In this talk we will discuss various recent claims of algorithms which solve certain instances of the elliptic curve discrete logarithm problem (ECDLP) over finite fields in sub-exponential time. In particular, we will discuss approaches which use Groebner basis algorithms to solve systems coming from summation polynomials. The complexity of these approaches relies on the so-called first fall degree assumption. We will raise doubt to this first fall degree assumption and hence to the claimed complexity.