s
Section 5.6 The Mighty Mod Function
429
Notice that if you intercept a message encoded using a Caesar cipher, you
need to try only 26 possible key values to break the code (actually only 25 since
k = 26 shifts each number back to itself). The Caesar cipher is not a very secure
code; it may have served well, however, in an era when not many people could
read at all. For additional security, the function g could be a less obvious bijection;
the number of possible bijections g is 26! This would mean a total of 26!*25 possibilities to try for g and k, too large a number even if you had your handy laptop
with you in Gaul. As a cryptanalyst, you would narrow down the possibilities by
making use of statistical characteristics of the language, such as individual letter
frequency, frequency of certain letter pairs, and so on.
example 52
In a Caesar cipher with k = 3,
E encodes to g
−1
( 34 + 3 4 mod 26) = g
−1
(7 mod 26) = g
−1
(7) = H
Y encodes to g
−1
( 324 + 3 4 mod 26) = g
−1
(27 mod 26) = g
−1
(1) = B
And for k = 3,
H decodes to g
−1
( 37 − 34 mod 26) = g
−1
(4 mod 26) = g
−1
(4) = E
B decodes to g
−1
( 31 − 3 4 mod 26) = g
−1
(−2 mod 26) = g
−1
(24) = Y
■
PRaCtiCe 45 Decode the following ciphertext that was encoded with a Caesar cipher using a key of 7:
AOLJHAPUAOLOHA.
The Caesar cipher is a simple substitution cipher, meaning that each plaintext character is coded consistently into the same single ciphertext character.
Encryption techniques where a single plaintext character contributes to several
ciphertext characters (and one ciphertext character is the result of several plaintext
characters) introduce diffusion. The advantage to diffusion is that it hides the frequency statistics of individual letters, making analysis of an intercepted ciphertext
message much more difficult.
Des (Data encryption standard) is an internationally standard encryption
algorithm developed in 1976. DES was developed to safeguard the security of
digital information, so we may consider the plaintext to be a string of bits (a string
of 0s and 1s). Unlike the Caesar cipher that encodes each single plaintext character individually, DES is a block cipher. A block of 64 plaintext bits is encoded as
a unit using a 56-bit key. This results in a block of 64 ciphertext bits. However,
changing one bit in the plaintext or one bit in the key changes about half of the
resulting 64 ciphertext bits, so DES exhibits high diffusion. One might expect
this effect to require some extremely complex mathematical encoding function,
but DES actually uses many simple operations over and over. The DES algorithm
calls for 16 “rounds” to be done; the original 56-bit key is modified from one
round to the next, as is the original 64-bit plaintext block. The modifications involve the following, among other things:
Section 5.6 The Mighty Mod Function
429
Notice that if you intercept a message encoded using a Caesar cipher, you
need to try only 26 possible key values to break the code (actually only 25 since
k = 26 shifts each number back to itself). The Caesar cipher is not a very secure
code; it may have served well, however, in an era when not many people could
read at all. For additional security, the function g could be a less obvious bijection;
the number of possible bijections g is 26! This would mean a total of 26!*25 possibilities to try for g and k, too large a number even if you had your handy laptop
with you in Gaul. As a cryptanalyst, you would narrow down the possibilities by
making use of statistical characteristics of the language, such as individual letter
frequency, frequency of certain letter pairs, and so on.
example 52
In a Caesar cipher with k = 3,
E encodes to g
−1
( 34 + 3 4 mod 26) = g
−1
(7 mod 26) = g
−1
(7) = H
Y encodes to g
−1
( 324 + 3 4 mod 26) = g
−1
(27 mod 26) = g
−1
(1) = B
And for k = 3,
H decodes to g
−1
( 37 − 34 mod 26) = g
−1
(4 mod 26) = g
−1
(4) = E
B decodes to g
−1
( 31 − 3 4 mod 26) = g
−1
(−2 mod 26) = g
−1
(24) = Y
■
PRaCtiCe 45 Decode the following ciphertext that was encoded with a Caesar cipher using a key of 7:
AOLJHAPUAOLOHA.
The Caesar cipher is a simple substitution cipher, meaning that each plaintext character is coded consistently into the same single ciphertext character.
Encryption techniques where a single plaintext character contributes to several
ciphertext characters (and one ciphertext character is the result of several plaintext
characters) introduce diffusion. The advantage to diffusion is that it hides the frequency statistics of individual letters, making analysis of an intercepted ciphertext
message much more difficult.
Des (Data encryption standard) is an internationally standard encryption
algorithm developed in 1976. DES was developed to safeguard the security of
digital information, so we may consider the plaintext to be a string of bits (a string
of 0s and 1s). Unlike the Caesar cipher that encodes each single plaintext character individually, DES is a block cipher. A block of 64 plaintext bits is encoded as
a unit using a 56-bit key. This results in a block of 64 ciphertext bits. However,
changing one bit in the plaintext or one bit in the key changes about half of the
resulting 64 ciphertext bits, so DES exhibits high diffusion. One might expect
this effect to require some extremely complex mathematical encoding function,
but DES actually uses many simple operations over and over. The DES algorithm
calls for 16 “rounds” to be done; the original 56-bit key is modified from one
round to the next, as is the original 64-bit plaintext block. The modifications involve the following, among other things: