Sufficient or Necessary? Translating from English (Weird cases)

Hello,

I was going over the concepts of necessary and sufficient conditions. Here are some terms I'm not sure how to interpret. Help would be appreciated:

"Assuming that A, B" or "B, assuming that A"

"As long as A, B" or "B as long as A" (also, "So long as")

"A provided that B"

"A on the condition that B"

Thanks!