Since kernel version 4.4.0-20, it was enforced that unsigned kernel modules will not be allowed to run with Secure Boot enabled. Because you want to keep Secure Boot, then the next logical step is to sign those modules.
So let's try it.
openssl req -new -x509 -newkey rsa:2048 -keyout MOK.priv -outform DER -out MOK.der -nodes -days 36500 -subj "/CN=Descriptive name/"
sudo /usr/src/linux-headers-$(uname -r)/scripts/sign-file sha256 ./MOK.priv ./MOK.der $(modinfo -n vboxdrv)
# also sign: vboxnetadp vboxnetflt vboxpci
sudo mokutil --import MOK.der
Supply a password for later use after reboot Reboot and follow instructions to Enroll MOK (Machine Owner Key). Here's a sample with pictures. The system will reboot one more time. After the reboot, you may also need to sudo modprobe vboxdrv to load the module. Please let me know if VirtualBox would run this way on Ubuntu 16.04 (on kernel 4.4.0-21 or higher, I believe).
http://gorka.eguileor.com/vbox-vmware-in-secureboot-linux/ https://github.com/Canonical-kernel/Ubuntu-kernel/blob/master/Documentation/module-signing.txt http://askubuntu.com/questions/760671/could-not-load-vboxdrv-after-upgrade-to-ubuntu-16-04-and-i-want-to-keep-secur
thank you so much you are a life saver. it worked for me ubuntu 20.04 5.4.0-33-generic