The runtime scales with the product of the sizes of the two arrays.

So? ;-) No matter what method you choose, worst case, the runtime will be at least proportional to the product of the sizes of the two array, if only because that can be the output size. See below.

    my %b_hash = map { $_ => 1 } @b_array;
    my @new_array = grep $b_hash{$_}, @a_array;
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You (and I think all of the others who answered this question) just failed question 1b. This will only work if the elements of @b_array are unique - but that's not given, and you shouldn't make that assumption; at least not without mentioning it.

I'd do something like:

    my %b_hash;
       $b_hash {$_} ++ for @b_array;
    my @c_array = map {($_) x ($b_hash {$_} || 0)} @a_array;
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which has a running time of O (sizeof (input) + sizeof (output)), which is asymptotical optimal.

Abigail