Index calculus in class groups of non-hyperelliptic curves of genus 3 from a full cost perspective

We consider the discrete logarithm problem (DLP) in degree 0 class groups of non-hyperelliptic curves of genus 3 over finite fields Fq. Using a recent index calculus algorithm with double large prime variation by the author, heuristically, one can solve this problem in a time of Õ(q). In this work, we study this problem from a full cost perspective. We argue that heuristically, using a 3-dimensional mesh, an instance of the problem can be solved with a full cost of Õ(q) (whereas the ρ-method has a full cost of Õ(q), and a full cost of Ω(q) if the group order is nearly prime). In fact, we argue that for n −→ ∞, one can even solve n instances of this DLP with q ≤ n with a full cost of Õ(n). Moreover, we argue that one can solve n instances of this DLP with q ≤ n with a full cost of Õ(n).