective loss of cholinergic neurons in the nucleus basalis[END_REF][START_REF] Whitehouse | Alzheimer's disease and senile dementia: loss of neurons in the basal forebrain[END_REF]. Cholinergic neurons are organized in dense nuclei with widespread projections that entirely cover the central nervous system. In particular, the cholinergic neurons, whose cell bodies are in the basal forebrain, send their long projections to the neocortex and hippocampus [START_REF] Bigl | Cholinergic projections from the basal forebrain to frontal, parietal, temporal, occipital, and cingulate cortices: A combined fluorescent tracer and acetylcholinesterase analysis[END_REF][START_REF] Mesulam | Cholinergic innervation of cortex by the basal forebrain: cytochemistry and cortical connections of the septal area, diagonal band nuclei, nucleus basalis (substantia innominata), and hypothalamus in the rhesus monkey[END_REF]. Several studies demonstrated the pivotal role of these cholinergic nuclei in cognitive functions. [START_REF] Woolf | A structural basis for memory storage in mammals[END_REF] proposed a model in which acetylcholine (ACh) release leads to the modulation of cortical circuitry that finally encodes for storage of long-term memory. The cholinergic system is also involved in attention processes [START_REF] Muir | Excitotoxic lesions of basal forebrain cholinergic neurons: effects on learning, memory and attention[END_REF][START_REF] Sarter | Cognitive functions of cortical acetylcholine: toward a unifying hypothesis[END_REF][START_REF] Wenk | The Nucleus Basalis Magnocellularis Cholinergic System: One Hundred Years of Progress[END_REF]. In a mouse model, the lack of ACh receptors in the prefrontal cortex (PFC) was demonstrated to be responsible for attention deficit, restored by the expression of the receptor in this area [START_REF] Guillem | Nicotinic acetylcholine receptor β2 subunits in the medial prefrontal cortex control attention[END_REF].

The neurotransmitter ACh binds to two families of receptors, nicotinic acetylcholine receptors (nAChRs) and muscarinic acetylcholine receptors (mAChRs). Both families of receptors regulate the cognitive processes mentioned above [START_REF] Ghoneim | Studies on human memory: the interactions of diazepam, scopolamine, and physostigmine[END_REF][START_REF] Petersen | Scopolamine induced learning failures in man[END_REF][START_REF] Sarter | Deficits in Attentional Control: Cholinergic Mechanisms and Circuitry-Based Treatment Approaches[END_REF], and are both affected in AD.

Binding studies performed with the use of [3H]-nicotine and [3H]-ACh showed a significant reduction in nicotine and ACh binding sites in cerebral cortex of patients suffering from AD, demonstrating a decrease of both nAChR and mAChR populations (Gotti et al., 2006a;[START_REF] Paterson | Neuronal nicotinic receptors in the human brain[END_REF][START_REF] Perry | Neurochemical activities in human temporal lobe related to aging and Alzheimer-type changes[END_REF][START_REF] Perry | Cholinergic correlates of cognitive impairment in Parkinson's disease: comparisons with Alzheimer's disease[END_REF][START_REF] Perry | Nicotinic receptor abnormalities in Alzheimer's and Parkinson's diseases[END_REF][START_REF] Perry | Amyloid precursor protein in senile plaques of Alzheimer disease[END_REF][START_REF] Shimohama | Changes in Nicotinic and Muscarinic Cholinergic Receptors in Alzheimer-Type Dementia[END_REF][START_REF] Whitehouse | Alzheimer disease: Evidence for selective loss of cholinergic neurons in the nucleus basalis[END_REF][START_REF] Whitehouse | Alzheimer's disease and senile dementia: loss of neurons in the basal forebrain[END_REF][START_REF] Whitehouse | Nicotinic acetylcholine binding sites in Alzheimer's disease[END_REF]. In addition to nAChRs and mAChRs, the enzyme choline acetyltransferase (ChAT), involved in ACh production, is also affected in AD. The activity of this ChAT enzyme, and consequently the synthesis of ACh, is decreased in AD brains. In addition, several authors observed a reduction in the activity of acetylcholinesterase (AChE), the enzyme that metabolises ACh after its release in the synaptic cleft [START_REF] Auld | Alzheimer's disease and the basal forebrain cholinergic system: relations to β-amyloid peptides, cognition, and treatment strategies[END_REF][START_REF] Bowen | Neurotransmitter-related enzymes and indices of hypoxia in senile dementia and other abiotrophies[END_REF][START_REF] Coyle | Alzheimer's disease: a disorder of cortical cholinergic innervation[END_REF][START_REF] Davies | Selective loss of central cholinergic neurons in Alzheimer's disease[END_REF][START_REF] Perry | Correlation of cholinergic abnormalities with senile plaques and mental test scores in senile dementia[END_REF].

The role of the cholinergic system in cognition and the modification observed in neurodegenerative diseases, and in particular in the case of AD, led to the formulation of the "cholinergic hypothesis" of geriatric disorders [START_REF] Bartus | The cholinergic hypothesis of geriatric memory dysfunction[END_REF][START_REF] Contestabile | The history of the cholinergic hypothesis[END_REF], according to which the reduction in cholinergic innervation is responsible for the cognitive decline observed in AD patients.

In this context, we will focus on nAChRs, since their involvement in AD has been largely demonstrated, while the contribution of mAChRs has been under-explored. The purpose of this review is to present evidence implicating the role of nAChRs in AD, discuss the data supporting their interaction with Aß, and the consequences of the perturbation of this interaction in murine models.

Brief overview of nAChR subtypes involved in AD

Nicotinic receptors are transmembrane pentameric proteins that belong to the "cys-loop" superfamily of ligand-gated ion channels together with GABAA, GABAC, glycine and 5hydroxytryptamine (5-HT3) ionotropic receptors [START_REF] Changeux | Allosteric Receptors after 30 Years[END_REF][START_REF] Le Novère | Molecular evolution of the nicotinic acetylcholine receptor: An example of multigene family in excitable cells[END_REF]. They are composed of a variety of α and β subunits, which determine the pharmacological and kinetic properties of the receptor [START_REF] Albuquerque | Mammalian Nicotinic Acetylcholine Receptors: From Structure to Function[END_REF][START_REF] Giniatullin | Desensitization of nicotinic ACh receptors: shaping cholinergic signaling[END_REF]. The five subunits that compose the receptor are assembled around a central hydrophilic pore that mediates the flow of the cations K + , Na + and Ca ++ . In the human nervous system, there are eight α subunits (α2-α7, α9, α10) and three ß subunits (ß2-ß4) that assemble in different combinations to generate a variety of nAChR subtypes with distinct electrophysiological properties and brain localization [START_REF] Albuquerque | Mammalian Nicotinic Acetylcholine Receptors: From Structure to Function[END_REF]Gotti et al., 2006b[START_REF] Gotti | Heterogeneity and complexity of native brain nicotinic receptors[END_REF][START_REF] Gotti | Structural and functional diversity of native brain neuronal nicotinic receptors[END_REF].

The use of radioactive ligands allowed nAChR classification into two distinct groups, αBungarotoxin sensitive and αBungarotoxin insensitive receptors [START_REF] Gotti | Neuronal nicotinic receptors: from structure to pathology[END_REF]. Homopentameric α7 nAChRs belong to the first class, while heteropentameric nAChRs containing the ß2 subunit belong to the second class. In this context we will focus on these α7 and ß2 subunits, that were shown to interact with Aß [START_REF] Liu | A Novel Nicotinic Acetylcholine Receptor Subtype in Basal Forebrain Cholinergic Neurons with High Sensitivity to Amyloid Peptides[END_REF][START_REF] Liu | Functional alpha7beta2 nicotinic acetylcholine receptors expressed in hippocampal interneurons exhibit high sensitivity to pathological level of amyloid ß peptides[END_REF][START_REF] Søderman | The nicotinic α7 acetylcholine receptor agonist ssr180711 is unable to activate limbic neurons in mice overexpressing human amyloid-β1-42[END_REF][START_REF] Sudweeks | Functional and molecular characterization of neuronal nicotinic ACh receptors in rat CA1 hippocampal neurons[END_REF]Wang et al., 2000aWang et al., , 2000b)).

The α7 homomeric receptor demonstrates a wide-spread localization in the brain and is characterized by a high calcium ion permeability and a fast desensitization rate [START_REF] Dani | Nicotinic Acetylcholine Receptors and Nicotinic Cholinergic Mechanisms of the Central Nervous System[END_REF][START_REF] Quick | Desensitization of neuronal nicotinic receptors[END_REF]. α7 nAChR on presynaptic terminals mediate release of others neurotransmitters [START_REF] Wonnacott | Nicotinic receptors modulate transmitter cross talk in the CNS: nicotinic modulation of transmitters[END_REF], while a postsynaptic or somatic localization elicits important changes in intracellular Ca ++ concentration, that can activate second messenger pathways mediating cellular processes such as neuronal survival and gene expression [START_REF] Berg | Nicotinic α7 receptors: Synaptic options and downstream signaling in neurons[END_REF][START_REF] Messi | Activation of α7 nicotinic acetylcholine receptor promotes survival of spinal cord motoneurons[END_REF][START_REF] Morley | Cholinergic receptors: dual roles in transduction and plasticity[END_REF]. Moreover, it was demonstrated that the activation of α7 nAChRs is important during development for the maturation of glutamatergic synapses [START_REF] Lozada | Glutamatergic Synapse Formation is Promoted by α7-Containing Nicotinic Acetylcholine Receptors[END_REF].

The role of α7 in memory and attention has been investigated for a long time. Knock-out (KO) mice for this subunit did not show a clear cognitive or attention deficit, except when the behavioural paradigm used implied prolonged sessions [START_REF] Young | Nicotine improves sustained attention in mice: evidence for involvement of the alpha7 nicotinic acetylcholine receptor[END_REF][START_REF] Young | Impaired attention is central to the cognitive deficits observed in alpha 7 deficient mice[END_REF].

