In this paper, we present a framework for computing dissimilarities between surfaces which is based on the mathematical model of normal cycle from geometric measure theory. This model allows to take into account all the curvature information of the surface without explicitely computing it. By defining kernel metrics on normal cycles, we define explicit distances between surfaces that are sensitive to curvature. This mathematical framework also has the advantage of encompassing both continuous and discrete surfaces (triangulated surfaces). We then use this distance as a data attachment term for shape matching, using the Large Deformation Diffeomorphic Metric Mapping (LDDMM) for modeling deformations. We also present an efficient numerical implementation of this problem in PyTorch, using the KeOps library, which allows both the use of auto-differentiation tools and a parallelization of GPU calculations without memory overflow. We show that this method can be scalable on data up to a million points, and we present several examples on surfaces, comparing the results with those obtained with the similar varifold framework.