Cryptography Reference
In-Depth Information
would need to count through all of the beads to determine the exact
location of one of the beads. This same effect happens in the math-
ematics. You would need to multiply numbers again and again to
determine if a particular number is the one you want.
The second property of the string of beads in this metaphor does
not make as much sense, but it can still be easily explained . If you
want to move along the string
beads, then you can jump there
almost instantaneously. You don't need to count each of the
k
beads
along the way. This allows you to encrypt and decrypt messages
using the public-key system.
The two special features are similar but they do not contradict
each other. The second says that it is easy to jump an arbitrary num-
ber of beads. The first says it's hard to count the number of pearls be-
tween the first bead and any particular bead. If you knew the count,
then you could use the second feature. But you don't so you have to
count by hand.
The combination of these two features makes it possible to en-
crypt and decrypt messages by jumping over large numbers of beads.
But it also makes it impossible for someone to break the system be-
cause they can't determine the number of steps in the jump without
counting.
This metaphor is not exactly correct, but it captures the spirit
of the system. Figure 2.3 illustrates it. Mathematically, the loop is
constructed by computing the powers of a number modulo some
other number. That is, the first element in the loop is the number.
The second is the square of the number, the third is the cube of the
number, and so on. In reality, the loop is more than one-dimension-
al, but the theme is consistent.
k
2.2.3 How Random Is the Noise?
How random is the output of a encryption function like DES or RSA?
Unfortunately, the best answer to that question is the philosophical
response, “What do you mean by random?” Mathematics is very
good at producing consistent results from well-defined questions,
but it has trouble accommodating capricious behavior.
At the highest level, the best approach is indirect. If there was
a black box that could look at the first
bits of a file and predict
the next set of bits with any luck, then it is clear that the file is not
completely random. Is there such a black box that can attack a file
encrypted with DES or AES? The best answer is that no one knows of
any black box that will do the job in any reasonable amount of time. A
brute-force attack is possible, but this requires a large machine and
n
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