Hacking the Xbox

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Hacking the Xbox: An Introduction to Reverse Engineering
over a single block of data as in a block cipher, or it may occur once as in
a stream cipher. All of the basic functions in a symmetric cipher are
computationally simple, so symmetric ciphers are the preferred method
for encrypting bulk data.
Typical examples of mixing functions are XORs, modular additions and
modular multiplications. The simplest function, XOR, has the property
that any number XOR itself is zero. The XOR operation is often denoted
with a ⊕ symbol. The XOR operation also has all the usual properties of
arithmetic (commutative, associative, distributive, etc.), so
(A ⊕ B) ⊕ B = A ⊕ (B ⊕ B) = A ⊕ 0 = A
Thus, if A were a message and B were a key, (A ⊕ B) would be the
ciphertext, and the plaintext can be recovered by simply performing an XOR
with B again.
A key schedule is an algorithm that takes a relatively short key and expands
its information over a long series of bits. Key schedules are used to help
diffuse the key data over a larger block of data so the relationship
between the ciphertext and the key is obscured.
Very Difficult Problems
Cryptographic functions are all based on mathematical algorithms
whose results are easy to compute given all the operands,
but whose operands are very difficult to compute given
just the result. The security of a cryptographic function is precisely
the difficulty of computing these operands given just
the results. Let us take a moment and explore what it means to
be very difficult.
Consider the symmetric cipher AES. It uses a 128-bit key, and
so far, it is strong against all known analytical cryptographic
attacks, such as differential and linear cryptanalysis. When I
say it is strong against analysis X, I mean that it will require at
least as many operations to recover the key or plaintext using
a brute-force search as it would using analysis X. A brute-force
search is when I take a very fast computer and try every one
of the 2 128 possible keys in order to recover the original data.
Most cryptographic algorithms in common use today are
strong against all known cryptanalysis techniques, so the important
number to understand is the strength of a brute-force
attack.
As it turns out, older algorithms such as DES, a 56-bit cipher is
not a very difficult problem. It is fairly easy to build a machine
using FPGAs (Field Programmable Gate Arrays) that can crack
keys at an economy of about 2 22 keys/second/dollar (2 22 is
about four million). Note that this number increases with time
at a rate equivalent to Moore’s Law. Today, if you are willing
to wait about a week for each key, you can recover them for
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