Backing up EFS and modem on the N7100/t03g

Reboot into recovery (you can do this with superuser while Android is running, but the filesystem will be dirty, so I don't recommend it);
The EFS partition on my Note 2 is at /dev/block/mmcblk0p3, and the modem is at /dev/block/mmcblk0p10, so do something like the following (annoyingly, it seems that trying to just redirect the output of adb shell to the local machine results in a corrupted file, so we need to copy locally on the phone and then pull it):

adb shell dd if=/dev/block/mmcblk0p3 of=/sdcard/efs.img
adb shell dd if=/dev/block/mmcblk0p10 of=/sdcard/modem.img
adb pull /sdcard/efs.img
adb pull /sdcard/modem.img

Make a copy of this image that we can test with (say, efs.img.verify);
Verify the EFS image is correct (it should look something like what you see below):

$ cp efs.img{,.verify}
$ file efs.img.verify
efs.img.verify: Linux rev 1.0 ext4 filesystem data (extents) (large files)
$ sudo mount efs.img.verify /mnt/efs
$ ls -la /mnt/efs
total 5188
drwxrwx--x  8  1001 chris    4096 Jan  1  2012 .
drwxr-xr-x 12 root  root     4096 Dec 13 03:09 ..
-rw-rw-r--  1  1001 sudo      152 Dec 11  2012 00000000.authtokcont
drwxrwxr-x  2  1001 chris    4096 Dec 11  2012 bluetooth
drwxrwxr-x  3  1019 chris    4096 Jan  1  2012 drm
drwxrwxr-x  2 chris chris    4096 Mar  5  2013 FactoryApp
drwxrwxr-x  5 root  root     4096 Jan  1  2012 .files
-rw-------  1 chris chris      12 Dec 11  2012 gyro_cal_data
-rw-r--r--  1  1001 sudo     1100 Jan  1  2012 h2k.dat
drwxrwxr-x  2  1001 chris    4096 Jan  1  2012 imei
-rwx------  1  1001 sudo  1048576 Jun 18 16:13 .nv_core.bak
-rwx------  1  1001 sudo       32 Jun 18 16:13 .nv_core.bak.md5
-rwx------  1  1001 sudo  2097152 Jun 18 16:13 .nv_data.bak
-rwx------  1  1001 sudo       32 Jun 18 16:13 .nv_data.bak.md5
-rwx------  1  1001 sudo  2097152 Dec 13 01:59 nv_data.bin
-rwx------  1  1001 sudo       32 Dec 13 01:59 nv_data.bin.md5
-rw-------  1  1001 sudo     1218 Jun 18 16:13 nv.log
-rw-------  1  1001 sudo        1 Dec 11  2012 .nv_state
drwxrwxr-x  2  1001 chris    4096 Jan  1  2012 wifi
-rw-r--r--  1  1001 sudo      220 Dec 11  2012 wv.keys

Back up the unmounted image (if you followed the instructions above, it's just called efs.img). I suggest you gzip -9 it, it will reduce in size dramatically;
Verify the modem image by looking at strings – you'll see a ton of embedded strings that quite obviously reference modem functionality.

To restore them, just do the inverse, writing to the same block devices.