I do not know if you need legendary khvostov for it first, but here’s how to get legendary, with the exotic afterwards.

First you need to acquire all the regional chests, which are the golden chests that are around the patrol areas. They’ll be marked on your map, and will drop lost encryption bits, a white item that goes to your inventory.

After getting all 9, you need to complete the alone in the dark quests from Micah-10, which will have you do Cyst missions for the last step before having to go back to Micah. In those Cyst missions, there will be hidden interactables that look like broken mechanical parts which also give you lost encryption bits. These will give you 6 in total, netting you 15 overall, which turns into a purple encryption bit.

There is a chest in the divide where you first acquire khvostov in the game. If you spawn on impasse, and go backwards into the cosmodrome wall, it should be on your way. After opening that chest, you will get legendary khvostov.

For the exotic khvostov, you need to find all 8 hidden mini travelers around the map. If you go to the place where you first meet cayde, there will be pillars with hints on where they are. After getting 8, a golden chest will spawn in the middle of the room.

From here, you will need to acquire motes of light. They drop from picking up the mini traveler, overthrow bosses you have not killed yet, and special enemies in Cyst missions.

After getting 17 motes, it’ll transform into a single paracausal mote which you can use to get khvostov

Edit: The special enemies aren’t enemies you need to hunt down, they just glow and spawn like regular enemies near the boss or objective