Nicolas Bernard

Jacobians of Genus-2 Curves with a Rational Point of Order 11

On the one hand, it is well known that Jacobians of (hyper) elliptic curves defined over ℚ having a rational point of order I can be used in many applications, for instance in the construction of class groups of quadratic fields with a nontrivial l-rank. On the other hand, it is also well known that 11 is the least prime number that is not the order of a rational point of an elliptic curve defined over ℚ. It is therefore interesting to look for curves of higher genus whose Jacobians have a rational point of order 11. This problem has already been addressed, and Flynn found such a family 𝔉 t of genus-2 curves. Now it turns out that the Jacobian J 0(23) of the modular genus-2 curve X 0(23) has the required property, but does not belong to 𝔉 t . The study of X 0(23) leads to a method giving a partial solution of the considered problem. Our approach allows us to recover X 0(23) and to construct another 18 distinct explicit curves of genus 2 defined over ℚ whose Jacobians have a rational point of order 11. Of these 19 curves, 10 do not have any rational Weierstrass point, and 9 have a rational Weierstrass point. None of these curves are ℚ̄-isomorphic to each other, nor ℚ̄-isomorphic to an element of Flynns family 𝔉 t . Finally, the Jacobians of these new curves are absolutely simple.

Hardened Bloom Filters, with an Application to Unobservability

Classical Bloom filters may be used to elegantly check if an element e belongs to a set S, and, if not, to add e to S. They do not store any data and only provide boolean answers regarding the membership of a given element in the set, with some probability of false positive answers. Bloom filters are often used in caching system to check that some requested data actually exist before doing a costly lookup to retrieve them. However, security issues may arise for some other applications where an active attacker is able to inject data crafted to degrade the filters’ algorithmic properties, resulting for instance in a Denial of Service (DoS) situation. This leads us to the concept of hardened Bloom filters, combining classical Bloom filters with cryptographic hash functions and secret nonces. We show how this approach is successfully used in the TrueNyms unobservability system and protects it against replay attacks.