Historically, the first nAChR subunit identified to interact with Aß was α7 (Wang et al., 2000a(Wang et al., , 2000b)). Later it was shown that Aß is able to activate also ß2*-nAChRs (ß2 subunitcontaining nAChRs). This subunit commonly forms heteropentameric receptors in combination with the α4 subunit. The pharmacological and functional characteristics of these heteromeric receptors are determined by both the contributing α and ß subunits. The subtype α4ß2 is characterized by lower calcium ion permeability and a slow desensitization rate compared to the homopentameric α7 nAChR [START_REF] Quick | Desensitization of neuronal nicotinic receptors[END_REF]. The "classic" highaffinity nAChR is composed of α4 and ß2 subunits [START_REF] Zoli | Identification of four classes of brain nicotinic receptors using beta2 mutant mice[END_REF]. In addition to the α4ß2 subtype, it was demonstrated that the α7 subunit is able to co-assemble with the ß2 subunit to form a heteropentameric receptor. This novel α7ß2 subtype was first investigated by expressing the heteromer in Xenopus oocytes. Recently, the α7ß2* nAChR subtype was found in basal forebrain cholinergic neurons and hippocampal interneurons of mouse brain, and in the human basal forebrain [START_REF] Moretti | The novel α7β2-nicotinic acetylcholine receptor subtype is expressed in mouse and human basal forebrain: biochemical and pharmacological characterization[END_REF]. This class of receptors seems to be particularly sensitive to Aß-induced toxicity [START_REF] Khiroug | Rat nicotinic ACh receptor 7 and 2 subunits co-assemble to form functional heteromeric nicotinic receptor channels[END_REF][START_REF] Liu | A Novel Nicotinic Acetylcholine Receptor Subtype in Basal Forebrain Cholinergic Neurons with High Sensitivity to Amyloid Peptides[END_REF][START_REF] Liu | Functional alpha7beta2 nicotinic acetylcholine receptors expressed in hippocampal interneurons exhibit high sensitivity to pathological level of amyloid ß peptides[END_REF]. The existence of this novel subtype was further confirmed in a human cell line (SHEP-1) transfected with the cDNA for α7 and ß2 subunits. Under these experimental conditions, α7 and ß2 are able to co-assemble into a functional receptor that localizes at the cell surface. The ability of α7 and ß2 subunits to form a functional receptor was confirmed in X. laevis oocytes.

This heteromer displayed only a modest difference in the electrophysiological response to pharmacological agents compared to the homomeric α7 nAChR [START_REF] Murray | α7β2 Nicotinic Acetylcholine Receptors Assemble, Function, and Are Activated Primarily via Their α7-α7 Interfaces[END_REF]. The exact stoichiometry of this recently discovered subtype was not defined. It is clear that functional ligand binding domains could only be formed at the α7-α7 interface. [START_REF] Murray | α7β2 Nicotinic Acetylcholine Receptors Assemble, Function, and Are Activated Primarily via Their α7-α7 Interfaces[END_REF] proposed a schematic model of all the possible stoichiometries for the α7ß2 subtype.

The importance of ß2 in maintaining brain homeostasis during normal ageing was highlighted in the KO mouse for this subunit. Aged ß2 null mutant mice have a thinner cortex compared to age-matched wild-type controls [START_REF] Zoli | Increased neurodegeneration during ageing in mice lacking high-affinity nicotine receptors[END_REF]. This work should be pursued further as it indicates a "neurotrophic" action of ß2 receptor activation by endogenous ACh (Zanardi et al., 2007). Null mutant ß2 mice were also tested to determine the role of this subunit in cognition. [START_REF] Guillem | Nicotinic acetylcholine receptor β2 subunits in the medial prefrontal cortex control attention[END_REF] showed that these mice exhibit an attention deficit which was restored by re-expression of this subunit with a lentiviral vector in the PFC.

We will now present and discuss the data demonstrating the existence of a physical interaction between nAChR and Aß, the functional consequences of this interaction and the intracellular pathways activated.

Interaction between nAChRs and Aß

The interaction between the α7 nAChR and Aß is widely demonstrated. The first indication of this interaction came from the experiments of Wang et al. (2000aWang et al. ( , 2000b)). They showed that α7 subunits co-localize with Aß 1-42 in senile plaques of brain slices obtained from patients that suffered from sporadic AD. In this context, no co-localization was found between Aß and the α4 subunit. The strong and specific association between Aß and α7, and no other subunits of the nAChRs, was further demonstrated with immunoprecipitation and Western Blot analysis. This set of experiments showed that Aß 1-42 is able to immunoprecipitate α7, which was not the case for other nAChR subunits such as α1, α3, α4, α8 or ß2. The same result was obtained with the reciprocal experiment, α7 immunoprecipitation and Aß 1-42 detection, meaning that the two proteins strongly interact. Experiments performed with fragments of Aß helped identify the sequence responsible for the interaction with α7, which corresponds to the amino acid residues 12-28 of the Aß sequence (Wang et al., 2000b). Subsequently, competition studies performed by incubating α7 nAChRs with Aß and αBungarotoxin showed that the application of αBungarotoxin is able to decrease the amount of Aß bound to α7 nAChRs, suggesting that both molecules compete for the same ligand binding domain (Wang et al., 2000b). Furthermore, studies using two selective nicotinic receptor ligands, namely

[3H]methyllycaconitine and [3H]cytisine, clearly showed that α7 nAChRs bind the less toxic peptide Aß 1-40 , but with lower affinity compared to the 1-42 form (Wang et al., 2000a). Direct interaction between Aß and α7 was further demonstrated in a transgenic mouse line, APPSwe/PS1ΔE9, where α7 was shown to immunoprecipitate with Aß 1-40/42 [START_REF] Søderman | The nicotinic α7 acetylcholine receptor agonist ssr180711 is unable to activate limbic neurons in mice overexpressing human amyloid-β1-42[END_REF]. Modulation of nAChRs by Aß was also found in ex vivo studies: Pettit and colleagues (2001) used rat hippocampal slices to show that Aß 1-42 incubation is able to reduce postsynaptic currents and open probability of both α7 and non-α7 nAChRs subtypes, demonstrating an interaction between Aß and other nAChR subunits. Even though studies have mainly focused on the interaction between α7 nAChRs and Aß and its functional consequences, it has been demonstrated that also ß2*-nAChRs bind to Aß. The evidence supporting this interaction comes mostly from electrophysiological studies [START_REF] Lamb | Inhibition of neuronal nicotinic acetylcholine receptor channels expressed in Xenopus oocytes by beta-amyloid1-42 peptide[END_REF][START_REF] Liu | A Novel Nicotinic Acetylcholine Receptor Subtype in Basal Forebrain Cholinergic Neurons with High Sensitivity to Amyloid Peptides[END_REF][START_REF] Liu | Functional alpha7beta2 nicotinic acetylcholine receptors expressed in hippocampal interneurons exhibit high sensitivity to pathological level of amyloid ß peptides[END_REF][START_REF] Wu | β-Amyloid Directly Inhibits Human α4β2-Nicotinic Acetylcholine Receptors Heterologously Expressed in Human SH-EP1 Cells[END_REF].

Several papers from the Yakel laboratory [START_REF] Lamb | Inhibition of neuronal nicotinic acetylcholine receptor channels expressed in Xenopus oocytes by beta-amyloid1-42 peptide[END_REF][START_REF] Pandya | Allosteric Modulator Desformylflustrabromine Relieves the Inhibition of α2ß2 and α4ß2 Nicotinic Acetylcholine Receptors by ß-Amyloid1-42 Peptide[END_REF][START_REF] Pettit | β-Amyloid1-42 Peptide Directly Modulates Nicotinic Receptors in the Rat Hippocampal Slice[END_REF] have investigated the action of Aß on these heteromeric receptors. In both hippocampal interneurons and oocytes they observed a block of ß2*-nAChRs, that could be prevented by the application of a positive allosteric modulator (PAM) specific for ß2 subunit containing receptors, desformylflustrabromine (dFBr) [START_REF] Pandya | Allosteric Modulator Desformylflustrabromine Relieves the Inhibition of α2ß2 and α4ß2 Nicotinic Acetylcholine Receptors by ß-Amyloid1-42 Peptide[END_REF]. These are the only conclusive studies to our knowledge, and it will probably be important to continue this dissection.

Functional consequences of nAChR-Aß interaction

In this paragraph we will summarize the extensive literature that covers nAChR functional activation or inhibition mediated by Aß, with emphasis on nAChR subtype, Aß concentration and experimental model used. The literature presents conflicting results about functional consequences of Aß binding on nAChRs, eliciting in some studies a receptor activation, and in others an inhibition [START_REF] Pettit | β-Amyloid1-42 Peptide Directly Modulates Nicotinic Receptors in the Rat Hippocampal Slice[END_REF][START_REF] Puzzo | Picomolar Amyloid-β Positively Modulates Synaptic Plasticity and Memory in Hippocampus[END_REF][START_REF] Tozaki | The inhibitory and facilitatory actions of amyloid-β peptides on nicotinic ACh receptors and AMPA receptors[END_REF]. In order to be able to compare and interpret these contrasting observations, we should be aware of the variability stemming from the different approaches used, such as the difference in receptor subtype (homo-or hetero-pentameric receptors), the system used for electrophysiological recordings (Xenopus oocytes that express transiently the subunits versus organotypic slices and cellular systems), and most importantly, the Aß preparation used (concentration and aggregation status). For instance, there is variability in the characteristics and toxicity of the oligomeric vs fibrillar species of Aß [START_REF] Selkoe | Alzheimer's disease[END_REF][START_REF] Walsh | Oligomers on the brain: the emerging role of soluble protein aggregates in neurodegeneration[END_REF]. Below, we will summarize the most remarkable findings over the past years.

