Algorithmic number theory dates back to the dawn of mathematics
itself, *cf.* Eratosthenes's sieve to enumerate consecutive prime numbers.
With the
arrival of computers, previously unsolvable problems have come into reach,
which has boosted the development of more or less practical algorithms
for essentially all number theoretic problems. The field is now mature
enough for a more computer science driven approach, taking into account
the theoretical complexities and practical running times of the algorithms.

Concerning the lower level multiprecision arithmetic, folklore has asserted for a long time that asymptotically fast algorithms such as Schönhage–Strassen multiplication are impractical; nowadays, however, they are used routinely. On a higher level, symbolic computation provides numerous asymptotically fast algorithms (such as for the simultaneous evaluation of a polynomial in many arguments or linear algebra on sparse matrices), which have only partially been exploited in computational number theory. Moreover, precise complexity analyses do not always exist, nor do sound studies to choose between different algorithms (an exponential algorithm may be preferable to a polynomial one for a large range of inputs); folklore cannot be trusted in a fast moving area such as computer science.

Another problem is the reliability of the computations; many number
theoretic algorithms err with a
small probability, depend on unknown constants or rely on a Riemann
hypothesis. The correctness of their output can either be ensured by a
special design of the algorithm itself (slowing it down) or by an *a
posteriori* verification. Ideally, the algorithm outputs a certificate,
providing an independent *fast* correctness proof. An example is integer
factorisation, where factors are hard to obtain but trivial to
check; primality proofs have initiated sophisticated generalisations.

One of the long term goals of the Lfant project team is to make an inventory of the major number theoretic algorithms, with an emphasis on algebraic number theory and arithmetic geometry, and to carry out complexity analyses. So far, most of these algorithms have been designed and tested over number fields of small degree and scale badly. A complexity analysis should naturally lead to improvements by identifying bottlenecks, systematically redesigning and incorporating modern asymptotically fast methods.

Reliability of the developed algorithms is a second long term goal of our project team. Short of proving the Riemann hypothesis, this could be achieved through the design of specialised, slower algorithms not relying on any unproven assumptions. We would prefer, however, to augment the fastest unproven algorithms with the creation of independently verifiable certificates. Ideally, it should not take longer to check the certificate than to generate it.

All theoretical results are complemented by concrete reference implementations in Pari/Gp, which allow to determine and tune the thresholds where the asymptotic complexity kicks in and help to evaluate practical performances on problem instances provided by the research community. Another important source for algorithmic problems treated by the Lfant project team is modern cryptology. Indeed, the security of all practically relevant public key cryptosystems relies on the difficulty of some number theoretic problem; on the other hand, implementing the systems and finding secure parameters require efficient algorithmic solutions to number theoretic problems.

Modern number theory has been introduced in the second half of the 19th
century by Dedekind, Kummer, Kronecker, Weber and others, motivated by
Fermat's conjecture: There is no non-trivial solution in integers to the
equation

The solution requires to augment the integers by *algebraic
numbers*, that are roots of polynomials in *number
field* consists of the rationals to which have been added finitely
many algebraic numbers together with their sums, differences, products
and quotients. It turns out that actually one generator suffices, and
any number field *algebraic integers*, “numbers without denominators”,
that are roots of a monic polynomial. For instance, *ring of integers* of

Unfortunately, elements in *ideals*, subsets of *principal*, that is,
generated by one element, so that ideals and numbers are essentially
the same. In particular, the unique factorisation of ideals then
implies the unique factorisation of numbers. In general, this is not
the case, and the *class group* *class number*

Using ideals introduces the additional difficulty of having to deal
with *fundamental units*. The *regulator*

One of the main concerns of algorithmic algebraic number theory is to
explicitly compute these invariants (

The *analytic class number formula* links the invariants
*generalised Riemann hypothesis
(GRH)*, which remains unproved even over the rationals, states that
any such

