One problem in “Bounds and constructions for unconditionally secure distributed key distribution schemes for general access structures”

In this paper "Bounds and constructions for unconditionally secure distributed key distribution schemes for general access structures" on page 286 it's written "The independence of the mappings $\lbrace \phi _j \rbrace _{j \in H_G}$, implies that $ker\varphi _h^0 + (\cap _{j \in H_G} ker \varphi _j^0 )=E_0^l$." May I ask for more explanation of this sentence please?

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 I haven't seen many cryptography questions on this forum, which I take as evidence that we don't draw much interest from that community. That doesn't mean we can't answer your question, but you might have to make it a bit easier on us. Can you write down a description of these maps $\phi$? Of $E$? Of $H_G$? Who knows? Maybe writing a longer question and trying to pinpoint exactly where your confusion is will lead you to the answer yourself. If you have no luck here, try the theoretical computer science overflow site (but add a link here to any post you make there). – David White Dec 6 at 14:55