Discrete logarithms in curves over finite fields

A survey on algorithms for computing discrete logarithms in Jacobians of curves over finite fields

On positive Matrices which have a Positive Smith Normal Form

It is known that any symmetric matrix $M$ with entries in $\R[x]$ and which is positive semi-definite for any substitution of $x\in\R$, has a Smith normal form whose diagonal coefficients are constant sign polynomials in $\R[x]$. We generalize this result by considering a symmetric matrix $M$ with entries in a formally real principal domain $A$, we assume that $M$ is positive semi-definite for any ordering on $A$ and, under one additionnal hypothesis concerning non-real primes, we show that the Smith normal of $M$ is positive, up to association. Counterexamples are given when this last hypothesis is not satisfied. We give also a partial extension of our results to the case of Dedekind domains

Computing isomorphisms between lattices

Let K be a number field, let A be a finite dimensional semisimple K-algebra and let Lambda be an O_K-order in A. It was shown in previous work that, under certain hypotheses on A, there exists an algorithm that for a given (left) Lambda-lattice X either computes a free basis of X over Lambda or shows that X is not free over Lambda. In the present article, we generalise this by showing that, under weaker hypotheses on A, there exists an algorithm that for two given Lambda-lattices X and Y either computes an isomorphism X -> Y or determines that X and Y are not isomorphic. The algorithm is implemented in Magma for A=Q[G], Lambda=Z[G] and Lambda-lattices X and Y contained in Q[G], where G is a finite group satisfying certain hypotheses. This is used to investigate the Galois module structure of rings of integers and ambiguous ideals of tamely ramified Galois extensions of Q with Galois group isomorphic to Q_8 x C_2, the direct product of the quaternion group of order 8 and the cyclic group of order 2.Comment: 30 pages; v3 revised and accepted version to appear in Mathematics of Computation; v2 has many minor corrections with additional explanation in section 1

Ring-LWE: Enhanced Foundations and Applications

Ring Learning With Errors assumption has become an important building block in many modern cryptographic applications, such as (fully) homomorphic encryption and post-quantum cryptosystems like the recently announced NIST CRYSTALS-Kyber public key encryption scheme. In this thesis, we provide an enhanced security foundation for Ring-LWE based cryptosystems and demonstrate their practical potential in real world applications. Enhanced Foundation. We extend the known pseudorandomness of Ring-LWE to be based on ideal lattices of non Dedekind domains. In earlier works of Lyubashevsky, Perkert and Regev (EUROCRYPT 2010), and Peikert, Regev and Stephens-Davidowitz (STOC 2017), the hardness of RLWE was established on ideal lattices of ring of integers of number fields, which are known to be Dedekind domains. These works extended Regev's (STOC 2005) quantum polynomial-time reduction for LWE, thus allowing more efficient and more structured cryptosystems. However, the additional algebraic structure of ideals of Dedekind domains leaves open the possibility that such ideal lattices are not as hard as general lattices. We show that, the Ring-LWE hardness can be based on the polynomial ring, which is potentially be a strict sub-ring of the ring of integers of a number field, and hence not be a Dedekind domain. We present a novel proof technique that builds an algebraic theory for general such rings that also include cyclotomic rings. We also recommend a ``twisted'' cyclotomic field as an alternative for the cyclotomic field used in CRYSTALS-Kyber, as it leads to a more efficient implementation and is based on hardness of ideals in a non Dedekind domain. We leverages the polynomial nature of Ring-LWE, and introduce XSPIR, a new symmetrically private information retrieval (SPIR) protocol, which provides a stronger security guarantee than existing efficient PIR protocols. Like other PIR protocol, XSPIR allows a client to retrieve a specific entry from a server's database without revealing which entry is retrieved. Moreover, the semi-honest client learns no additional information about the database except for the retrieved entry. We demonstrate through analyses and experiments that XSPIR has only a slight overhead compared to state-of-the-art PIR protocols, and provides a stronger security guarantee while enabling the client to perform more complicated queries than simple retrievals

Disciplines and Styles in Pure Mathematics, 1800-2000

This workshop addressed issues of discipline and style in number theory, algebra, geometry, topology, analysis, and mathematical physics. Most speakers presented case studies, but some offered global surveys of how stylistic shifts informed the transition and transformation of special research fields. Older traditions in established research communities were considered alongside newer trends, including changing views regarding the role of proof