In , G. Castagnos, F. Laguillaumie and I. Tucker revisit a recent cryptographic primitive called Functional encryption for inner products (FE4IP).

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 $G$ where the decision Diffie-Hellman assumption holds together with a subgroup $F$ of $G$ where the discrete logarithm problem is easy. This setting allows to encode information in the exponent of the subgroup $F$, which can be efficiently recovered whatever its size.

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 $p$ that do not restrict the size of the inputs or of the resulting inner product, which are the most efficient such schemes to date.

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 $p$, there are 3-dimensional drums that sound the same but have different $p$-torsion in their homology. This completes previous work  by proving that the behaviour observed by computer experimentation was indeed a general phenomenon.

More precisely: if $M$ is a manifold with an action of a group $G$, then the homology group $H_1(M,\mathbb{Q})$ is naturally a $\mathbb{Q}\left[G\right]$-module, where $\mathbb{Q}\left[G\right]$ denotes the rational group ring. Bartel and Page prove that for every finite group $G$, and for every $\mathbb{Q}\left[G\right]$-module $V$, there exists a closed hyperbolic 3-manifold $M$ with a free $G$-action such that the $\mathbb{Q}\left[G\right]$-module $H_1(M,\mathbb{Q})$ is isomorphic to $V$. They give an application to spectral geometry: for every finite set $P$ of prime numbers, there exist hyperbolic 3-manifolds $N$ and $N^{\text{'}}$ that are strongly isospectral such that for all $p\in P$, the $p$-power torsion subgroups of $H_1(N,\mathbb{Z})$ and of $H_1(N^{\text{'}},\mathbb{Z})$ have different orders. They also show that, in a certain precise sense, the rational homology of oriented Riemannian 3-manifolds with a $G$-action "knows" nothing about the fixed point structure under $G$, in contrast to the 2-dimensional case. The main geometric techniques are Dehn surgery and, for the spectral application, the Cheeger-Müller formula, but they also make use of tools from different branches of algebra, most notably of regulator constants, a representation theoretic tool that was originally developed in the context of elliptic curves.

In , C. Maire and A. Page revisit a construction due to Lenstra and Guruswami by generalising them to unit groups of division algebras.

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.

In , L. De Feo, J. Kieffer and B. Smith revisit the ordinary isogeny-graph based cryptosystems of Couveignes and Rostovtsev–Stolbunov, long dismissed as impractical.

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).

In , A. Enge, F. Johansson and their coauthor W. Hart consider the problem of numerically evaluating one-dimensional $\theta$-functions and the elliptic $\eta$-function. They construct short addition sequences reaching an optimal number of $N+o\left(N\right)$ multiplications for evaluating the function as a sparse series with $N$ terms. The proof relies on the representability of specific quadratic progressions of integers as sums of smaller numbers of the same kind. For example, they show that every generalised pentagonal number $c>5$ can be written as $c=2a+b$, where $a$, $b$ are smaller generalised pentagonal numbers. They then give a baby-step giant-step algorithm that breaks through the theoretical barrier achievable with addition sequences, and which uses only $O(N/{\left(logN\right)}^r)$ multiplications for any $r>0$. These theoretical improvements also lead to an interesting speed-up in practice, and they have been integrated into the CM and the ARB software.

In , X. Caruso and A. Durand define a new family of linear codes which is a common generalization of Reed–Solomon codes on the one hand and Gabidulin codes on the other hand. Their construction works over an arbitrary field (not necessarily finite) equipped with an automorphism of finite order and a twisted derivation whose subfield of constants is sufficiently large. This setting allows for example the base field to be ${𝔽}_q\left(t\right)$ equipped with its natural derivation and then provides a new large family of interesting codes. Caruso and Durand then compute the minimal distance of their codes and design an efficient algorithm for decoding up to the half of the minimal distance.

In , F. Johansson and I. Blagouchine devise an efficient algorithm to compute the generalized Stieltjes constants ${\gamma }_n\left(a\right)$ to arbitrary precision with rigorous error bounds, for the first time achieving this with low complexity with respect to the order $n$. The algorithm consists of locating an approximate steepest descent contour and then evaluating the integral numerically in ball arithmetic using the Petras algorithm with a Taylor expansion for bounds near the saddle point. An implementation is provided in the Arb library.

In , F. Johansson describes algorithms to compute elliptic functions and their relatives (Jacobi theta functions, modular forms, elliptic integrals, and the arithmetic-geometric mean) numerically to arbitrary precision with rigorous error bounds for arbitrary complex variables. Implementations in ball arithmetic are available in the Arb library. This overview article discusses the standard algorithms from a concrete implementation point of view, and also presents some improvements.

In , F. Johansson describes an implementation of arbitrary-precision numerical integration with rigorous error bounds in the Arb library. Rapid convergence is ensured for piecewise complex analytic integrals by use of the Petras algorithm, which combines adaptive bisection with adaptive Gaussian quadrature where error bounds are determined via complex magnitudes without evaluating derivatives. The code is general, easy to use, and efficient, often outperforming existing non-rigorous software.

In , F. Johansson and M. Mezzarobba describe a strategy for rigorous arbitrary-precision evaluation of Legendre polynomials on the unit interval and its application in the generation of Gauss-Legendre quadrature rules. The focus is on making the evaluation practical for a wide range of realistic parameters, corresponding to the requirements of numerical integration to an accuracy of about 100 to 100 000 bits. The algorithm combines the summation by rectangular splitting of several types of expansions in terms of hypergeometric series with a fixed-point implementation of Bonnet's three-term recurrence relation. Rigorous enclosures of the Gauss-Legendre nodes and weights are then computed using the interval Newton method. The work provides rigorous error bounds for all steps of the algorithm. The approach is validated by an implementation in the Arb library, which achieves order-of-magnitude speedups over previous code for computing Gauss-Legendre rules with simultaneous high degree and precision.

In , Bill Allombert with Nicolas Brisebarre and Alain Lasjaunias describe a noteworthy transcendental continued fraction in the field of power series over $\mathbb{Q}$, having irrationality measure equal to 3. This article has been published in The Ramanujan Journal.

The article  by Nicolas Mascot, on the Certification of modular Galois representations has been published in Mathematics of Computation.

In , K. Belabas and H. Cohen give theoretical and practical information on the Pari/GP modular forms package, using the formalism of trace formulas. This huge package (about 70 exported public functions) handles standard operations on classical modular forms in $M_k({\Gamma }_0\left(N\right),\chi )$, also in weight 1 and non-integral weight (which are not cohomological, hence not directly handled by trace formulas). It is the first publicly available package which can compute Fourier expansions at any cusps, evaluate modular forms near the real axis, evaluate L-functions of non-eigenforms, and compute general Petersson scalar products.

In , H. Cohen explained how to compute Fourier expansions at all cusps of any modular form of integral or half-integral weight.

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 $\mathbb{Q}$.

In , H. Cohen gives an overview of Computational Number Theory in Relation with L-Functions, both in the local case (counting points on varieties over finite fields, involving in particular a detailed study of Gauss and Jacobi sums), and in the global case (for instance Dirichlet L-functions, involving in particular the study of inverse Mellin transforms). He also gives a number of little-known but very useful numerical methods, usually but not always related to the computation of L-functions.

In https://hal.inria.fr/hal-01379473/, H. Cohen and F. Thorne give explicit formulas for the Dirichlet series generating function of $D_{\ell }$-extensions of odd prime degree $\ell$ with given quadratic resolvent.