When

Algebraic curves over finite fields are used to build the currently
most competitive public key cryptosystems. Such a curve is given by
a bivariate equation *elliptic curves* of equation
*hyperelliptic curves* of
equation

The cryptosystem is implemented in an associated finite
abelian group, the *Jacobian* *rational function field* with subring *function field* of *coordinate ring*

The size of the Jacobian group, the main security parameter of the
cryptosystem, is given by an *genus*

The security of the cryptosystem requires more precisely that the
*discrete logarithm problem* (DLP) be difficult in the underlying
group; that is, given elements

For any integer *Weil pairing* *Tate-Lichtenbaum pairing*, that is more difficult to define,
but more efficient to implement, has similar properties. From a
constructive point of view, the last few years have seen a wealth of
cryptosystems with attractive novel properties relying on pairings.

For a random curve, the parameter

Complex multiplication provides a link between number fields and
algebraic curves; for a concise introduction in the elliptic curve case,
see Sect. 1.1, for more background material,
. In fact, for most curves *CM field*. The CM field
of an elliptic curve is an imaginary-quadratic field *Hilbert class field*

Algebraically, *Galois* if *Galois group* *abelian* extension is a Galois extension with abelian Galois
group.

Analytically, in the elliptic case *singular value* *modular* function

The same theory can be used to develop algorithms that, given an
arbitrary curve over a finite field, compute its

A generalisation is provided by *ray class fields*; these are
still abelian, but allow for some well-controlled ramification. The tools
for explicitly constructing such class fields are similar to those used
for Hilbert class fields.

Chloe Martindale defended her PhD thesis on *Isogeny Graphs, Modular Polynomials, and Applications*.

Antonin Riffaut defended his PhD thesis on *Effective computation of
special points*.

A new release of Pari/Gp, 2.11.0, has been published. This is a major stable release ending a development cycle which started in November 2016; it includes among others an extensive new package for modular forms.

2018 was also a year with more workshops on Pari/Gp than ever:
Besides two general workshops uniting developers and users, organised
together with the universities of Besançon and Rome in the respective
cities, the team participated with lectures on Pari/Gp at the
École jeunes chercheurs en théorie des nombres à Besançon
(https://

*Another Pairing Implementation in PARI*

Keywords: Cryptography - Computational number theory

Scientific Description: Apip , Another Pairing Implementation in PARI, is a library for computing standard and optimised variants of most cryptographic pairings.

The following pairings are available: Weil, Tate, ate and twisted ate, optimised versions (à la Vercauteren–Hess) of ate and twisted ate for selected curve families.

The following methods to compute the Miller part are implemented: standard Miller double-and-add method, standard Miller using a non-adjacent form, Boxall et al. version, Boxall et al. version using a non-adjacent form.

The final exponentiation part can be computed using one of the following variants: naive exponentiation, interleaved method, Avanzi–Mihailescu's method, Kato et al.'s method, Scott et al.'s method.

Part of the library has been included into Pari/Gp proper.

Functional Description: APIP is a library for computing standard and optimised variants of most cryptographic pairings.

Participant: Jérôme Milan

Contact: Andreas Enge

URL: http://

*Abelian Varieties and Isogenies*

Keywords: Computational number theory - Cryptography

Functional Description: AVIsogenies is a Magma package for working with abelian varieties, with a particular emphasis on explicit isogeny computation.

Its prominent feature is the computation of (l,l)-isogenies between Jacobian varieties of genus-two hyperelliptic curves over finite fields of characteristic coprime to l, practical runs have used values of l in the hundreds.

It can also be used to compute endomorphism rings of abelian surfaces, and find complete addition laws on them.

Participants: Damien Robert, Gaëtan Bisson and Romain Cosset

Contact: Damien Robert

Keyword: Arithmetic

Functional Description: The Cm software implements the construction of ring class fields of imaginary quadratic number fields and of elliptic curves with complex multiplication via floating point approximations. It consists of libraries that can be called from within a C program and of executable command line applications.

