Air carrier flight crew reported receiving a GPWS terrain alert and a low altitude alert from ATC on approach to BOI when they misunderstood their descent clearance.

Reporter 1

I was Pilot Flying and PIC (Pilot in Command) on this flight and we were cleared to descend via our arrival that we had briefed. The bottom altitude of the arrival was 8000ft so 8000 was set in the altitude selector and confirmed by both myself and the FO (First Officer) who was PM (Pilot Monitoring). The aircraft was in LNAV/VNAV modes and descending on profile. The weather at our destination had not been good all morning and the current conditions had 1 1/4 visibility, gusty winds, and snow. I had briefed the existence of terrain in the north east vicinity of the airport and the highest MSA of 9500ft. I had a terrain map page displayed on my MFD (Multi-Function Display). The runway condition codes had just been updated in the ATIS and I was trying to figure out if I could use ILS CAT I mins because I was still on high PIC mins. We were handed off to Approach Control who then gave us a heading of 130 and to descend and maintain 9000ft. I selected heading mode and selected 130. I remember my FO (First Officer) reading back 8000 for the altitude and I saw that 8000 was still bugged. We both paused and confirmed 8000ft at the same time. We continued descending and built the approach straight out in the downwind, anticipating the base turn. I noticed afterwards that I was high on profile and deployed speed brakes to return to profile. At around 8600ft, ATC issued an altitude alert and advised us to check our altitude at once. A few seconds later, we get a terrain caution, followed by an EGPWS warning at the same time ATC demanded we return to 9000ft. I executed the CFIT escape maneuver immediately and leveled at 9300ft. After regaining control of the aircraft we returned to 9000 and reengaged the automation. I was later informed the lowest ATC recorded us was 8200ft. We joined the approach and landed without further incident. The primary cause of this event was task saturation on both pilots during a critical phase of flight while receiving ATC instructions. Factors also include poor weather. Suggestions: Immediate confirmation of ATC instructions and asking for clarification when there is any doubt from either crew member.

Reporter 2

We were originally cleared on the arrival to 8000, but then we were vectored off and then told to descend 9000 instead. As Pilot Monitoring I read back 9000 to ATC. We got distracted with other tasks right after I read 9000, and the Pilot Flying forgot to set 9000. We then looked up from distractions and saw 8000 and just said 8000 out loud for our altitude. We descended below 9000 and got a warning from ATC about altitude, and then we got a EGPWS warning from terrain and the Pilot Flying immediately flew the CFIT escape maneuver correctly. We got back up to 9000 and flew the rest of the flight normally The cause of event would probably be high task saturation with being on the descent, and about to turn to final. We also both got distracted with other tasks getting ready for the approach. The weather was also a strong snow storm which played a factor. I would suggest anytime a new altitude is assigned, that unless it is an emergency situation or maneuver, to always read back the altitude out loud right away. If unsure ask ATC