The composer, Yukumi Tetsuya is a member of an indies band named LOVER SOUND TRACK. Maybe Ayumi liked the melody and got his permission to use it. This band has become a little famous now in Japan because of Ayumi effect. This is their homepage.
LINK
"Kumo" has two meanings, "spider" and "cloud". But we know it's "spider" in this case, because it's written in Kanji.