What is Cryptography?
1. Cryptography
The word cryptography comes from the
Greek words κρυπτο (hidden or secret) and γραφη
(writing). Oddly enough,
cryptography is the art of secret writing. More generally,
people think of cryptography as the
art of mangling information into apparent unintelligibility
in a manner allowing a secret method
of unmingling. The basic service provided by
cryptography is the ability to send
information between participants in a way that
prevents others from reading it. In
this book we will concentrate on the kind of
cryptography that is based on
representing information as numbers and mathematically
manipulating those numbers. This
kind of cryptography can provide other services,
such as • integrity
checking—reassuring the recipient of a message that the message
has not been altered since it was
generated by a legitimate source • authentication—verifying
someone’s (or something’s) identity
But back to the traditional use of cryptography.
A message in its original form is
known as plaintext or clear text. The mangled
information is known as ciphertext.
The process for producing ciphertext from plaintext
is known as encryption. The reverse
of encryption is called decryption. While cryptographers
invent clever secret codes,
cryptanalysts attempt to break these codes. These two
disciplines constantly try to keep
ahead of each other.
Cryptographic systems tend to involve both an algorithm and a secret value. The
secret value is known as the key. The reason for having a key in addition to an
algorithm is that it is difficult to keep devising new algorithms that will allow
reversible scrambling of information, and it is difficult to quickly explain a newly
devised algorithm to the person with whom you’d like to start communicating securely.
With a good cryptographic scheme it is perfectly OK to have everyone, including
the bad guys (and the cryptanalysts) know the algorithm because knowledge of the
algorithm without the key does not help unmingle the information. The concept of
a key is analogous to the combination for a combination lock. Although the concept
of a combination lock is well known (you dial in the secret numbers in the correct
sequence and the lock opens), you can’t open a combination lock easily without knowing
the combination.
2. Computational Difficulty
It is important for cryptographic algorithms to be reasonably efficient for the
good guys to compute. The good guys are the ones with knowledge of the keys.1 Cryptographic
algorithms are not impossible to break without the key. A bad guy can simply try
all possible keys until one works. The security of a cryptographic scheme depends
on how much work it is for the bad guy to break it. If the best possible scheme
will take 10 million years to break using all of the computers in the world, then
it can be considered reasonably secure. Going back to the combination lock example,
a typical combination might consist of three numbers, each a number between 1 and
40. Let’s say it takes 10 seconds to dial in a combination. That’s reasonably convenient
for the good guy. How much work is it for the bad guy?
There are 403 possible combinations, which is 64000. At 10 seconds per try, it would
take a week to try all combinations, though on average it would only take half that
long (even though the right number is always the last one you try!). Often a scheme
can be made more secure by making the key longer. In the combination lock analogy,
making the key longer would consist of requiring four numbers to be dialed in. This
would make a little more work for the good guy. It might now take 13 seconds to
dial in the combination. But the bad guy has 40 times as many combinations to try,
at 13 seconds each, so it would take a year to try all combinations. (And if it
took that long, he might want to stop to eat or sleep).
With cryptography, computers can be used to exhaustively try keys. Computers are
a lot faster than people, and they don’t get tired, so thousands or millions of
keys can be tried per second. Also, lots of keys can be tried in parallel if you
have multiple computers, so time can be saved by spending money on more computers.
Sometimes a cryptographic algorithm has a variable-length key. It can be made more
secure by increasing the length of the key. Increasing the length of the key by
one bit makes the good guy’s job just a little bit harder, but makes the bad guy’s
job up to twice as hard (because the number of possible keys doubles). Some cryptographic
algorithms have a fixed-length key, but a similar algorithm with a longer key can
be devised if necessary. If computers get 1000 times faster, so that the bad guy’s
job becomes reasonably practical, making the key 10 bits longer will make the bad
guy’s job as hard as it was before the advance in computer speed. However, it will
be much easier for the good guys (because their computer speed increase far outweighs
the increment in key length). So the faster computers get, the better life gets
for the good guys. Keep in mind that breaking the cryptographic scheme is often
only one way of getting what you want. For instance, a bolt cutter works no matter
how many digits are in the combination.
3. Secret Codes
We use the terms secret code and
cipher interchangeably to mean any method of encrypting
data. Some people draw a subtle
distinction between these terms that we don’t find
useful. The earliest documented
cipher is attributed to Julius Caesar. The way the
Caesar cipher would work if the
message were in English is as follows. Substitute
for each letter of the message, the
letter which is 3 letters later in the alphabet
(and wrap around to A from Z). Thus
an A would become a D, and so forth. For instance,
DOZEN would become GRCHQ. Once you
figure out what’s going on, it is very easy to
read messages encrypted this way
(unless, of course, the original message was in
Greek). A slight enhancement to the
Caesar cipher was distributed as a premium with
Oval tine in the 1940s as Captain
Midnight Secret Decoder rings. (Where this done
today, Oval tine would probably be
in violation of export controls for distributing
cryptographic hardware!) The variant
is to pick a secret number n between 1 and
25, instead of always using 3.
Substitute for each letter of the message, the letter
which is n higher (and wrap around
to A from Z of course). Thus if the secret number
was 1, an A would become a B, and so
forth. For instance HAL would become IBM. If
the secret number was 25, then IBM
would become HAL. Regardless of the value of
n, since there are only 26 possible
ns to try, it is still very easy to break this
cipher if you know it’s being used
and you can recognize a message once it’s decrypted.
The next type of cryptographic
system developed is known as a monoalphabetic cipher,
which consists of an arbitrary
mapping of one letter to another letter. There are
26! possible pairings of letters,
which is approximately 4×1026. [Remember, n!,
which reads “n factorial”, means
n(n−1)(n−2)⋅⋅⋅1.] This might seem secure, because
to try all possibilities, if it took
1 microsecond to try each one, would take about
10 trillion years. However, by
statistical analysis of language (knowing that certain
letters and letter combinations are
more common than others), it turns out to be
fairly easy to break. For instance,
many daily newspapers have a daily cryptogram,
which is a monoalphabetic cipher,
and can be broken by people who enjoy that sort
of thing during their subway ride to
work. An example is Cflqr’xsxsnyctm n eqxxqgsyiqulqfwdcpeqqh,
erllqrxqgtiqul!
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