Air Carrier Captain reported a track deviation occurred in the vicinity of the Black Sea as a result of GPS jamming and an FMC malfunction.

Narrative

While operating from ZZZZ to ZZZZ1 we experienced a navigation system anomaly/malfunction which resulted in multiple 'unable RNP' EICAS messages and a possible small momentary route deviation after being assigned a route offset by ATC (Urumqi) upon entry into China at point SARIN. System issues appear to have begun while transiting the Black Sea area; South of Crimea when we received an EICAS message for Transponder Right; referenced the checklist; referenced the position reference pages of the FMC; noted GPS was intermittent; and then tried the left transponder with similar results. Per the QRH we determined the issue was related to a GPS outage/jamming issue causing ADS-B to be inoperative (experienced on this route previously). A short time later we received status messages for ATC L and ATC R along with a complete loss of GPS position input to the FMCs however Actual Navigation Performance (ANP) remained well below 1.0 NM while the Required Navigation Performance (RNP) curiously defaulted to 1.0 which is uncommon for most high altitude airway route segments. Approaching point BODSI we initiated a crew swap with the Relief Pilots; conducted a thorough changeover brief (specifically addressing the GPS/Transponder situation) and entered rest (CA (Captain) and FO (First Officer)). Approximately 20 minutes after I started my rest period; the relief FO asked me to come back to the Flight Deck. The aircraft had just given the relief pilots an 'Unable RNP' EICAS message due to an increasing ANP (upon my arrival on flight deck it was 2.5 NM). As a crew; we ran the checklist; checked for any popped circuit breakers and verified aircraft position using available conventional NAVAIDs; which the FMC also began using to lower RNP to approximately 0.38 NM. While verifying position we noticed GPS position inputs had returned; however both the GPS position and the IRS position computed by the FMC showed ANP of 16 or more (however; contradicting displayed ANP; GPS coordinates appeared to match radio generated coordinates); at this time the radio position updating indicated an RNP under 1. After discussing the situation for a few minutes I initiated a SATCOM call with the company to seek additional technical expertise; determine/validate proper RNP for the route through China and discuss options/contingencies/recommendations for continuing or possibly diverting prior to entry into China given some of our past concerns/issues in that airspace. We conducted a lengthy conference call with Dispatch; Maintenance; the Duty Pilot and a technical expert. The end result was the recommendation to continue since the ANP issues were attributed to the proximity to GPS jamming mentioned earlier; the ANP was well within limits at this time; route RNP was validated by Dispatch and if we encountered another system problem we would notify ATC of the intermittent navigation system accuracy malfunction and contact the Company as required. I reentered rest at this point; approximately 20-30 minutes prior to our entry into Chinese airspace. Upon entry into Urumqi FIR at point SARIN; ATC assigned a 3 NM right offset. While executing this clearance the Relief Crew encountered a short-lived unable RNP; a rapid rise in ANP to 20 NM while turning to the offset and notified ATC of the degraded navigation capability at that moment. ATC apparently did not seem concerned; notified the crew they were showing the aircraft left of track (unsure if they meant the airway or the offset track) while the crew showed the aircraft was 3 NM right of the airway as directed. ATC approved a request to return to centerline; the ANP reduced to within tolerances and the relief crew continued without further incident; notifying each ATC sector of the degraded navigation accuracy/capability. The FO and I returned from rest 1 hr 30 minutes prior to our ZZZ ETA; were briefed on what occurred and began preparations for the arrival. As part of our contingency planning we discussed how we would handle an unable RNP message on arrival/approach. As a side note; it was apparent ATC was passing from sector to sector that we had encountered a Nav accuracy issue as each sector queried us upon each radio frequency change. Our reply was that the inaccuracy issue was intermittent. The arrival/approach was normal until we received another unable RNP message while established on the ILS; below 2;000 ft. AGL. The FO was flying; I immediately notified Approach Control and requested alternate missed approach instructions in the event of a missed approach. ATC issued runway heading to 5;000 and we landed uneventfully. We conducted a crew debrief as we went and a brief recap after block in. I told the crew I would take care of reporting; if they felt the need they could also submit [reports] and I recommended we all submit reports. GPS Jamming and a FMC malfunction.Suggestion: Change routing to increase buffer between our routes and conflict related airspace so we are unaffected by GPS jamming. Consider adding FMC reset or de-powering the aircraft to the maintenance transit/Daily checks to clear/reboot FMC memory. Provide additional guidance/checklist info in QRH for handling an unable RNP message during cruise.