Why unit vectors should be encrypted bit per bit in that case?

At this work at section $2.2$ concerning a possible application for the BGN cryposystem the author points out that if you want to encrypt a unit vector $\overrightarrow{u_l}$ of size $l$ then the naive way is to encrypt each bit separately. And by following the idea of BGN the size of the ciphertext is reduced to $2\sqrt{l}$ from $l$. My question is why the naive way is to encrypt each bit seperately and not encrypt at once as a number in the specified group $\mathbb{G}$ as long as the value can let you efficiently compute the $DL$ as the discrete logarithm is needed for decryption as described at the original paper?

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Perhaps there isn't a one-to-one mapping from every possible vector to field elements? I need to look at the math in more detail. –  mikeazo Apr 11 '13 at 18:59