Important evidence for a functional interaction between nAChRs and Aß came from the work of [START_REF] Pettit | β-Amyloid1-42 Peptide Directly Modulates Nicotinic Receptors in the Rat Hippocampal Slice[END_REF]. They demonstrated that Aß 1-42 drives a reversible inhibition of nAChRmediated currents in hippocampal GABAergic neurons recorded from rat slices. In these experimental conditions the most effective Aß 1-42 concentration was 500nM, but inhibition was found also at the lower concentration of 100nM. With the use of selective antagonists it was possible to determine that inhibition operates on both α7 and non-α7 receptors [START_REF] Pettit | β-Amyloid1-42 Peptide Directly Modulates Nicotinic Receptors in the Rat Hippocampal Slice[END_REF]. Experiments performed on Xenopus laevis oocytes transiently transfected with α7 or α4ß2 cDNA showed a concentration dependent effect of Aß on receptor inhibition. In this case the peptide used was Aß 1-40 and the concentrations adopted were between 0.1 and 10 µM, with increased Aß concentration resulting in a bigger inhibition of the receptor [START_REF] Tozaki | The inhibitory and facilitatory actions of amyloid-β peptides on nicotinic ACh receptors and AMPA receptors[END_REF]. A different set of experiments demonstrated that Aß enhances ACh activation of the α4ß2 nAChRs expressed in oocytes, this first activation of the receptor was followed by its inhibition [START_REF] Pym | Subtype-specific actions of β-amyloid peptides on recombinant human neuronal nicotinic acetylcholine receptors (α7, α4β2, α3β4) expressed in Xenopus laevis oocytes[END_REF]. However, α7 nAChR activation was observed in X. laevis oocytes when a range of Aß concentration spanning from 1 to 100 pM was applied [START_REF] Dineley | β-Amyloid Peptide Activates α7 Nicotinic Acetylcholine Receptors Expressed in Xenopus Oocytes[END_REF]. Using a different experimental model, [START_REF] Liu | β-Amyloid peptide blocks the response of α7containing nicotinic receptors on hippocampal neurons[END_REF] obtained a different outcome. The incubation of cultured rat hippocampal neurons with Aß 1-42 resulted in inhibition of α7 nAChRs, more precisely of both somato-dendritic and presynaptic populations of receptors. In this case, non-α7 nAChRs were insensitive to Aß inhibition [START_REF] Liu | β-Amyloid peptide blocks the response of α7containing nicotinic receptors on hippocampal neurons[END_REF]. Further studies showed an inhibitory effect of Aß 1-42 on human α4ß2 nAChRs trasnfected in the cell line (SHEP1) [START_REF] Wu | β-Amyloid Directly Inhibits Human α4β2-Nicotinic Acetylcholine Receptors Heterologously Expressed in Human SH-EP1 Cells[END_REF]. Other work performed in cellular systems on human nAChRs showed receptor activation. The oligomeric form of Aß 1-40 was able to activate α7 nAChR expressed in SH-SY5Y cell line [START_REF] Lilja | Functional Interactions of Fibrillar and Oligomeric Amyloid-β with Alpha7 Nicotinic Receptors in Alzheimer's Disease[END_REF]. [START_REF] Arora | Impact of Sustained Exposure to β-Amyloid on Calcium Homeostasis and Neuronal Integrity in Model Nerve Cell System Expressing α4β2 Nicotinic Acetylcholine Receptors[END_REF] investigated, in a cellular system, the effect of prolonged Aß exposure on nAChR function. The rodent neuroblastoma cell line NG108-15 was transfected with α4ß2 nAChRs and treated for three days with 100 nM Aß. The following acute stimulation with Aß and nicotine led to receptor activation that caused a perturbation of intracellular calcium homeostasis followed by mitochondrial dysfunction and increased oxidative stress [START_REF] Arora | Impact of Sustained Exposure to β-Amyloid on Calcium Homeostasis and Neuronal Integrity in Model Nerve Cell System Expressing α4β2 Nicotinic Acetylcholine Receptors[END_REF]. In a different study, rat hippocampus and cortex were investigated and the activation of both α7 and non-α7 receptors was obtained with an enhancement of Ca ++ influx into the neuron following the application of picomolar (pM) concentrations of Aß 1-42 . In this case, a distinct response of homomeric and heteromeric receptors was found, specifically, α7 are activated at pM concentrations of Aß, while non-α7 receptors were responsive at nM Aß concentrations [START_REF] Dougherty | β-Amyloid Regulation of Presynaptic Nicotinic Receptors in Rat Hippocampus and Neocortex[END_REF]. A study performed on nerve endings with application of Aß demonstrated a non competitive action of Aß 1-40 on α4ß2 nAChR [START_REF] Olivero | Effects of soluble β-amyloid on the release of neurotransmitters from rat brain synaptosomes[END_REF].

In conclusion, the outcome of receptor activation or inhibition depends on the system used and on Aß concentration. In general, it is possible to summarize that short incubation and lower concentration lead to receptor activation [START_REF] Dineley | β-Amyloid Peptide Activates α7 Nicotinic Acetylcholine Receptors Expressed in Xenopus Oocytes[END_REF][START_REF] Dougherty | β-Amyloid Regulation of Presynaptic Nicotinic Receptors in Rat Hippocampus and Neocortex[END_REF][START_REF] Puzzo | Picomolar Amyloid-β Positively Modulates Synaptic Plasticity and Memory in Hippocampus[END_REF], while longer incubation period and higher peptide concentrations give rise to an inhibitory effect [START_REF] Dineley | β-Amyloid Activates the Mitogen-Activated Protein Kinase Cascade via Hippocampal α7 Nicotinic Acetylcholine Receptors:In Vitro and In Vivo Mechanisms Related to Alzheimer's Disease[END_REF][START_REF] Parri | Research update: Alpha7 nicotinic acetylcholine receptor mechanisms in Alzheimer's disease[END_REF][START_REF] Pettit | β-Amyloid1-42 Peptide Directly Modulates Nicotinic Receptors in the Rat Hippocampal Slice[END_REF]. However, due to the variability in the experimental models used, the concentration and aggregation form of Aß (see Table 1), and in the obtained data, it is not possible to definitively conclude whether Aß binding on nAChRs exerts an inhibitory or excitatory effect in a physiological context.

However, it is clear that nAChR-Aß interaction initiates intracellular signalling implicating a set of transduction cascades. Akt phosphorylation mediates the downstream activation of an anti-apoptotic pathway, which is also activated by nicotine treatment [START_REF] Kihara | α7 Nicotinic Receptor Transduces Signals to Phosphatidylinositol 3-Kinase to Block A β-Amyloid-induced Neurotoxicity[END_REF].

The molecular pathways activated are associated with neuroprotection, synaptic plasticity, learning and memory [START_REF] Plant | The Production of Amyloid β Peptide Is a Critical Requirement for the Viability of Central Neurons[END_REF][START_REF] Puzzo | Picomolar Amyloid-β Positively Modulates Synaptic Plasticity and Memory in Hippocampus[END_REF]. Subsequent to incubation with pM concentrations of Aß 1-42 monomers and oligomers, an increase of hippocampal LTP was observed. This enhancement of synaptic plasticity and the activation of intracellular pathways are mediated by the activation of α7 nAChRs [START_REF] Dineley | β-Amyloid Activates the Mitogen-Activated Protein Kinase Cascade via Hippocampal α7 Nicotinic Acetylcholine Receptors:In Vitro and In Vivo Mechanisms Related to Alzheimer's Disease[END_REF][START_REF] Parri | Research update: Alpha7 nicotinic acetylcholine receptor mechanisms in Alzheimer's disease[END_REF][START_REF] Plant | The Production of Amyloid β Peptide Is a Critical Requirement for the Viability of Central Neurons[END_REF][START_REF] Puzzo | Picomolar Amyloid-β Positively Modulates Synaptic Plasticity and Memory in Hippocampus[END_REF]. An in vivo Aß infusion in mice was able to enhance hippocampal dependent memory, highlighted with memory tasks such as the Morris water maze and contextual fear conditioning, which are both hippocampus dependent behavioural tasks [START_REF] Puzzo | Picomolar Amyloid-β Positively Modulates Synaptic Plasticity and Memory in Hippocampus[END_REF]. On the other hand, an opposite effect was shown with Aß-nAChR interaction being responsible for inhibition of survival pathways. In this system enhancement of Akt phosphorylation and activation of ERK pathway was observed following α7 agonist treatment, suggesting that Aß inhibits the neuroprotective effect of α7 nAChR activation [START_REF] Zhi | Simvastatin Exerts Antiamnesic Effect in Aβ25-35-Injected Mice[END_REF]. Even though some key differences in Aß concentration used among the different studies are present, it is possible to conclude that in certain conditions Aß activates survival pathways. It was then postulated that Aß-nAChR interaction has a physiological role in neuronal homeostasis that is disrupted when Aß concentrations increase in a pathological context, leading to receptor inhibition and possible cellular toxicity [START_REF] Dineley | β-Amyloid Activates the Mitogen-Activated Protein Kinase Cascade via Hippocampal α7 Nicotinic Acetylcholine Receptors:In Vitro and In Vivo Mechanisms Related to Alzheimer's Disease[END_REF][START_REF] Parri | Research update: Alpha7 nicotinic acetylcholine receptor mechanisms in Alzheimer's disease[END_REF].

nAChR null-mutants in AD

An interesting approach to study the role of nAChRs in AD in vivo is to generate KO transgenic mice for nAChRs subunits expressing fAD (familial AD) associated mutations.

This method was used to investigate the role of the α7 subunit in AD, a strategy that has so far not been used for the other nAChR subunits. Two examples from the literature showed strikingly opposite outcomes [START_REF] Dziewczapolski | Deletion of the α7 Nicotinic Acetylcholine Receptor Gene Improves Cognitive Deficits and Synaptic Pathology in a Mouse Model of Alzheimer's Disease[END_REF][START_REF] Hernandez | Loss of α7 Nicotinic Receptors Enhances β-Amyloid Oligomer Accumulation, Exacerbating Early-Stage Cognitive Decline and Septohippocampal Pathology in a Mouse Model of Alzheimer's Disease[END_REF]. In the first study conducted by [START_REF] Dziewczapolski | Deletion of the α7 Nicotinic Acetylcholine Receptor Gene Improves Cognitive Deficits and Synaptic Pathology in a Mouse Model of Alzheimer's Disease[END_REF], a mouse model, that we will refer to as APP-α7KO, was generated by crossing an AD model that expresses the human APP (Amyloid precursor protein) with the Swedish (KM670/671NL) and the Indiana (V717F) mutations [START_REF] Mucke | High-Level Neuronal Expression of Aβ1-42 in Wild-Type Human Amyloid Protein Precursor Transgenic Mice: Synaptotoxicity without Plaque Formation[END_REF], with the α7 nAChR null-mutant [START_REF] Orr-Urtreger | Mice Deficient in the α7 Neuronal Nicotinic Acetylcholine Receptor Lack α-Bungarotoxin Binding Sites and Hippocampal Fast Nicotinic Currents[END_REF].

This AD model displays spatial memory deficit at 13-16 months of age, while APP-α7KO mice did not exhibit any memory deficit, suggesting that the absence of the α7 subunit of the nicotinic receptor protects against the behavioural deficit caused by expression of the mutated forms of APP in this AD model. It is known that AD mice display some pathological modifications, like loss of synaptic and dendritic markers, such as synaptophysin and MAP2 [START_REF] Games | Alzheimer-type neuropathology in transgenic mice overexpressing V717F β-amyloid precursor protein[END_REF][START_REF] Hsia | Plaque-independent disruption of neural circuits in Alzheimer's disease mouse models[END_REF]. In the APP-α7KO line the lack of α7 was sufficient to preserve synaptic terminals and dendrites, rescuing levels of synaptophysin and MAP2 to reach that of aged-matched WT controls. To investigate whether the difference in APP expression in the different mouse lines was responsible of the lack of neuropathology, Western Blot and ELISA analyses were performed. These experiments confirmed that the level of expression of APP and the consequent synthesis of Aß were comparable between mouse lines, demonstrating that the difference in cognitive deficit and neuropathology were mediated exclusively by the nicotinic receptor. In addition, APP mice demonstrated a deficit in LTP while APP-α7KO mice did not display this phenotype. The authors postulated that the absence of α7 could prevent Aß intracellular accumulation ameliorating the cognitive neuropathology and its phenotypic association [START_REF] Dziewczapolski | Deletion of the α7 Nicotinic Acetylcholine Receptor Gene Improves Cognitive Deficits and Synaptic Pathology in a Mouse Model of Alzheimer's Disease[END_REF].

A second study was published by [START_REF] Hernandez | Loss of α7 Nicotinic Receptors Enhances β-Amyloid Oligomer Accumulation, Exacerbating Early-Stage Cognitive Decline and Septohippocampal Pathology in a Mouse Model of Alzheimer's Disease[END_REF]. In their work, the transgenic AD model line Tg2576 expressing the human APP sequence with the Swedish mutation [START_REF] Hsiao | Correlative memory deficits, Abeta elevation, and amyloid plaques in transgenic mice[END_REF] was crossed with the α7 KO mouse [START_REF] Orr-Urtreger | Mice Deficient in the α7 Neuronal Nicotinic Acetylcholine Receptor Lack α-Bungarotoxin Binding Sites and Hippocampal Fast Nicotinic Currents[END_REF] to establish a different APP-α7KO line. In this series of experiments the mice were tested with a panel of behavioural tasks at five months of age, investigating the role of α7 in early AD. The contextual fear conditioning and the novel object recognition tasks both showed that the cognitive deficits worsen when α7 is absent. In the hippocampus, it was shown that APP-α7KO mice had high levels of Aß, although significantly less than APP mice, an effect which is not due to modification of the APP expression level, equivalent in the two lines. As a consequence of the lower Aß concentration, the plaque load was clearly reduced in APP-α7KO mice. Signs of neuropathology were found in APP-α7KO illustrated by loss of MAP2 immunoreactivity in the hippocampus. In conclusion, this study demonstrated the protective role of α7 nAChRs. The mechanism proposed is that the Aß-α7 nAChR interaction could activate neuroprotective downstream pathways [START_REF] Parri | Research update: Alpha7 nicotinic acetylcholine receptor mechanisms in Alzheimer's disease[END_REF], and that at the same time the interaction engages Aß preventing its aggregation. With the progression of the disease the amount of Aß increases, it starts to accumulate, and becomes toxic for the neurons [START_REF] Hernandez | Loss of α7 Nicotinic Receptors Enhances β-Amyloid Oligomer Accumulation, Exacerbating Early-Stage Cognitive Decline and Septohippocampal Pathology in a Mouse Model of Alzheimer's Disease[END_REF].

In conclusion, α7 nAChRs seem to have a dual effect depending on the age of the mice tested and probably on Aß concentration and aggregation state. However, the AD model used to generate the APP-α7KO line express different fAD associated mutations (Swedish and Indiana mutations in [START_REF] Dziewczapolski | Deletion of the α7 Nicotinic Acetylcholine Receptor Gene Improves Cognitive Deficits and Synaptic Pathology in a Mouse Model of Alzheimer's Disease[END_REF]Swedish mutation in Hernandez et al., 2010). The different mutations cause AD pathology to develop in distinct ways, making it difficult to directly compare the results (see Table 2).

Pharmacological interference between α7 nAChR-Aß binding

A possible therapeutic approach for AD treatment is the use of nAChR agonists or antagonists to interfere with nAChR-Aß interaction. Numerous molecules with agonist or antagonist activity on nAChRs exist. The protective role of these molecules against Aß toxicity was widely demonstrated using many experimental models and approaches. [START_REF] Gao | Evaluation of nicotine and cotinine analogs as potential neuroprotective agents for Alzheimer's disease[END_REF] investigated the neuroprotective effect of cotinine, nicotine and their analogs, in vitro on primary cultures of rat cortical and hippocampal neurons. Cotinine is a nicotine metabolite known for its positive effects on memory and attention and lower toxicity compared to nicotine [START_REF] Hatsukami | Safety of cotinine in humans: physiologic, subjective, and cognitive effects[END_REF]. Methyllycaconitine (MLA), an α7 nAChR antagonist, showed neuroprotective effect on mouse and rat primary cell culture [START_REF] Martin | The α7 nicotinic acetylcholine receptor-selective antagonist, methyllycaconitine, partially protects against β-amyloid1-42 toxicity in primary neuron-enriched cultures[END_REF].

Another molecule investigated was 2-[2-(4-bromophenyl)-2-oxoethyl]-1-methyl pyridinium (S 24795), a partial α7 nAChR agonist. When this molecule was applied to synaptosomal preparations from rat frontal cortex and post mortem human AD samples it was able to dissociate Aß in a concentration dependent manner. The incubation with S 24795 was able to normalize Ca ++ influx mediated by both α7 nAChR and NMDAR [START_REF] Wang | Dissociating β-Amyloid from α7 Nicotinic Acetylcholine Receptor by a Novel Therapeutic Agent, S 24795, Normalizes α7 Nicotinic Acetylcholine and NMDA Receptor Function in Alzheimer's Disease Brain[END_REF][START_REF] Wang | S 24795 Limits β-Amyloid-α7 Nicotinic Receptor Interaction and Reduces Alzheimer's Disease-Like Pathologies[END_REF].

Many other examples of neuroprotective effects elicited by nAChR agonists exist in the literature. However, here we will focus on the results obtained with in vivo experimentation.

The first attempt to treat an AD mouse model with nicotine was performed by [START_REF] Nordberg | Chronic nicotine treatment reduces β-amyloidosis in the brain of a mouse model of Alzheimer's disease (APPsw)[END_REF]. The transgenic line harbouring Swedish double mutations [START_REF] Hsiao | Correlative memory deficits, Abeta elevation, and amyloid plaques in transgenic mice[END_REF] was treated for 5.5 months with nicotine in the drinking water. APP Swe mice showed a reduction in plaque load of 80% compared to sucrose treated mice. The analysis of the Aß fraction reduced by nicotine showed that mainly insoluble Aß 1-40/42 was affected while there was no change in soluble Aß [START_REF] Nordberg | Chronic nicotine treatment reduces β-amyloidosis in the brain of a mouse model of Alzheimer's disease (APPsw)[END_REF]. Another study investigated the effect of short and longterm nicotine treatment in the same APP Swe line. Nicotine was administered through subcutaneous injections for 10 consecutive days on 9 month old transgenic mice. For longterm administration, mice were treated with nicotine in the drinking water for a period of 5.5 months. The short-term treatment of 10 days showed a significant reduction in cortical insoluble Aß 1-40/42 . Long-term nicotine administration elicited a reduction in Aß deposits in blood vessel. APP Swe mice at 14.5 months have fewer αBungarotoxin binding sites, while in transgenic mice treated with nicotine the number of αBungarotoxin binding sites was recovered and comparable to non transgenic age-matched control mice, suggesting that there was an increase in the population of α7 nAChRs [START_REF] Hellström-Lindahl | Nicotine reduces Aβ in the brain and cerebral vessels of APPsw mice[END_REF]. [START_REF] Inestrosa | Nicotine Prevents Synaptic Impairment Induced by Amyloid-β Oligomers Through α7-Nicotinic Acetylcholine Receptor Activation[END_REF] then investigated the effect of intraperitoneal nicotine injections in the APP/PS1 double transgenic line. This line expresses fAD associated mutations in both APP and Presenilin 1 proteins. Mice aged 6 months were treated for one month with nicotine injections, which led to an improvment in working and episodic memory compared to nontreated transgenic mice. This study investigated the effect of nicotine delivery on both young and old mice. For this purpose, 12 month old mice were treated with nicotine injections, and then tested for spatial memory. Like the young mice, they also displayed an improvement in spatial memory, demonstrating that nicotine enhances memory in both young and old mice.

The amount of Aß was quantified, and following nicotine injections a reduction in Aß, particularly in the oligomeric form, was found. This work confirmed the observation that AD mouse models display a reduction of α7 binding sites and PSD-95 puncta reflecting a synaptic deficit in this line. Both α7 binding sites and PSD-95 puncta were increased following nicotine treatment. The explanation proposed by the authors is that α7 nAChR activation through nicotine binding could promote survival pathways and recover the synaptic damage caused by Aß [START_REF] Inestrosa | Nicotine Prevents Synaptic Impairment Induced by Amyloid-β Oligomers Through α7-Nicotinic Acetylcholine Receptor Activation[END_REF]. [START_REF] Oddo | Chronic nicotine administration exacerbates tau pathology in a transgenic model of Alzheimer's disease[END_REF] investigated the consequences of nAChR activation on tau pathology in AD model 3xTg-AD. This line was chosen because it showed cholinergic pathology with a decrease of α7 nAChRs in brain regions affected by Aß deposition and tau aggregation. Chronic nicotine administration was performed in drinking water for 5 months. The long-term nicotine treatment caused faster tau aggregation in CA1 pyramidal neurons. The possible mechanism by which nicotine enhances the aggregation of tau is through the activation of p38-MAP kinase. This kinase is sensitive to Ca ++ , whose levels are increased following nAChR activation [START_REF] Oddo | Chronic nicotine administration exacerbates tau pathology in a transgenic model of Alzheimer's disease[END_REF]. Even though nicotine showed a positive effect reducing plaque load [START_REF] Hellström-Lindahl | Nicotine reduces Aβ in the brain and cerebral vessels of APPsw mice[END_REF][START_REF] Inestrosa | Nicotine Prevents Synaptic Impairment Induced by Amyloid-β Oligomers Through α7-Nicotinic Acetylcholine Receptor Activation[END_REF][START_REF] Nordberg | Chronic nicotine treatment reduces β-amyloidosis in the brain of a mouse model of Alzheimer's disease (APPsw)[END_REF], its use in AD treatment should be limited due to its toxic effect on tau pathology. [START_REF] Shim | Nicotine Leads to Improvements in Behavioral Impairment and an Increase in the Nicotine Acetylcholine Receptor in Transgenic Mice[END_REF] investigated the effect of nicotine administration in an AD mouse model harbouring the Swedish mutation. 12 month old mice were treated with three different doses of nicotine (5 mg/kg, 30 mg/kg and 180 mg/kg) in drinking water for 6 months. Nicotine treatment improved the memory deficit, highlighted with the Morris water maze task.

Surprisingly, this study showed a dose dependent increase of α7 nAChR, a result that is in contrast with the literature [START_REF] Oddo | Chronic nicotine administration exacerbates tau pathology in a transgenic model of Alzheimer's disease[END_REF]. In the group that received the higher dose of nicotine, the level of α7 nAChR was restored to the level found in wild-type animals [START_REF] Shim | Nicotine Leads to Improvements in Behavioral Impairment and an Increase in the Nicotine Acetylcholine Receptor in Transgenic Mice[END_REF]. However, nicotine is addictive and has numerous side effects, for example on the cardiovascular system. The use of cotinine for the treatment of AD was also investigated. We already cited an in vitro study on the protective effect of cotinine against Aß toxicity [START_REF] Gao | Evaluation of nicotine and cotinine analogs as potential neuroprotective agents for Alzheimer's disease[END_REF]. In a mouse model of AD, cotinine treatment decreased the plaque load and was able to activate the Akt pathway, that was shown to be neuroprotective [START_REF] Echeverria | Cotinine reduces amyloid-β aggregation and improves memory in Alzheimer's disease mice[END_REF]. Cotinine is an α7 nAChR PAM (positive allosteric modulator). It was already demonstrated that this molecule does not display side effects when administrated in humans [START_REF] Hatsukami | Safety of cotinine in humans: physiologic, subjective, and cognitive effects[END_REF]. Thus, cotinine is an interesting molecule for AD treatment (for a review see [START_REF] Echeverria | Cotinine: A Potential New Therapeutic Agent against Alzheimer's disease[END_REF].

The neuroprotective effect of 4OH-GTS-21, an α7 nAChR agonist, was also investigated. The molecule was administered to PS1 and APP/PS1trangenic mice following a lesion to the fimbria fornix region (FFX). This novel model displayed deficits in spatial memory and reduced neuronal density. Whilst the spatial memory deficit was restored by 4OH-GTS-21 treatment, this molecule had no effect on neuronal density [START_REF] Ren | The α7 nicotinic receptor agonist 4OH-GTS-21 protects axotomized septohippocampal cholinergic neurons in wild type but not amyloid-overexpressing transgenic mice[END_REF]. Wild-type mice treated with nicotine or with SSR180711, another partial agonist of α7 [START_REF] Biton | SSR180711, a Novel Selective α7 Nicotinic Receptor Partial Agonist: (1) Binding and Functional Profile[END_REF], showed increased LTP, while the transgenic AD model APPSwe/PS1ΔE9 showed no effect on LTP following SSR180711 treatment. The authors performed an autoradiographic study to investigate the number of α7 nAChR binding sites. Surprisingly, no decrease was found when transgenic and wild-type mice were compared, a result in contrast with work from other authors [START_REF] Shim | Nicotine Leads to Improvements in Behavioral Impairment and an Increase in the Nicotine Acetylcholine Receptor in Transgenic Mice[END_REF]. The molecule SSR180711 seems to be effective in enhancing LTP in wild-type mice but is not able to interfere with Aß-nAChR binding, thus not being considered for AD treatment [START_REF] Söderman | Activation of nicotinic α7 acetylcholine receptor enhances long term potentation in wild type mice but not in APPswe/PS1ΔE9 mice[END_REF].

Pregnenolone sulfate (PREGS) is an endogenous steroid known to ameliorate cognitive performance in animals. PREGS is a modulator of synaptic plasticity, acting on the activation of glutamatergic transmission [START_REF] Smith | Pregnenolone sulfate as a modulator of synaptic plasticity[END_REF]. It was proposed that the positive action of PREGS is mediated by α7 nAChR. In order to investigate this hypothesis, [START_REF] Yang | Anti-amnesic effect of neurosteroid PREGS in Aβ25-35-injected mice through σ1 receptor-and α7nAChRmediated neuroprotection[END_REF] treated a mouse model obtained with injection of Aß 25-35 for 7 days whit PREGS. The treatment improved spatial memory and reduced apoptosis in CA1 pyramidal cells. Even though the protective effect of PREGS is evident, less clear is whether or not it implies α7 nAChR activation [START_REF] Yang | Anti-amnesic effect of neurosteroid PREGS in Aβ25-35-injected mice through σ1 receptor-and α7nAChRmediated neuroprotection[END_REF]. Simvastatin (SV) is a statin commonly used in the clinic to control cholesterol levels and it was shown to improve cognitive function in AD patients [START_REF] Simons | Treatment with simvastatin in normocholesterolemic patients with Alzheimer's disease: A 26-week randomized, placebocontrolled, double-blind trial[END_REF][START_REF] Sparks | Statin therapy in Alzheimer's disease[END_REF]. In a study conducted by [START_REF] Zhi | Simvastatin Exerts Antiamnesic Effect in Aβ25-35-Injected Mice[END_REF], this molecule was used to treat a mouse model obtained with injections of the peptide Aß 25-35 . Administration for 11 days of SV improved memory performance in the Morris water maze task and promoted survival of CA1 pyramidal cells. This effect was proposed to be promoted by α7 nAChRs, since it is blocked by MLA administration [START_REF] Zhi | Simvastatin Exerts Antiamnesic Effect in Aβ25-35-Injected Mice[END_REF].

A caveat with these pharmacological studies directed at α7 nAChRs is the contribution of potential off-target effects, hence other approaches have been investigated. The intracellular signaling initiated by the binding of Aß to nAChR at the cell surface requires the recruitment of Filamin A (FLNA), a scaffold protein that is known to crosslink actin, and in addition could also have a function in certain intracellular pathways [START_REF] Stossel | Filamins as integrators of cell mechanics and signalling[END_REF]. Wang et al. showed that the association between FLNA and the α7 subunit is elevated in AD samples compared to age matched controls. A novel molecule called PTI-125 was used to interfere with the interaction of FLNA and α7. The treatment with PTI-125 prevents FLNA binding to α7 and as consequence reduces the affinity of Aß for nAChRs, attenuating the toxic effect of Aß [START_REF] Wang | Reducing Amyloid-Related Alzheimer's Disease Pathogenesis by a Small Molecule Targeting Filamin A[END_REF].

Conclusion

In this review we present evidence supporting the existence of a direct interaction between nAChRs and Aß. The effect of Aß on nAChR physiology is complex. It was indeed shown that this interaction can lead to either activation of survival pathways or to a toxic effect. Under different experimental conditions, Aß can act like an agonist or an antagonist on AChRs, which underlies the observed variability. Thus, nAChR activation, for example with nicotine, can have a protective effect against Aß exerted toxicity, with the clearance of the peptide. However, this interaction should be studied further to understand the cellular mechanism(s) underlying the activation of survival pathways and Aß clearance, observed following nicotine administration. Moreover, the use of different α7 agonists to target this nAChR subtype is a promising approach that can be translated into AD therapy. Despite the fact that mainly the α7 subunit was investigated in the AD field, some data demonstrating the functional interaction between Aß and non-α nAChRs also exist. We believe that other nAChR subunits could be a target of Aß toxicity and their contribution to AD pathology should be elucidated.

Agonist molecules are responsible for the activation of the receptor at the surface of the cell, and scaffold proteins anchoring nicotinic receptors mediate and regulate the transduction of the intracellular signal. These proteins that interact with the receptors, or directly linked interacting proteins, are largely unknown. In addition to nAChRs, these proteins also present future targets for the study of neurodegeneration research and potential treatment.

Finally, the possibility to generate multiple transgenic models expressing fAD associated mutations and KOs for nAChR subunits was explored. This approach showed apparently contrasting results that however could reflect a different role (protective or not) of the nicotinic receptors depending on Aβ concentration.

Significant evidence suggesting a potential role of nAChRs as therapeutic target in AD exists.

These studies provide support for pursuing the investigations to define the detailed mechanism of nAChR neuroprotection and design nAChR agonists, antagonists or allosteric modulators for clinical translation. 9Rescue(of(memory(deficit( 9Rescue(of(synapEc(deficit( 9Rescue(of(LTP(impairment( • (13916(months( • (19922(months( • (not(specified( Not(protecEve( Dziewczapolski(et(al.,(2009( Swedish( (KM670/671NL)( 9Memory(deficit( 9SynapEc(deficit( 9Increased(plaque(load( 9Decreased(Aβ( (5(and(12(months( ProtecEve( Hernandez(et(al.,(2012(

Table Legends

Table 1 Action of Aß on nAChRs: The incubation with Aß activates or inhibits nAChRs. Here we summarised the results from the literature with particular attention on the Aß fragment and concentrations used, the model in which the experiments were performed and the nAChR subtype investigated.

Table 2

Conflicting data on α7 KO crossed with AD model: Comparison of two published studies showing the AD model used for the generation of the APP-α7KO line, the phenotype of the obtained APP-α7KO with the age of tested mice and the overall effect of α7 nAChRs.

Table 1 Reference' System'used' Type'of'Aβ' Aβ'concentra5on' nAChR'subtype' Effect'elicited''

 1

	Inhibi#on%		Inhibi#on%	Inhibi#on%	Insensi#ve%to%	inhibi#on%	Inhibi#on%	Inhibi#on%	First%ac#va#on,%	aOer%inhibi#on%%	Ac#va#on%	Ac#va#on%	Ac#va#on%	Ac#va#on%	Ac#va#on%	Inhibi#on%
	α7%%	Non7α7%	α7%7%α4β2%	α7%%	α4β2%		α4β2%	α7%%	α4β2%		α7%%	Non7α7%	α7%	α7%	α4β2%	α4β2%
	100%nM%7%1μM%	0.1710%μM%		100%nM%		1nM%		10%nM%		100%pM%	100%nM%	17100%pM%	10 78 %-%10 77 %M %	100%nM%	100%nM%
	Aβ 1742%	Aβ 1740%	Aβ 1742% 7%Aβ 1740%		Aβ 1742%		Aβ 1742% 7%Aβ 1740%		Aβ 1742%	Aβ 1742%	Aβ 1740%	% (oligomeric)	Aβ 1742%	Aβ 1740%
	Pe#t%et%al.,%2001% Rat%Hippocampal% slices%	Tozaki%et%al.,%2002% X.#leavis%oocytes%	Liu%et%al.,%2001% Rat%hippocampal% neurons%	Wu%et%al.,%2004% SHEP1%cell%line%		Pym%et%al.,%2005% X.#leavis#oocytes%		Dougherty%et%al.,% Rat%synaptosomes% (hippocampus%and% 2003% cortex)%	Dineley%et%al.,%2002% X.#leavis%oocytes%	Lilja%et%al.,%2011% SH7SY5Y%cells%	Arora%et%al.,%2013% NG108715%cells%	Oliviero%et%al.,%2014% Rat%synaptosomes%%

Acknowledgments

SL was supported by EU FP7 "BRAINTRAIN", Fondation pour la Recherche Médicale FRM and Institut Pasteur programme "PasteurInnov" 2012.


Nadia El Mrabet
email: 8-elmrabet@univ-paris8.fr

Side Channel Attacks against Pairing over Theta Functions

Keywords: pairing based cryptography, Theta function, side channel attacks, differential power analysis, fault attacks

, Lubicz and Robert generalized the Tate pairing over any abelian variety and more precisely over Theta functions. The security of the new algorithms is an important issue for the use of practical cryptography. Side channel attacks are powerful attacks, using the leakage of information to reveal sensitive data. The pairings over elliptic curves were sensitive to side channel attacks. In this article, we study the weaknesses of the Tate pairing over Theta functions when submitted to side channel attacks.

Introduction

Since they appeared in cryptography, the efficient computation of pairings is a very active area of research. Originally defined over elliptic curves in Weierstrass model [START_REF] Miller | The Weil pairing, and its efficient calculation[END_REF], pairings have been computed in other models of elliptic curves (for example Edwards [START_REF] Ionica | Another approach to pairing computation in Edwards coordinates[END_REF], Huff [START_REF] Joye | Huff's model for elliptic curves[END_REF], Jacobi [START_REF] Duquesne | Tate pairing computation on jacobi's elliptic curves[END_REF]). They have also been studied in different systems of coordinates such as affine [START_REF] Lauter | An analysis of affine coordinates for pairing computation[END_REF], Jacobian, projective, Chudnovsky [START_REF] Costello | Faster pairing computations on curves with high-degree twists[END_REF] or in original representation of finite fields RNS [START_REF] Ray | Fpga implementation of pairings using residue number system and lazy reduction[END_REF]. The main algorithm to compute pairings is the Miller algorithm [START_REF] Miller | The Weil pairing, and its efficient calculation[END_REF]. It is based on a double and add scheme. Several works aimed to reduce the number of iterations of Miller's algorithm and to develop the notion of optimal pairings [START_REF] Hess | The Eta Pairing Revisited[END_REF]. In both Optimal Pairings [START_REF] Vercauteren | Optimal pairings[END_REF] and Pairings Lattices [START_REF] Hess | Pairing Lattices[END_REF] the authors present methods to find the Miller algorithm with the smallest number of iterations. All these works deal with a computation of pairing over elliptic (or hyper elliptic) curves.

The latest improvement in the computation of pairing was the description of efficient pairing computation in a more general case for any algebraic variety; and in particular pairings over Theta functions. In [START_REF] Lubicz | Efficient pairing computation with theta functions[END_REF], Lubicz and Robert generalize the notion of the Weil and the Tate pairings to any abelian variety. To do so, they made an explicit link between the Weil and the Tate pairings and the intersection pairing on the degree 1 homology of an abelian variety. The result is a general definition of pairings and they explicit the formulas for the case of level 2 and 4 Theta functions in order to obtain the most efficient algorithm, considering time and memory consumption. Their algorithm to compute pairing is based on a Montgomery Ladder's approach.

Each time new formulas for pairing are proposed, the security and implementation of the new algorithms are an important issue for the use in practical cryptography. As for every cryptographic protocol constructed nowadays, the size of groups involved in pairing computation are chosen to be large enough to avoid the discrete logarithm attack. Consequently, pairing implementations are secured against mathematical attacks. Nevertheless, considering side channel attacks, we cannot predict if an algorithm is more or less secure than another given the representation of the groups. Weaknesses to side channel attacks of pairing based cryptography over elliptic curve have been highlighted [START_REF] Page | A fault attack on pairing-based cryptography[END_REF][START_REF] Whelan | Side channel analysis of practical pairing implementation: Which path is more secure? In VietCrypt[END_REF][START_REF] Whelan | The importance of the final exponentiation in pairings when considering fault attacks[END_REF][START_REF] Mrabet | A practical differential power analysis attack against the miller algorithm[END_REF][START_REF] Mrabet | Fault attacks on pairing based cryptography: A state of the art[END_REF]. Then, wondering if a pairing implemented in Theta function would be vulnerable to side channel attacks is an important issue for pairing based cryptography and this is the main objective of this contribution. The remaining of the article is organized as follows. The Section 2 is devoted to the definition of pairings over Theta functions. In Section 3 we describe the application of side channel attacks to pairing over Theta functions, we highlight the weaknesses of the pairing and provide countermeasures to secure the computation. We conclude in Section 4.

Pairings over Theta Function

This Section is a brief review of the results in [START_REF] Lubicz | Efficient pairing computation with theta functions[END_REF]. We present the notations and background of Theta functions in Section 2.1. We give the definition of the Weil and the Tate pairings and of the algorithm to compute the Tate pairing in Section 2.2.

Background on Theta function

Let H g be the g dimensional Siegel upper-half space which is the set of g × g symmetric matrices Ω whose imaginary part is positive definite. For Ω ∈ H g , let Λ Ω = ΩZ g × Z g the lattice of C g defined by Ω. If A is an abelian variety of dimension g over the number field K with a principal polarization then A is analytically isomorphic to C g /Λ Ω . Let Π : C → C g /Λ Ω = A be the canonical projection. The classical theory of Theta functions gives a lot of functions on C g that are pseudo-periodic with respect to Λ Ω and can be used as a projective coordinate system for A. For a, b ∈ Q g , the Theta function with rational characteristics (a, b) is an analytic function on C g × H g given by

θ a b (z, Ω) = n∈Z g exp Πi t (n + a).Ω.(n + a) + 2Πi t (n + a).(z + b) ,
where t represents the transpose of a vector. In order to describe the pseudo-periodicity relations verified by the Theta function, we introduce a certain pairing on C g . We have that C g is isomorphic to R 2g via the map

R 2g -→ C g (x 1 , x 2 ) -→ Ωx 1 + x 2 . For α, β ∈ R 2g , let α = (α 1 , α 2 ) and β = (β 1 , β 2 ), we define e Ω : R 2g → C by e Ω (α, β) = exp (2iΠ(α 1 β 2 -α 2 β 1 )).
The pseudo periodicity of θ is given by

θ a b (z + Ω.m + n, Ω) = e Ω (Ωa + b, Ωm + n) × e (-Πi t mΩm-2Πi t mz) × θ a b (z, Ω). A function f on C g is Λ Ω -Theta-periodic of level l ∈ N if for all z ∈ C g and m ∈ Z g , we have f (z + m) = f (z), f (z + Ω.m) = exp(-Πil t m.Ω.m -2Πil t z.m)f (z).
For any l ∈ N , the set H Ω,l of Λ Ω -quasi-periodic functions of level l is a finite dimensional C-vector space whose basis can be given by the Theta functions with characteristics θ

0 b/l (z, l -1 .Ω) b∈[0,...,l-1] g . If l = k 2 , then an alternative basis of H Ω,l is θ a/k b/k (kz, Ω) a,b∈[0,...,k-1] g .
Once the level l ∈ N is fixed, the following conventions are adopted Z(l) = (Z/lZ) g and for a point z P ∈ C g and i ∈ Z(l) let θ i (z

P ) = θ 0 i/l (z P , Ω/l). If l = k 2 , for i, j ∈ Z(k), let θ i,j (z P ) = θ i/k j/k (k.z P , Ω).
Let P denote the element of A l g (C) with coordinates P i = θ i (z P ). Let P be the associated point of A that will be be considered depending on the situation as embedded in P l g -1 or as a point on the analytic variety C g /Λ Ω . For n, l ∈ N, if n divides l then Z(n) will be considered as a subgroup of Z(l) via the morphism x → (l/n).x. Let Ξ be the Theta divisor of level l on A, i.e. Ξ is the divisor of zero of θ 0 0 (z, l -1 Ω) . There is an isogeny φ l : A → A = Pic 0 A , defined by x → τ *

x Ξ l -Ξ l where τ x is the translation by x morphism on A. Let A[l] be the kernel of φ l . Let K(A) be the function field of A and (f ) be the divisor of a function f ∈ K(A). We then present the definition of the Weil and theTate pairing.

Definition and computation of pairings over Theta function

The Weil pairing For Ω ∈ H g , let A = C g /Λ Ω be the complex abelian variety and denote by π : C g → A the natural projection. Let l be a positive integer and µ l be the subgroup of C of l th roots of unity. For z P , z Q ∈ C g , let P , Q be the associated points of A. The Weil pairing is the map e W : A[l] × A[l] → µ l , (P, Q) → e Ω (z P , z Q ) l . The value e W (P, Q) does not depend on the choice of z P and z Q representing P and Q and e W is a non-degenerate skew linear form. This pairing can be expressed using certain Theta functions.

Definition 1. Let Ω ∈ H g , a, b ∈ Q g , l be a positive integer and let z P , z Q ∈ C g be such that l.z P = l.z Q = 0 mod Λ Ω . Let P = π(z P ) and Q = π(z Q ). Let L(z P , z Q ) = θ a b (l.z P + z Q , Ω) θ a b (z Q , Ω) θ a b (0, Ω) θ a b (l.z P , Ω) , R(z P , z Q ) = θ a b (l.z Q + z P , Ω) θ a b (z P , Ω) θ a b (0, Ω) θ a b (l.z Q , Ω)
.

If L(z P , z Q ) and R(z P , z Q ) are well defined and non null, then

e W (P, Q) = L(z P , z Q ) -1 .R(z P , z Q ) = e Ω (z P , z Q ) l .
The algorithm to compute the Weil pairing is composed of four calls to the function ScalarMult.

Theorem 1. Suppose that n and l are relatively prime. For X, Y ∈ A(K), denote by X, Y , X + Y any affine lifts of X, Y and

X + Y . For i ∈ Z(n), let X i be the i th coordinate of X. For ∈ N and i ∈ Z(n), let f T ( X, Y , X + Y , 0, l, i) = ScalarMult( X + Y , X, Y , 0, l) i ScalarMult( X, X, 0, 0, l) i 0 i Y i . Then for P , Q ∈ A[l] and i ∈ Z(n), we have e W (P, Q) n = f T ( P , Q, P + Q, 0, l, i) -1 f T ( Q, P , P + Q, 0, l, i),
whenever the right hand side is well defined.

The Tate pairing For efficiency reasons, the pairing that will be implemented is the Tate pairing (or a variant of the Tate pairing) so we only consider the side channel attacks against the Tate pairing. Let K be a number field and suppose that A is defined over K. Recall that l ∈ N is the level of the Theta function and it is fixed once for all. In this section, we suppose that µ l ⊂ K and that A[l] is rational over K. Let K be the algebraic closure of K and G = Gal(K/K). Let

δ 1 : K /K l → Hom(G, µ l ) (respectively δ 2 : A(K)/[l]A(K) → Hom(G, A[l]
)) be the connecting morphism of the Galois cohomology long exact sequence associated to the Kummer exact sequence (respectively to the short exact sequence 0

→ A[l] → A(K) → A(K) → 0)
. There exists a bilinear application often referred to as the Tate pairing e T :

A(K)/[l]A(K) × A[l] → K /K l such that for (P, Q) ∈ A(K)/[l]A(K) × A[l], e W (δ 2 (P ), Q) = δ 1 (e T (P, Q))
, where e W is the Weil pairing over Theta functions.

Definition 2. Let K be a number field and let A be a dimension g abelian variety over K.

Let Ω ∈ H g be such that A is analytically isomorphic to

C g /Λ Ω . Let a, b ∈ Q g and l be a positive integer. Let P ∈ A(K)/[l]A(K), Q ∈ A[l](K) and z P , z Q ∈ C g such that π(z P ) = P and π(z Q ) = Q where π : C g → A is the natural projection 1 . Suppose that z P , z Q and z P +Q are chosen such that θ 0 0 (z P + z Q , Ω) θ 0 0 (z P , Ω) θ 0 0 (0, Ω) θ 0 0 (z Q , Ω) ∈ K , then 1
By abuse of notation we use P , Q to denote the corresponding points of an algebraic and analytic model of A.

e T (P,

Q) = θ 0 0 (l.z Q + z P , Ω) θ 0 0 (z P , Ω) θ 0 0 (0, Ω) θ 0 0 (l.z Q , Ω)
.

The algorithm for computation of pairings over Theta functions Let n, l ∈ N and assume that 2 divides n and that gcd(n, l) = 1. Let A be an abelian variety over C with period matrix Ω. We represent A as a closed subvariety of P n g -1 by the way of level n Theta functions and suppose that this embedding is defined over K. Let A be the pullback of A via the natural projection κ : A n g → P n g -1 . For P ∈ A, let P be an affine lift of P that is a point of A n g such that κ( P ) = P . Important ingredients of the algorithm in [START_REF] Lubicz | Efficient pairing computation with theta functions[END_REF] are the Riemann addition formulas. Suppose that the Theta null point 0 = (θ i (0)) i∈Z(n) is known.

From [17, Theorem 1], we can construct an algorithm that takes as input

P = P i i∈Z(n) , Q = Q i i∈Z(n) and P -Q = ( P -Q) i i∈Z(n)
and outputs

P + Q = ( P + Q) i i∈Z(n)
.

Let P + Q = PseudoAdd( P , Q, P -Q).
Using the Riemann addition formulas, if n = 4, the projective point P +Q can be recovered from P and Q. As a consequence, with the knowledge of P , Q and P -Q there is a unique affine point P + Q above P +Q that satisfies the addition formulas from [17, Theorem 1]. The result is extended in [START_REF] Lubicz | Efficient pairing computation with theta functions[END_REF] for n = 2.

Chaining the algorithm PseudoAdd in a classical Montgomery Ladder yields an algorithm that takes as inputs Q, P + Q, P , 0 and an integer l and outputs P + lQ.

Let P + lQ = ScalarMult( P + Q, Q, P , 0, l). In particular, l P = ScalarMult( P , P , 0, 0, l). The output of ScalarMult is independent on the particular chain of PseudoAdd calls it uses.

Theorem 2. Suppose that n and l are relatively prime. For X, Y ∈ A(K), denote by X, Y , X + Y any affine lifts of X, Y and X + Y . For i ∈ Z(n), let X i be the i th coordinate of X. For ∈ N and i ∈ Z(n), let

f T ( X, Y , X + Y , 0, l, i) = ScalarMult( X + Y , X, Y , 0, l) i ScalarMult( X, X, 0, 0, l) i 0 i Y i .
Then, for

P ∈ A(K)/[l]A(K), Q ∈ A[l],
if we suppose that 0, P , Q and P + Q are affine lifts of 0, P , Q and P + Q with coordinates in K, then we have for i ∈ Z(n),

e T (P, Q) n = f T ( Q, P , P + Q, 0, l, i),
whenever the right hand side is well defined.

For example, let E be an elliptic curve defined by Ω ∈ H 1 and Ω = Ω/2. Put

a = θ 0 0 (0, Ω ); b = 0 1/2 (0, Ω ); A = θ 0 0 (0, 2Ω ); B = θ 1/2 0 (0, 2Ω ).
The algorithm ScalarMult is composed by a doubling algorithm and a differential addition algorithm given in Figure 1.

Doubling Algorithm

Differential Addition Algorithm Input: A point P = (xP : zP ).

Input: Two points P = (xP : zP ) and The general scheme of an identity based encryption is recalled in [START_REF] Galbraith | Pairings in Advances in Elliptic Curve Cryptography[END_REF]. The important point is that to decipher a message using an Identity Based Protocol, a computation of a pairing between a private key and a public message is performed. Side channel attacks are powerful attacks using the leakage of information during the execution of a cryptographic protocol.

Q = (xQ, zQ) on E, R = (xR : zR) = P -Q, with xRzR = 0. Output: The double 2P = (x2P : z2P ) Output: The point P + Q = (xP +Q : zP +Q) 1. x0 = (x 2 P + z 2 P ) 2 1. x0 = (x 2 P + z 2 P )(x 2 Q + z 2 Q ) 2. z0 = A 2 B 2 (x 2 P -z 2 P ) 2 2. z0 = A 2 B 2 (x 2 P -z 2 P )(x 2 Q -z 2 Q ) 3. x2P = x0 + z0 3. xP +Q = (x0 + z0)/xR 4. z2P = a b (x0 -z0) 4
As soon as the algorithm involves a computation between a secret and a public data, side channel attacks can be applied in order to reveal the secret, or information about the secret. The particularity of identity based cryptography is that an attacker can know the algorithm used, the number of iterations and the exponent. The secret is only one of the arguments of the pairing. We describe here two attacks, namely the Differential Power Analysis (DPA) and the fault attack. There are other side channel attacks, but the popular ones are either a generalization of the DPA (DEMA, CPA) or fault attacks.

The possible targets

If we compare the efficiency of the Tate and of the Weil pairings, the former is more efficient than the later at least for the security levels considering today, when pairings are computed using a Miller's algorithm. In the case of Theta functions, the algorithmic complexity of the Tate pairing consists in two applications of the function ScalarMult, while the Weil pairing consists in four applications of this function described in Appendix 4. It is quite evident that the Tate pairing over Theta function will always be more efficient than the Weil pairing over the Theta function. So we study only the weakness of the Tate pairing considering side channel attacks. Nevertheless, the attacks described for the Tate pairing can easily be adapted to the Weil pairing. As a consequence the countermeasure proposed here must be considered also for the implementation of the Weil pairing. The Tate pairing is composed of two applications of ScalarMult. First of all, we focus on side channel attacks against one application of ScalarMult and after that we will consider side channel attacks against the Tate pairing. The same argument can provide the result of side channel attacks against the Weil pairing, or any optimizations of the Tate pairing namely Ate, twisted Ate or optimal pairings. The function ScalarMult is a Montgomery Ladder composed by the Doubling and Differential Addition algorithms at each step. When the secret is the exponent this algorithm is an efficient countermeasure to side channel attacks. In the case of pairing based cryptography, the secret is not the exponent but one of the parameters of the Mongtomery Ladder algorithm. Consequently, the analysis considering side channel attacks against Montgomery's Ladder for the classical use in cryptography (efficient exponentiation) is no more available. We analyze the weaknesses of the algorithm to compute pairing using Theta functions. We will focus on the DPA and on the fault attack. The consideration of the DPA includes also the consideration of the Correlation Power Analysis (CPA) and the Differential Electromagnetic Attack (DEMA). Indeed the DEMA works exactly like the DPA and the CPA is an improvement of the DPA.

Differential Power Analysis Attack and generalization

In order to simplify the explanation, we describe here only the differential power analysis (DPA) attack. As the concept is the same for all differential attacks, we include in the same family the differential power analysis (DPA) and the differential electromagnetic attack (DEMA) [START_REF] Mulder | Differential power and electromagnetic attacks on a FPGA implementation of elliptic curve cryptosystems[END_REF]. Further on, correlation attack [START_REF] Whelan | Side channel analysis of practical pairing implementation: Which path is more secure? In VietCrypt[END_REF] is just a form of DPA using the particular side-channel distinguisher i.e. Pearson correlation.

We now introduce some theoretical issues that allow the reader to understand the principle underlying the DPA attack, more details can be found in [START_REF] Paul | Differential power analysis[END_REF][START_REF] Mangard | DPA book[END_REF]. We consider the output of a gate whose state depends on both the plain text to be ciphered (primary inputs) and the secret key. It is called the target node. We consider now a sequence of input patterns P 0 , P 1 , . . . , P n that generate the transitions T 1 (P 0 → P 1 ), T 2 (P 1 → P 2 ), . . . , T n (P n-1 → P n ) on the circuit primary inputs. A logic simulation of the circuit while monitoring the target node allows classifying these input transitions in two sets, according to a guess on the key:

• P A , composed by the transitions that make the target node to commute from 0 to 1 and therefore that make the target gate to consume current; • P B , composed by the transitions that do not lead the target gate to participate to the power consumed by the circuit (i.e., transitions from 0 to 0, 1 to 1 and 1 to 0 on the target node).

Figure 2 represents the power consumption of the device when stimulated by numerous input vectors. We assume here that the guess on the secret key is correct. In other word, the simulation is performed with the key actually used in the circuit from which power consumptions are collected. Each rectangle represents the total power consumed by the circuit when a new vector is applied to the inputs. In this figure and just for clarity of explanation, the power consumption is represented by a rectangle corresponding to the average of the consumption over the transition time. The set of transitions on the circuit inputs is splitted in the two sets: in the left part there are the P A transitions and the related consumptions while in the right part there are the P B transitions and their corresponding consumptions. A part of the power consumption related to the transitions belonging to P A is due to the power consumed by the target gate (shaded rectangles). Obviously, the commutation from 0 to 1 of non-target nodes also contributes to the power consumption of the circuit but input transitions that lead to such commutations are assumed to be evenly distributed to sets P A and P B . If a large number of transitions are considered, mean consumptions related to sets P A and P B are almost equal, except for the contribution of the target node. In other words, since the two sets are classified in such a way that the set P A always leads to a component of power consumption that is not present in the set P B , the difference between the two mean powers computed from set P A and set P B must show a noticeable difference.

Fig. 2. Power consumption after pattern partitioning

During a DPA attack, the target node is chosen in such a way that it depends on a small part of the key only, so that all the key guesses can be considered.

For each key guess, the two sets P A and P B are created according to the results of the logic simulation and the key guess under evaluation. The power mean values are calculated for each set using the simulated power traces of the circuit under attack for each transition. Finally, the differences of the mean values of the two sets are calculated. When the key guess is correct (and only in this case), P A actually includes the input transitions that lead to a transition 0 to 1 on the target node while P B does not include any of these transitions. The difference between the mean power obtained from P A and P B can be observed in this case. On the contrary, when the curves are classed in P A or P B independently from the actual value of the secret key, the two average curves do not present any noticeable difference. The classification process is illustrated in Figure 3 where K x is assumed to be the correct key, the one actually used during ciphering.

DPA attack

The computations sensitive to the DPA attack are the one involving the coordinates of the points P and Q. As a consequence, the DPA attack could only be done in the addition step. Indeed, the doubling step does not involve any computation between the coordinates of P and Q, the operations are multiplications by constant.

Without protection, the DPA attack is a threat against the addition step, whenever the secret is (point P or Q). According to the algorithm of ScalarMult, the argument Q of the pairing is in fact a multiple of the point Q and the point P is fixed.

The target of the DPA attack is the computation of x 0 and z 0 in the differential addition algorithm. In order to compute x 0 , we have to perform a multiplication between (x 2 P + z 2 P ) and (x 2 Q + z 2 Q ). Suppose that the point P is public and that Q is secret, we know the value of (x 2 P + z 2 P ) and the value (

x 2 Q + z 2 Q ) is secret.
We perform the DPA attack against the multiplication (x 2

P + z 2 P ) × (x 2 Q + z 2 Q ).
Assuming that the multiplication is implemented using the Schoolbook method, the guesses on the value (x 2 Q + z 2 Q ) can be done by words of 32 or 64 bits and begin by the less significant bits. The result of the DPA attack against x 0 is (x 2 Q + z 2 Q ). During the computation of x 0 the Differential Addition Algorithm would give us the value

x Q 2 + z Q 2 .
In parallel, another DPA attack during the computation of z 0 would give the value

x Q 2 -z Q 2 .
Once we have these two values, it is easy to extract x Q 2 and z Q 2 , which gives 4 possible couples for the coordinates of point Q.

As the Differential Addition Algorithm is symmetric in the coordinates of P and Q, the same attack is efficient if the point P is secret and Q is public.

Classical countermeasures presented in the case of pairing over elliptic curves can easily be adapted for the computation of pairings over Theta functions [START_REF] Mrabet | Fault attacks on pairing based cryptography: A state of the art[END_REF]. A native countermeasure is the homogeneity of the projective coordinates. Indeed, the point P = (x P : z P ) is also the point (λx P : λz Q ), for λ a non zero integer and Q = (βx Q , βy Q ), for β a non zero integer. The main hypothesis of a DPA attack is that the secret is the same for several executions of the algorithm. So, if we modify the coordinates of the secret for each execution the DPA attack can no longer be performed.

An efficient countermeasure consists in multiplying the coordinates of P by a random non zero integer λ and the coordinates Q by a random β for every iteration of the ScalarMult algorithm. This countermeasure is a good protection against any differential attack (power or electromagnetic) and consequently a protection for the Tate (and the Weil) pairing.

Description of the fault attack

The goal of a fault injection attack is to provoke mistakes during the calculation of an algorithm, for example by modifying the internal memory, in order to reveal sensitive data. This attack needs a very precise positioning and an expensive apparatus to be performed. Nevertheless, current technologies could allow for this attack [START_REF] Habing | The use of lasers to simulate radiation-induced transients in semiconductor devices and circuits[END_REF]. the faults can be performed using a laser or electromagnetic emissions [START_REF] Mulder | Differential power and electromagnetic attacks on a FPGA implementation of elliptic curve cryptosystems[END_REF].

We follow the scheme of attack described in [START_REF] Page | A fault attack on pairing-based cryptography[END_REF] and completed in [START_REF] Mrabet | What about vulnerability to a fault attack of the miller's algorithm during an identity based protocol[END_REF]. We assume that the pairing is used during an Identity Based Protocol, the secret point is introduced in a smart card or an electronic device and is a parameter of the pairing. In order to find the secret, we modify the number of iterations in the Tate pairing algorithm by the following way.

First of all, we have to find the flip-flops belonging to the counter of the number of iterations (i.e. log 2 (s)) in the Tate pairing algorithm. This step can be done by using reverse engineering procedures. In classical architecture, the counter is divided into small pieces of 32 or 64 bits (according to the size of a word). To find it, we make one normal execution of the algorithm, without any fault. Then we choose one piece of the counter and provoke disturbances in order to modify it and consequently the number of iterations of the algorithm. For example the disturbance can be induced by a laser [START_REF] Anderson | Tamper resistance: a cautionary note[END_REF]. Nowadays lasers are thin enough to make this attack realistic [START_REF] Habing | The use of lasers to simulate radiation-induced transients in semiconductor devices and circuits[END_REF]. Counting the clock cycles, we are able to know how many iterations the Tate pairing loop has done. Each time, we record the value of the pairing loop and the number of iterations we made.

Fault attack

State of the art The principle of fault attacks in pairing based cryptography consists to force the algorithm to stop by reducing the number of iterations and by finding the results of two consecutive iterations τ and τ +1. The results of these two executions give equations that allow to find the secret. In the case of pairings over Theta function, the fault attack consists in finding one the coordinates involved during the computation of ScalarMult( P + Q, Q, P ). The ScalarMult algorithm is composed by the doubling and differential addition algorithms, As the coordinates of P are such that x P z P = 0, we can consider that α = 1 z P and write the equation λ = β((x P ) 2 + 1) + γ((x P ) 2 -1), which leads to

(x P ) 2 = λ -β + γ β -γ .
Up to the sign, we find one coordinate of a representative of the point P and from that point we can find the secret.

If the secret is the point Q The formulas are symmetric in the coordinates of P and jQ. Following the same scheme, we obtain z 1 for j not equal to the order of Q and that gives the coordinates of a representative of jQ, knowing j. To find the coordinates of Q, we just have to compute the inverse of j mod (l) and after that we can recover the coordinates of the point Q.

The condition to perform the fault attack when Q is secret is to stop the computation before j = l, as Q is a point of order l. This is a simplification of the fault attack against the pairing considering Miller's algorithm, because we only need one faulty execution of ScalarMult.

Considering the computation of the Tate pairing Recall that the algorithm to compute the Tate pairing is

e T = ScalarMult( P + Q, Q, P , l) i ScalarMult( Q, Q, 0, l) i 0 i P i .
The attacks described above for ScalarMult can be directly adapted to the Tate pairing (and also to the Weil pairing). For efficiency reasons, the computation of ScalarMult( P + Q, Q, P , l) i and ScalarMult( Q, Q, 0, l) i would certainly be implemented in parallel. As a consequence, the fault attack forces the algorithm to stop after the same number of iterations and the result ScalarMult( P + Q, Q, P , j) i and ScalarMult( Q, Q, 0, j) i , for the same integer j. For both cases, either P secret or either Q, the homogeneity of projective coordinates is a trapdoor that gives information about the secret. Let P be the secret point and Q be public, then the coordinate P i is also secret, but the homogeneity of the projective coordinates allows us to consider that for example the z coordinate is set to 1, exactly like in the attack described above. We just have to be careful to set the same coordinate to 1 in both calls to ScalarMult, the z one for example. The Equation (1) would give a slightly different system but linear and easily solvable. The method is the same if the point Q is secret.

Countermeasure to the fault attack Considering that the fault attack uses the homogeneity of the coordinates, the countermeasure to the DPA attack is clearly not sufficient. We have to present another countermeasure and this countermeasure must protect the pairing algorithm from the fault and the DPA attacks. So, we have to modify the coordinates of the point P and Q for every pairing computation. A solution would be to use the bilinearity of the pairing [START_REF] Mrabet | Fault attacks on pairing based cryptography: A state of the art[END_REF]. Indeed, if we compute the Tate pairing between the points P and Q, the bilinearity induces that e T (P, Q) = e T δP, (δ -1 mod (l))Q , for a non zero integer δ. The cost of this countermeasure consists in two exponentiations over the variety A(K).

Conclusion

We analyze the weaknesses of the pairings over Theta function with respect to side channel attacks. We consider the differential power analysis and the fault attack. The scheme of the differential power analysis embraces the differential electromagnetic attack and the correlation power analysis. The ScalarMult algorithm is sensitive to the DPA attack, but the homogeneity of the projective coordinates provides a native countermeasure. Unfortunately, the homogeneity is a trapdoor for the fault attack. The fault attack against pairing over Theta functions is easier than in the case of pairings using the Miller's algorithm. We only need one fault to recover the secret. As the homogeneity of the coordinates is no longer a countermeasure, we present an alternative countermeasure. This countermeasure relies on the bilinearity of pairings and is efficient for all side channel attacks.

Fig. 1 .

 1 Fig. 1. Doubling and Differential Addition Algorithms

Fig. 3 .

 3 Fig. 3. Pattern classification for several key guesses

Acknowledgment

The author wishes to thank the anonymous referees for their helpful remarks and comments.

The author wishes to acknowledge support from French project ANR INS 2012 SIMPATIC.

the result of of ScalarMult are the coordinates of P + lQ. The fault attack consists in reducing the number of iteration of ScalarMult. To do so, we can use a laser or electromagnetic emissions to locally modify the register storing l. The target of this attack is then a smart card or a FPGA. Let τ be the reduced number of iteration performed by ScalarMult. In practice τ can be recovered using the number of clock cycles made by the algorithm. Indeed, we know the binary decomposition of l, we are then able to find when the algorithm stops and how many iterations were done. Let j be the integer composed by the τ most significant bits of l, which is public. The fault attack for pairing over Theta function is easier than the classical fault attack in pairing based cryptography. We need only one fault and the result of this faulty execution to find the secret involved in the ScalarMult algorithm

The result of the pairing is the coordinates of the point P + lQ. We can suppose that we obtain one of the two coordinates, for example the coordinate z. With the z coordinate of the result, we are able to recover the secret argument of the pairing computations.

Suppose that we can recover the coordinate z of the point P + jQ, for j < l. As the points P and Q are of order l by construction, the result of the pairing itself cannot give us information. That is why we need to provoke a fault reducing the number of iterations of the ScalarMult algorithm.

Let z 1 = z P +jQ , where j is a known integer. The equation of z 1 is the following

where • P = (x P , z P ) = (x, z) (with the notations introduced above)

• A and B are constants.

We first describe the attack of the algorithm ScalarMult, before considering the fault attack against the whole Tate pairing algorithm.

If the secret is the point P Suppose that the point P is secret. The fault attack provide us z 1 , the values A, B, x j and z j are public. All together, they verify the equation

, where the data in bold (λ, β, γ) are known. The coordinates x P and z P are the values we are looking for.

The point P is given in projective coordinates, this equality is correct for any representative of the point P , i.e. for any α = 0 we have that λ(αz P ) = β((αx P ) 2 + (αz P ) 2 ) + γ((αx P ) 2 -(αz P ) 2 ).