Finding low-weight multiples of a binary polynomial is a difficult problem arising in the context of stream ciphers cryptanalysis. The best algorithms to solve this problem are based on a time memory tradeoff. Staying in this category, we will present a new approach using discrete logarithm rather than a direct representation of the involved polynomials. This provides an alternative to the previously known algorithms which improves in some case the computational complexity.

We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic$2$. We provide new characterizations of APN functions and of APN permutations by means of their component functions. We generalize some results of Nyberg (1994) and strengthen a conjecture on the upper bound of nonlinearity of APN functions. We also focus on the case of quadratic functions. We contribute to the current works on APN quadratic functions by proving that a large class of quadratic functions cannot be APN.

We investigate some open problems on almost perfect nonlinear (APN) functions over a finite field of characteristic 2. We provide a new characterization of APN mappings and of APN permutations by means of their component functions. We also focus on the case of quadratic functions. Most notably, we prove that a class of quadratic functions cannot be APN. Our result strengthens the conjecture that all quadratic APN functions are power functions, up to equivalence