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We define cohomological complexes of locally compact abelian groups associated with varieties over p-adic fields and prove a duality theorem under some assumption. Our duality takes the form of Pontryagin duality between locally compact motivic cohomology groups.

Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gr{\"o}bner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gr{\"o}bner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gr{\"o}bner bases. In this article, we investigate how the FGLM change of ordering algorithm can be adapted to the tropical setting. As the valuations of the polynomial coefficients are taken into account, the classical FGLM algorithm's incremental way, monomo-mial by monomial, to compute the multiplication matrices and the change of basis matrix can not be transposed at all to the tropical setting. We mitigate this issue by developing new linear algebra algorithms and apply them to our new tropical FGLM algorithms. Motivations are twofold. Firstly, to compute tropical varieties, one usually goes through the computation of many tropical Gr{\"o}bner bases defined for varying weights (and then varying term orders). For an ideal of dimension 0, the tropical FGLM algorithm provides an efficient way to go from a tropical Gr{\"o}bner basis from one weight to one for another weight. Secondly, the FGLM strategy can be applied to go from a tropical Gr{\"o}bner basis to a classical Gr{\"o}bner basis. We provide tools to chain the stable computation of a tropical Gr{\"o}bner basis (for weight [0,. .. , 0]) with the p-adic stabilized variants of FGLM of [RV16] to compute a lexicographical or shape position basis. All our algorithms have been implemented into SageMath. We provide numerical examples to illustrate time-complexity. We then illustrate the superiority of our strategy regarding to the stability of p-adic numerical computations.

International audience; We start with some general remarks related to Julian Schwinger’s works in, and attitude to, physics, in particular in relation with Moshe Flato and me. Then we recall some notable events with Moshe (70s to 90s) and (since the 80s) with Noriko Sakurai, and also with Clarice Schwinger and me this millennium. A few photos illustrated the talk, most of which are reproduced at the end of this contribution.

This article is an extended version of: https://hal.archives-ouvertes.fr/hal-01792165; International audience; Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. Because of the use of the valuation, the theory of tropical Gröbner bases has proved to provide settings for computations over polynomial rings over a p-adic field that are more stable than that of classical Gröbner bases. Beforehand, these strategies were only available for homogeneous polynomi-als. In this article, we extend the F5 strategy to a new definition of tropical Gröbner bases in an affine setting. We also provide a competitor with an adaptation of the F4 strategy to tropical Gröbner bases computations. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm. We also illustrate its merits as a first step before an FGLM algorithm to compute (classical) lex bases over p-adics.

International audience; Let K be a field equipped with a valuation. Tropical varieties over K can be defined with a theory of Gröbner bases taking into account the valuation of K. While generalizing the classical theory of Gröbner bases, it is not clear how modern algorithms for computing Gröbner bases can be adapted to the tropical case. Among them, one of the most efficient is the celebrated F5 Algorithm of Faugère. In this article, we prove that, for homogeneous ideals, it can be adapted to the tropical case. We prove termination and correctness. Because of the use of the valuation, the theory of tropical Gröb-ner bases is promising for stable computations over polynomial rings over a p-adic field. We provide numerical examples to illustrate time-complexity and p-adic stability of this tropical F5 algorithm.

Part 1: Modern Cryptography; International audience; Among many approaches for privacy-preserving biometric authentication, we focus on the approach with homomorphic encryption, which is public key encryption supporting some operations on encrypted data. In biometric authentication, the Hamming distance is often used as a metric to compare two biometric feature vectors. In this paper, we propose an efficient method to compute the Hamming distance on encrypted data using the homomorphic encryption based on ideal lattices. In our implementation of secure Hamming distance of 2048-bit binary vectors with a lattice of 4096 dimension, encryption of a vector, secure Hamming distance, and decryption respectively take about 19.89, 18.10, and 9.08 milliseconds (ms) on an Intel Xeon X3480 at 3.07 GHz. We also propose a privacy-preserving biometric authentication protocol using our method, and compare it with related protocols. Our protocol has faster performance and shorter ciphertext size than the state-of-the-art prior work using homomorphic encryption.

International audience; In this article, we present new results for efficient arithmetic operations in a number field K represented by successive extensions. These results are based on multi-modular and evaluation-interpolation techniques. We show how to use intrinsic symmetries in order to increase the efficiency of these techniques. Applications to splitting fields of univariate polynomials are presented.

International audience; In this article, we present new results about the computation of a general shape of a triangular basis generating the splitting ideal of an irreducible polynomial given with the permutation representation of its Galois group G. We provide some theoretical results and a new general algorithm based on the study of the non redundant bases of permutation groups. These new results deeply increase the efficiency of the computation of the splitting field of a polynomial.

International audience; We establish an isomorphism between the quintic cyclic polynomi-als discovered by Hashimoto-Tsunogai and those arising from Kummer theory for certain algebraic tori. This enables us to solve the isomorphism problem for Hashimoto-Tsunogai polynomials and also Brumer's quintic polynomials.

International audience; Let f be a univariate monic integral polynomial of degree n and let (α1, ..., αn) be an n-tuple of its roots in an algebraic closure Q of Q. Obtaining an algebraic representation of the splitting field Q(α1, ..., αn) of f is a question of first importance in effective Galois theory. For instance, it allows us to manipulate symbolically the roots of f. In this paper, we propose a new method based on multi-modular strategy. Actually, we provide algorithms for this task which return a triangular set encoding the splitting ideal of f. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

International audience; We provide a modular method for computing the splitting field $K_f$ of an integral polynomial $f$ by suitable use of the byproduct of computation of its Galois group $G_f$ by $p$-adic Stauduhar’s method. This method uses the knowledge of $G_f$ with its action on the roots of $f$ over a $p$-adic number field, and it reduces the computation of $K_f$ to solving systems of linear equations modulo some powers of $p$ and Hensel liftings. We provide a careful treatment on reducing computational difficulty. We examine the ability/practicality of the method by experiments on a real computer and study its complexity.

Professor T. Mitsui set the problem of finding conditions that a square matrix with integral entries is a quadratic residue modulo a prime. We solve this problem in the case of order 2 and further consider some related topics.