I'm running 12.04 Unity on my HP laptop.

My Package Manager is not working. First of all, there's an icon on the upper panel. Clicking on it reveals a message (see attached image file nopackage2.jpg) that says there was an error opening the cache. It also recommends that I run Package Manager or apt-get to find more information.

Package Manager says it couldn't initialize the package information (see attached image file nopackage.jpg).

Both the warning icon and Package Manager say "E:Encountered a section with no Package: header, Eroblem with MergeList /var/lib/apt/list/us.archive.ubuntu.com_ubuntu_dists_precise-updates_main_binary-amd64_Packages, E:The package list or status file could not be parsed or opened."

If I run sudo apt-get update I get many errors associated with GPG keys and signature verifications.

Code:
Reading package lists... Error!
W: GPG error: http://us.archive.ubuntu.com precise Release: The following signatures were invalid: BADSIG 40976EAF437D05B5 Ubuntu Archive Automatic Signing Key <ftpmaster@ubuntu.com>
W: GPG error: http://ppa.launchpad.net precise Release: The following signatures were invalid: BADSIG 0D18601A1EE8660B Launchpad PPA for Darik Horn
W: A error occurred during the signature verification. The repository is not updated and the previous index files will be used. GPG error: http://extras.ubuntu.com precise Release: The following signatures were invalid: BADSIG 16126D3A3E5C1192 Ubuntu Extras Archive Automatic Signing Key <ftpmaster@ubuntu.com>

W: A error occurred during the signature verification. The repository is not updated and the previous index files will be used. GPG error: http://us.archive.ubuntu.com precise-backports Release: The following signatures were invalid: BADSIG 40976EAF437D05B5 Ubuntu Archive Automatic Signing Key <ftpmaster@ubuntu.com>

W: Failed to fetch http://us.archive.ubuntu.com/ubuntu/dists/precise-backports/Release  

W: Failed to fetch http://extras.ubuntu.com/ubuntu/dists/precise/Release  

W: Some index files failed to download. They have been ignored, or old ones used instead.
E: Encountered a section with no Package: header
E: Problem with MergeList /var/lib/apt/lists/us.archive.ubuntu.com_ubuntu_dists_precise-backports_multiverse_binary-amd64_Packages
E: The package lists or status file could not be parsed or opened.
My other computer, which also runs 12.04 Unity, is set up almost identically and isn't having this problem. Any ideas?