Requires installation of untrusted packages?

I tried installing VLC Player yesterday, and I got the Requires installation of untrusted packages error. I searched for hours for a solution to this problem, and finally came upon a suggestion that said to change servers for downloading updates from. I did this, and it gave me a 404 error when I clicked on "Check" in the Update Manager. I tried switching my server back to the main one (and after that, several other ones), but I am now still getting a 404 error:

And the other error I get when downloading or updating:

Edit: Output of cat /etc/apt/sources.list:

deb-src http://us.archive.ubuntu.com/ubuntu/ precise main restricted #Added by software-properties

# See http://help.ubuntu.com/community/UpgradeNotes for how to upgrade to
# newer versions of the distribution.
deb http://us.archive.ubuntu.com/ubuntu/ precise main restricted multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ precise multiverse universe #Added by software-properties

## Major bug fix updates produced after the final release of the
## distribution.
deb http://us.archive.ubuntu.com/ubuntu/ precise-updates main restricted multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ precise-updates restricted main multiverse universe #Added by software-properties

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu
## team. Also, please note that software in universe WILL NOT receive any
## review or updates from the Ubuntu security team.
deb http://us.archive.ubuntu.com/ubuntu/ precise universe
deb http://us.archive.ubuntu.com/ubuntu/ precise-updates universe

## N.B. software from this repository is ENTIRELY UNSUPPORTED by the Ubuntu 
## team, and may not be under a free licence. Please satisfy yourself as to 
## your rights to use the software. Also, please note that software in 
## multiverse WILL NOT receive any review or updates from the Ubuntu
## security team.

## N.B. software from this repository may not have been tested as
## extensively as that contained in the main release, although it includes
## newer versions of some applications which may provide useful features.
## Also, please note that software in backports WILL NOT receive any review
## or updates from the Ubuntu security team.
deb http://us.archive.ubuntu.com/ubuntu/ precise-backports main restricted universe multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ precise-backports main restricted universe multiverse #Added by software-properties

deb http://us.archive.ubuntu.com/ubuntu/ precise-security main restricted multiverse
deb-src http://us.archive.ubuntu.com/ubuntu/ precise-security restricted main multiverse universe #Added by software-properties
deb http://us.archive.ubuntu.com/ubuntu/ precise-security universe

## Uncomment the following two lines to add software from Canonical's
## 'partner' repository.
## This software is not part of Ubuntu, but is offered by Canonical and the
## respective vendors as a service to Ubuntu users.
deb http://archive.canonical.com/ubuntu precise partner
deb-src http://archive.canonical.com/ubuntu precise partner

## This software is not part of Ubuntu, but is offered by third-party
## developers who want to ship their latest software.

Solution 1:

Based on the errors you pasted in the comments section of my previous answer, I have another possible solution:

sudo apt-get clean
cd /var/lib/apt
sudo mv lists lists.old
sudo mkdir -p lists/partial
sudo apt-get clean 
sudo apt-get update

This will rebuild the cache.
Courtesy:http://ubuntuforums.org/showthread.php?t=1983220#8

Solution 2:

This link may help. It addresses a very similar problem.

Basically, you open a terminal and type sudo apt-get update && sudo apt-get upgrade. That did it for me.