Summer Number Theory Day; Session 3

SPEAKER: Francois Rodier
TITLE: Asymptotic nonlinearity of Boolean functions

ABSTRACT:
The nonlinearity of Boolean functions on the space Fm2 is important in cryptography. It is used to measure the strength of cryptosystems when facing linear attacks. In the case low degree of approximation attacks, we examine the nonlinearity of order r of a Boolean function, which equals the number of necessary substitutions in its truth table needed to change it into a function of degree at most r. Studies aimed at the distribution of Boolean functions according to the r-th order nonlinearity. Asymptotically, a lower bound is established in the higher order cases for almost all Boolean functions, whereas a concentration point is shown in the first and second order nonlinearity case. In the case of vectorial Boolean functions, a concentration point is shown in the first order nonlinearity case.

SPEAKER: Sorina Ionica
TITLE: Pairing-based methods for genus 2 curve jacobians with maximal endomorphism ring

ABSTRACT:
Algorithms for constructing jacobians of genus 2 curves with nice cryptographic properties involve the computation of Igusa class polynomials for CM quartic fields. The CRT method used to compute these polynomials needs to find first a jacobian with maximal endomorphism ring over a finite field, and then enumerates all others jacobians having maximal endomorphism ring using horizontal isogenies. For ℓ 2, we use Galois cohomology and the Tate pairing to compute the action of the Frobenius on the ℓ-torsion. In view of application to Igusa class polynomials computation, we deduce an algorithm to verify whether the jacobian of a genus 2 curve has locally maximal endomorphism ring at ℓ. Moreover, we derive a method to construct horizontal isogenies starting from a jacobian with maximal endomorphism ring.