Release Functional Description: Features - Precisions beyond 300000 bits are now supported by an addition chain of variable length for the -function. Dependencies - The minimal version number of Mpfr has been increased to 3.0.0, that of Mpc to 1.0.0 and that of Pari to 2.7.0.

Participant: Andreas Enge

Contact: Andreas Enge

*Computation of Igusa Class Polynomials*

Keywords: Mathematics - Cryptography - Number theory

Functional Description: Cmh computes Igusa class polynomials, parameterising two-dimensional abelian varieties (or, equivalently, Jacobians of hyperelliptic curves of genus 2) with given complex multiplication.

Participants: Andreas Enge, Emmanuel Thomé and Regis Dupont

Contact: Emmanuel Thomé

Keyword: Number theory

Functional Description: Cubic is a stand-alone program that prints out generating equations for cubic fields of either signature and bounded discriminant. It depends on the Pari library. The algorithm has quasi-linear time complexity in the size of the output.

Participant: Karim Belabas

Contact: Karim Belabas

URL: http://

Keyword: Number theory

Functional Description: Euclid is a program to compute the Euclidean minimum of a number field. It is the practical implementation of the algorithm described in [38] . Some corresponding tables built with the algorithm are also available. Euclid is a stand-alone program depending on the PARI library.

Participants: Jean-Paul Cerri and Pierre Lezowski

Contact: Jean-Paul Cerri

URL: http://

Keywords: Computational geometry - Computational number theory

Functional Description: KleinianGroups is a Magma package that computes fundamental domains of arithmetic Kleinian groups.

Participant: Aurel Page

Contact: Aurel Page

URL: http://

Keyword: Arithmetic

Functional Description: Mpc is a C library for the arithmetic of complex numbers with arbitrarily high precision and correct rounding of the result. It is built upon and follows the same principles as Mpfr. The library is written by Andreas Enge, Philippe Théveny and Paul Zimmermann.

Release Functional Description: Fixed `mp\_pow`, see
http://`\#18257`: Switched to libtool 2.4.5.

Participants: Andreas Enge, Mickaël Gastineau, Paul Zimmermann and Philippe Théveny

Contact: Andreas Enge

Keyword: Arithmetic

Functional Description: Mpfrcx is a library for the arithmetic of univariate polynomials over arbitrary precision real (Mpfr ) or complex (Mpc ) numbers, without control on the rounding. For the time being, only the few functions needed to implement the floating point approach to complex multiplication are implemented. On the other hand, these comprise asymptotically fast multiplication routines such as Toom-Cook and the FFT.

Release Functional Description: - new function `produc\_an\_hecke`
- improved memory consumption for unbalanced FFT multiplications

Participant: Andreas Enge

Contact: Andreas Enge

Keyword: Computational number theory

Functional Description: Pari/Gp is a widely used computer algebra system designed for fast computations in number theory (factorisation, algebraic number theory, elliptic curves, modular forms ...), but it also contains a large number of other useful functions to compute with mathematical entities such as matrices, polynomials, power series, algebraic numbers, etc., and many transcendental functions.

Participants: Andreas Enge, Hamish Ivey-Law, Henri Cohen and Karim Belabas

Partner: CNRS

Contact: Karim Belabas

Functional encryption (FE) is an advanced cryptographic primitive which allows, for a single encrypted message, to finely control how much information on the encrypted data each receiver can recover. To this end many functional secret keys are derived from a master secret key. Each functional secret key allows, for a ciphertext encrypted under the associated public key, to recover a specific function of the underlying plaintext.

Since constructions for general FE that appear in the past five years are far from practical, the problem arose of building efficient FE schemes for restricted classes of functions; and in particular for linear functions, (i.e. the inner product functionality). Such constructions yield many practical applications, while developing our understanding of FE.

Though such schemes had already been conceived in the past three years (Abdalla *et al.* 2015, Agrawal *et al.* 2016), they all suffered of practical drawbacks. Namely the computation of inner products modulo a prime are restricted, in that they require that the resulting inner product be small for decryption to be efficient. The only existing scheme that overcame this constraint suffered of poor efficiency due in part to very large ciphertexts.
This work overcomes these limitations and we build the first FE schemes for inner products modulo a prime that are both efficient and recover the result whatever its size.

To this end, Castagnos *et al.* introduce two new cryptographic assumptions. These are variants of the assumptions used for the Castagnos-Laguillaumie encryption of 2015. This supposes the existence of a cyclic group

From these assumptions Castagnos *et al.* construct generic, linearly homomorphic encryption schemes over a field of prime order which are semantically secure under chosen plaintext attacks.
They then use the homomorphic properties of the above schemes to construct generic inner product FE schemes over the integers and over fields of prime order. They thereby provide constructions for inner product FE modulo a prime

This paper was presented at the ASIACRYPT Conference 2018, and is part of the Alambic project.

In collaboration with Pierre Lezowski, Jean-Paul Cerri has studied norm-Euclidean properties of totally definite quaternion fields over number fields. Building on their previous work about number fields, they have proved that the Euclidean minimum and the inhomogeneous minimum of orders in such quaternion fields are always equal. Besides, they are rational under the hypothesis that the base number field is not quadratic. This single remaning open case corresponds to the similar open case remaining for real number fields.

They also have extended Cerri's algorithm for the computation of the upper part of the norm-Euclidean spectrum of a number field to this noncommutative context. This algorithm has allowed to compute the exact value of the norm-Euclidean minimum of orders in totally definite quaternion fields over a quadratic number field. This has provided the first known values of this minimum when the base number field has degree strictly greater than 1.

Consequently, both theoretical and practical
milestones set in the previous quadrennial report
were reached. These results are
presented in , due to appear in
*International Journal of Number Theory*.

In , A. Bartel and A. Page describe all possible
actions of groups of automorphisms on the homology of 3-manifolds, and prove
that for every prime

More precisely: if

Lenstra and Guruswami described number field analogues of the algebraic geometry codes of Goppa. Recently, Maire and Oggier generalised these constructions to other arithmetic groups: unit groups in number fields and orders in division algebras; they suggested to use unit groups in quaternion algebras but could not completely analyse the resulting codes. Maire and Page prove that the noncommutative unit group construction yields asymptotically good families of codes for division algebras of any degree, and estimate the smallest possible size of the alphabet in terms of the degree of the algebra.

De Feo, Kieffer and Smith give algorithmic improvements that accelerate key exchange in this framework, and explore the problem of generating suitable system parameters for contemporary pre-and post-quantum security that take advantage of these new algorithms. They prove the session-key security of this key exchange in the Canetti-Krawczyk model, and the IND-CPA security of the related public-key encryption scheme, under reasonable assumptions on the hardness of computing isogeny walks. This system admits efficient key-validation techniques that yield CCA-secure encryption, thus providing an important step towards efficient post-quantum non-interactive key exchange (NIKE).

A complementary package using modular symbols is used in
by
Karim Belabas, Dominique Bernardi and Bernadette Perrin-Riou to
compute Manin's constant and the modular degree of elliptic curves
defined over

In https://

https://

The Alambic project is a research project formed by members of the Inria Project-Team CASCADE of ENS Paris, members of the AriC Inria project-team of ENS Lyon, and members of the CRYPTIS of the university of Limoges. G. Castagnos is an external member of the team of Lyon for this project.

Non-malleability is a security notion for public key cryptographic encryption schemes that ensures that it is infeasible for an adversary to modify ciphertexts into other ciphertexts of messages which are related to the decryption of the first ones. On the other hand, it has been realized that, in specific settings, malleability in cryptographic protocols can actually be a very useful feature. For example, the notion of homomorphic encryption allows specific types of computations to be carried out on ciphertexts and generate an encrypted result which, when decrypted, matches the result of operations performed on the plaintexts. The homomorphic property can be used to create secure voting systems, collision-resistant hash functions, private information retrieval schemes, and for fully homomorphic encryption enables widespread use of cloud computing by ensuring the confidentiality of processed data.

The aim of the Alambic project to investigate further theoretical and practical applications of malleability in cryptography. More precisely, this project focuses on three different aspects: secure computation outsourcing and server-aided cryptography, homomorphic encryption and applications and << paradoxical >> applications of malleability.

The

The CLap–CLap ANR project aims at accelerating the expansion of the

draw a conjectural picture of the (still hypothetical)

compute many deformation spaces of Galois representations and make the bridge with deformation spaces of representations of reductive groups,

design new algorithms for computations with Hilbert and Siegel modular forms and their associated Galois representations.

This project will also be the opportunity to contribute to the development of the mathematical software SageMath and to the expansion of computational methodologies.

Title: OpenDreamKit

Program: H2020

Duration: January 2016 - December 2020

Coordinator: Nicolas Thiéry

Inria contact: Karim Belabas

Description
http://

OpenDreamKit is a Horizon 2020 European Research Infrastructure project (#676541) that will run for four years, starting from September 2015. It provides substantial funding to the open source computational mathematics ecosystem, and in particular popular tools such as LinBox, MPIR, SageMath, GAP, Pari/GP, LMFDB, Singular, MathHub, and the IPython/Jupyter interactive computing environment.

**International Laboratory for Research in Computer Science and Applied Mathematics**

Associate Team involved in the International Lab:

Title: (Harder Better) FAster STronger cryptography

International Partner

Université des Sciences et Techniques de Masuku (Gabon) - Tony Ezome and the PRMAIS project

Start year: 2017

The project aims to develop better algorithms for elliptic curve cryptography with prospect of the two challenges ahead: - securing the internet of things - preparing towards quantum computers.

Elliptic curves are currently the fastest public-key cryptosystem (with a key size that can fit on embeded devices) while still through a different mode of operation beeing (possibly) able to resist quantum based computers.

Activities for this year involved:

Tony Ezome organised a Cimpa school on Courbes algébriques pour une arithmétique efficace des corps finis from 17/11/2018 - 30/11/2018 in Ziguinchor (Sénégal), Institution Université Assane Seck de Ziguinchor.

Abdoul Asiz Ciss and Damien Robert represented the team at the Journées du Lirima. One of the suggestion was to find industrial collaborations in Africa, especially in Senegal. Ongoing work is done by the team to find such a collaboration, especially on the new challenges of post-quantum cryptography.

Abdoulaye Maiga visited in Bordeaux to work with Damien Robert from 22/10/2018 to 18/01/2019. Tony Ezome and Mohamadou Sall visited from 08/12/2018 to 22/12/2018.

Activities for this year involved
the funding of Luca De Feo to speak at the EMA
“Mathématiques pour la Cryptographie Post-quantique et
Mathématiques pour le Traitement du Signal”,
organised by Djiby Sow and Abdoul Asiz Ciss organised an EMA
at the
École Polytechnique de Thiès (Sénégal) from May 10 to May 23,
about
“Cryptographie à base d'isogénies”;
the visit of Abdoulaye Maiga to the LFANT team where he worked
with Damien Robert to find absolute invariants of good reduction modulo 2
for abelian surfaces; and the organisation by
Damien Robert of a workshop in Bordeaux with most of the team
members from September 04 to September 08.
The slides or proceedings are available at
https://

The team is used to collaborate with Leiden University through the ALGANT program for PhD joint supervision.

Eduardo Friedman (U. of Chile), long term collaborator of K. Belabas and H. Cohen is a regular visitor in Bordeaux (about 1 month every year).

Nicolas Mascot (American University of Beirut, Lebanon) visited the team for a week (8-12/01/2018).

Alex Bartel (University of Glasgow, UK) visited the team for two weeks (27/03/2018 to 07/04/2018).

Takashi Fukuda (Nihon University, Japan) visited the team for two months (20/01/2018 to 25/03/2018)

Tony Ezome (Université des Sciences et Techniques de Masuku) and Mohamadou Sall (Dakar) visited the team for two weeks in December. Abdoul Aziz (Dakar) visited the team for one week in September.

