So, in my case, I used cygwin with windows, and I got:
* `private_key_path` file must exist:
C:\cygwin64\home\basic.user/.vagrant.d/insecure_private_key
After several minutes of investigation, I realized that the environment variable VAGRANT_HOME was not normal, so exporting the correct environment variable completed the task:
VAGRANT_HOME=/cygdrive/c/Users/basic.user
export VAGRANT_HOME