GPS points and streets intersection

My goal is to find an intersect between streets and GPS points in QGIS. The output should represent all of the GPS points (GpsLongitude, GpsLatitude) intersecting with streets. So basically in the end i would like to export a CSV file containing points which intersect with streets.

What have I done so far:

1. I imported a street map of the country where the vehicle was driving; I created a buffer and create spatial index for it. Name of the layer is Buffered (light blue lines on the picture).

2. I imported a CSV file with around 6 million GPS points, created a shapefile, imported it and created spatial index for it. Name of the layer is 28_10_shapefile_telematics (pink dots on the picture).

3. Then i choose Vector –> Geoprocessing Tools –> Intersection and fill it out as on the second picture.

The problem is, that it is processing for nearly 12 hours! Is there a way to to this better/faster/more efficient? Thank you!