Abdoulaye Maiga visited the team for three months, from October to January 2019.

Researchers visiting the team to give a talk to the team seminar include Elie Eid (Université de Rennes), Jean-François Biasse (University of South Florida), Francesco Battistoni (University of Milan), Alex Bartel (Glasgow University), Tristan Vaccon (Université de Limoges), and Takashi Fukuda (Nihon University).

A. Page visited Alex Bartel (University of Glasgow, UK) for two weeks (16-27/07/2018) and Michael Lipnowski (McGill University, Montreal, Canada) for two weeks (10-23/11/2018).

A. Page and Alex Bartel did a research stay in Oberwolfach (Allemagne) with the Research In Pairs programme for three weeks (14/10/2018-3/11/2018).

K. Belabas acts on the editorial board of
*Journal de Théorie des Nombres de Bordeaux* since 2005
and of *Archiv der Mathematik* since 2006.

X. Caruso is an editor and one of the founder of the journal
*Annales Henri Lebesgue*.

H. Cohen is an editor for the Springer book series
*Algorithms and Computations in Mathematics (ACM)*.

J.-M. Couveignes is a member of the editorial board (scientific committee)
of the *Publications mathématiques de Besançon* since 2010.

From January 2015 to September 2018 J.-M. Couveignes was a member of the scientific council of the Fondation Mathématique de Paris.

A. Enge is an editor of *Designs, Codes and Cryptography*
since 2004.

A. Page: *Algorithms for the cohomology of compact arithmetic
manifolds and Hecke operators* in the Simons collaboration conference
*Arithmetic Geometry, Number Theory, and Computation*, MIT (Boston,
US), August 20-24, 2018.

K. Belabas is a member of the 'conseil scientifique' of the Société Mathématique de France

Since January 2017, A. Enge is “délégué scientifique” of the Inria research centre Bordeaux–Sud-Ouest. As such, he is also a designated member of the “commission d'évaluation” of Inria.

Since January 2015, K. Belabas is vice-head of the Math Institute (IMB). He also leads the computer science support service (“cellule informatique”) of IMB and coordinates the participation of the institute in the regional computation cluster PlaFRIM.

He is an elected member of “commission de la recherche” in the academic senate of Bordeaux University.

He is a member of the “Conseil National des Université” (25th section, pure mathematics).

J.-P. Cerri is an elected member of the scientific council of the Mathematics Institute of Bordeaux (IMB) and responsible for the bachelor programme in mathematics and informatics.

From January 2015 until January 2019, J.-M. Couveignes was the head of the Math Institute (IMB). He is head of the Scientific Committee of the Albatros (ALliance Bordeaux universities And Thales Research in AviOnicS) long term cooperation between Inria, Bordeaux-INP, Université de Bordeaux and CNRS.

Master : G. Castagnos, *Cryptanalyse*,
60h, M2, University of Bordeaux, France;

Master : G. Castagnos, *Cryptologie avancée*,
30h, M2, University of Bordeaux, France;

Master : G. Castagnos, *Courbes elliptiques*,
60h, M2, University of Bordeaux, France;

Master : D. Robert, *Courbes elliptiques*,
60h, M2, University of Bordeaux, France;

Master : K. Belabas, *Computer Algebra*,
91h, M2, University of Bordeaux, France;

Master : J.-M. Couveignes, *Algorithmic Arithmetic*,
30h, M2, University of Bordeaux, France;

Master : J.-M. Couveignes, *Modules, espaces quadratiques*,
30h, M1, University of Bordeaux, France;

Licence : Jean-Paul Cerri, Algèbre linéaire 2, 51h TD, L2, Université de Bordeaux, France

Licence : Jean-Paul Cerri, Arithmétique et Cryptologie, 24h TD, L3, Université de Bordeaux, France

Licence : Jean-Paul Cerri, Structures algébriques 2, 35h TD, L3, Université de Bordeaux, France

Master : Jean-Paul Cerri, Cryptologie, 60h TD, M1, Université de Bordeaux, France

Master : Jean-Paul Cerri, 3 TER, Université de Bordeaux, France

Licence : Jean Kieffer, Mathématiques pour la biologie, 64h TD, L1, Université de Bordeaux, France

PhD: Chloe Martindale, *Isogeny graphs, modular polynomials, and applications*,
defended in 2018, supervised by A. Enge and Marco Streng (Universiteit Leiden).

