2026-09-29
11:00
Salle 2
Geometric characterization of p-exceptional monomial GAPN functions
On finite fields of characteristic $p$, PN (perfect nonlinear) functions for odd $p$ and APN (almost perfect nonlinear) functions for even $p$ are well-known classes of highly nonlinear functions. GAPN (generalized almost perfect nonlinear) functions were introduced as a generalization of APN functions for even $p$ to all $p$. One of the main targets of studies on such highly nonlinear functions is their classification. While $p$-exceptional monomial PN and 2-exceptional monomial APN functions have been classified, the corresponding problem for GAPN functions remains open. Here, a polynomial over $\mathbb{F}_p$ is called a $p$-exceptional PN (resp. APN, resp. GAPN) function if it is a PN (resp. APN, resp. GAPN) function on $\mathbb{F}_{p^n}$ for infinitely many positive integers $n$. In this talk, we give a geometric characterization of $p$-exceptional monomial GAPN functions. To this end, for a prime power $q$, we establish a geometric characterization of non-zero polynomials in $\mathbb{F}_q[x, y]$ having no absolutely irreducible factor defined over $\mathbb{F}_q$.
2026-09-22
11:00
Salle 2
A unified reduction from RLWE to MP-LWE
Ring Learning With Errors (RLWE) is a central assumption in lattice-based cryptography, but its hardness depends on the underlying number field. Middle-Product Learning With Errors (MP-LWE) offers a structured alternative whose definition is independent of any particular number field. Previous reductions from RLWE to MP-LWE cover different families of fields, and neither of the two main approaches subsumes the other. In this talk, I will present a unified reduction framework for both primal and dual RLWE, parameterized by a pair of field elements, which recovers both previous approaches. Using geometry of numbers, we prove the existence of parameters giving polynomial noise growth for every monic irreducible defining polynomial whose coefficients are polynomially bounded in its degree. The result holds for sufficiently large prime moduli coprime to the polynomial discriminant, extending the families covered by previous reductions. I will explain the algebraic mechanism and the geometry behind the noise bound. The reduction is non-uniform: suitable parameters are proved to exist, while finding them in polynomial time cost in general remains open.
