Given the hash:
how do I construct a rainbow table to crack this password?
closed as too localized by rook, Scott Pack, Iszi, Hendrik Brummermann♦ Oct 25 '12 at 20:34
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One doesn't construct a rainbow table to crack a specific password. If you want to crack a specific password it is more efficient to simply brute force it then to generate a rainbow table. One constructs a rainbow table to be able to crack a lot of passwords that share certain characteristics such as the password hash function and the password dictionary from which the passwords are taken.
If someone has already built a rainbow table for passwords with the same characteristics as the password you're trying to crack then you can use it. But I'd be surprised if anyone has ever built a rainbow table for a password dictionary of any six hex bytes and a "hash function" in which the password is padded with zeroes and AES encrypted with a key of zeroes.
BTW, I wrote hash function in quotation marks because this isn't a hash function - it's an encryption function. If instead the (padded) password was used as a key to AES encrypt a block of zeroes - that would have been a hash function.
To build the table as is described in the article, well, you just follow the description which is in the article. That's the point of the exercise: to learn how to read a scientific article and implement it. With a 24-bit space of possible passwords, you will just need about 256 tables, each with 256 chains, which is totally within range of a non-optimized implementation. Just like what a student can be reasonably expected to produce as part of homework.
Of course, it makes little sense to build such tables for cracking a single password; precomputed tables, including the special kind which Hellman describes, are worth the effort only when you want to crack at least two passwords. Also, a 24-bit space is a piece of cake and could be cracked with brute force in less than one second on average, if implemented properly (and substantially less if implemented properly with the AES-NI instructions).
Note that Hellman's time-memory trade-off is not a rainbow table, but its immediate predecessor. Rainbow tables are an optimization designed by Oechslin in 2003. I invite you to read the first parts of this article, which recalls Hellman's trade-off and might help you reach enlightenment (which is why you do the homework in the first place).