A300 Captain reported while in a descent to their destination encountering weather that included two lightning strikes to the radome resulting in several systems failing. The flight crew diverted to their alternate airport.

Narrative

On descent into ZZZ weather was reported as IFR Lightning All Quads. Our dispatcher advised that the tower reported the storm was clearing to the east and we should have no problem getting in. Attempted TWIP (Terminal Weather Information for Pilots system) but did not work. Cloud layers started below 15;000. Passing approximately 13k we were handed off from ZZZ [ARTCC] to ZZZ APCH. Our first check in resulted in no comm. Second attempt ZZZ [TRACON] advised 'ATC Zero and contact ZZZ Center XXX.X' We went back to ZZZ [ARTCC]. They were not sure what ATC Zero meant but they cleared us Direct ZZZZZ at or above 9;000.' We accepted the clearance and asked for deviations south because our Weather radar showed that there were yellow returns to the north and green only south. We had previously encountered moderate rain; but were now out of precip but passing approximately 11;0000 were were at -2 degrees SAT (Static Air Temperature). We were then hit directly in the radome with the first of two lightning bolts. The first kicked off Auto Throttles and Autopilot and started a continuous warning to STOP RUDDER INPUTS with a red light and audible warning. The first officer immediately noted 'No AP (Autopilot) and No auto throttles.' I asked him to coordinate a climb away from the weather to the east while I had flew the airplane. He also assisted in throttle monitoring during the climb to ensure that we did not get slow. He coordinated the climb and I asked him to tell ATC that we needed to start proceeding to ZZZ1 (our alternate) while we assessed the situation. The FO coordinated the climb and when we were out of the immediate weather situation; he was able to re-establish auto throttles and AP although Flight Director 1 had failed. After we were established in the climb and cleared to ZZZ1; we exchanged flying duties so that I could coordinate the divert with ATC and our dispatcher and further trouble shoot systems problems. We also communicated our intentions to our Jumpseater and asked him whether he could see the wings; for any potential issues. He saw none. We continued the flight with a STOP RUDDER inputs light illuminated and FD (Flight Director) 1 inop but had no further issues into ZZZ1. We were later informed that ATC Zero was declared because of smoke or electrical fumes smell in ZZZ tower.Cause: Weather: Lightning strikeSuggestions: No. We used all onboard and off-board systems to evaluate the weather and design and brief a plan to mitigate the weather threat.