I always though I understood UMTS AKA (authentication and key agreement), including the re-synchronization procedure. It's been years since I wrote tools like osmo-sim-auth which you can use to perform UMTS AKA with a SIM card inserted into a PC reader, i.e. simulate what happens between the AUC (authentication center) in a network and the USIM card.

However, it is only now as the sysmocom team works on 3G support of the dedicated OsmoHLR (outside of OsmoNITB!), that I seem to understand all the nasty little details.

I always thought for re-synchronization it is sufficient to simply increment the SQN (sequence number). It turns out, it isn't as there is a MSB-portion called SEQ and a lower-bit portion called IND, used for some fancy array indexing scheme of buckets of highest-used-SEQ within that IND bucket.

If you're interested in all the dirty details and associated spec references (the always hide the important parts in some Annex) see the discussion between Neels and me in Osmocom redmine issue 1965.