PhD: Antonin Riffaut
*Calcul effectif de points spéciaux*, defended in 2018,
supervised by Y. Bilu and K. Belabas.

PhD in progress : Ida Tucker, *Design of new advanced cryptosystems from homomorphic building blocks*, since October 2017, supervised by Guilhem Castagnos and Fabien Laguillaumie

PhD in progress: Abdoulaye Maiga,
*Computing canonical lift of genus 2 hyperelliptic curves*,
University Dakar,
supervised by Djiby Sow, Abdoul Aziz Ciss and D. Robert.

PhD in progress: Jared Asuncion, *Class fields of complex multiplication fields*, since September 2017, supervised by A. Enge and Marco Streng (Universiteit Leiden).

PhD in progress: Emmanouil Tzortzakis
*Algorithms for $\mathbb{Q}$-curves*,
supervised by K. Belabas, P. Bruin and B. Edixhoven.

PhD in progress: Pavel Solomatin
*Topics on $L$-functions*,
supervised by B. de Smit and K. Belabas.

PhD in progress: Jean Kieffer
*Isogénies et endomorphismes de variétés abéliennes*,
supervised by D. Robert and A. Page.

Master thesis: Amandine Malonguemfo Teagho *Algorithms for
isometries of lattices*, supervised by A. Page.

Master thesis: William Dallaporta *Bhargava's theory
and parametrization of algebraic structures*, supervised by K. Belabas.

X. Caruso has written a report for the doctoral dissertation by
Robin Bartlett, King's College in London:
*On the reductions of some crystalline representations*.

A. Enge has written a report for the doctoral dissertation by
Benjamin Wesolowski, École polytechnique fédérale de Lausanne:
*Arithmetic & Geometric Structures in Cryptography*.

A. Enge has written a report for the professorial dissertation by
Luca De Feo, Université de Versailles–Saint Quentin:
*Exploring Isogeny Graphs*.

X. Caruso published an article entitled *Polynômes tordus*
in the journal *Au fil des maths de la maternelle à l'université...*
edited by APMEP.

H. Cohen wrote in an introduction to Modular forms, which has been published in the book Notes from the International School on Computational Number Theory.

D. Robert is a member of the jury of Agregations de Mathematiques. He is also the codirector with Alain Couvreur of the option “calcul formel” of the Modelisation part of the oral examination.

24/02/2018 in Olot (Spain), A. Page, with the other participants of Sage Days 93: one day for 20 local high school students to explore mathematical problems.

24/05/2018, A. Page: Unithé ou café on the mathematics of wireless
communications: *Méthodes algébriques et géométriques pour les
communications sans fil : comment l'espace hyperbolique peut-il améliorer
vos appels téléphoniques ?*

30/05/2018, A. Page: in Poitiers half a day meeting with junior school students who took part in the Al-Kindi competition; introduction to cryptography.

27/09/2018 D. Robert and A. Page: demonstration stand on graph-based cryptography at the Inria BSO Party Day.

9-11/10/201 A. Page: Fête de la Science at Inria Bordeaux, activity on cryptography (7 groups of students).

13/10/2018 D. Robert and A. Page: demonstration stand on graph-based cryptography at the Inria BSO Open Day.

11/12/2018 A. Page: talk at the Inria BSO Comité des Projets
*Variations arithmétiques et algorithmiques sur le thème << Peut-on
entendre la forme d'un tambour? >>*