- 1.
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Cryptology and MathematicsAn Exploration in Substitution
CiphersUNC-Charlotte Senior projectbyColin Garth UtleyAndUnder the
Direction ofJohn Russell Taylor, Ph.D.
When Julius Caesar learned Quintus Cicero was besieged in the land
of the Nervii, he wanted to communicate with him but was unable to
do so in person.So Caesar used a method that was simple and
efficient; a cipher.He created a message by substituting Greek
letters in the place of Roman letters and sent it to Cicero by
messenger.When the messenger was unable to reach Cicero, he tied
the message to a spear and threw it into Ciceros camp.A few days
later, when the spear was noticed lodged in the side of a tower, it
was brought down and the message was read; Veni, Vidi, Vici, I
came, I saw, I conquered.1This is the first time a substitution
cipher was used for a military purpose in documented history.
To begin our discussion on cryptology, ciphers, encryption,
decryption, and code theory we should begin by defining the terms
which will be used.Cryptology also known as cryptography is the
study or analysis of codes and coding methods.5 A cipher is a
written code in which the letters of a text are replaced with
others according to a system.5A code is a system of letters,
numbers, or symbols into which normal language is converted to
allow information to be communicated secretly, briefly, or
electronically.5To encrypt is to convert a text into code or
cipher.5And, as one might expect, to decrypt/decode is to transform
an encoded message into an understandable form.5
To prepare a solid foundation for discussion of the General
Substitution Cipher and the Caesar Cipher, a special case of the
General Substitution Cipher, we will begin with a brief review of
Modern Algebra and Modular Arithmetic.To begin we will remind
ourselves of several important definitions.The first definition is
what it means to divide in Modern Algebra.let a and b be integers
with b 0.We say that b divides a if a = bc for some integer
c.(Hungerford, 7) Next we need to be reminded of congruence.Let a,
b, n be integers with n > 0.Then a is congruent to b modulo n,
provided that n divides a - b. (Hungerford, 24)And finally, the
congruence class of a(modulo n), denoted [a], is the set of all
those integers that are congruent to a(modulo n), that is, [a] =
[b] implies b a(mod n). (Hungerford, 26).The next topic we need to
cover in order to discuss our ciphers is Ring Structure.There are
several different types of structures that fall into the Ring
category; regular Rings, Commutative Rings, Commutative Rings with
Identity, a Commutative Ring with Identity that has no
Zero-Divisors which is also known as an Integral Domain, and
finally Fields.All of these different types of rings have the same
basic substructure in that they all fit the definition of a regular
Ring.So what is a ring?
A ring R is a nonempty set equipped with two operations, addition
and multiplication, that satisfy the following axioms for every a,
b, c R:
If a, b R then a + b R
If a + (b + c) R then (a + b) + c R
a + b = b + a
There exists 0R R such that a + 0R = a = 0R + a for every a R
For every a R the equation a + x = 0R has a solution in R
If a, b R; then ab R
a(bc) = (ab)c
If a(b + c) = ab + ac; then (a + b)c = ac + bc.
Thus if all of the above properties are met we can safely say that
a nonempty set is a ring R.
To go one step further a Commutative Ring is a ring such that given
a ring R and a, b R; (ab) = (ba).To refine the ring yet another
step; a Commutative Ring with Identity is a Commutative Ring with
an element, 1R R such that given a R, a(1R) = a = (1R)a.Before
giving the definition for an Integral Domain we should know what it
means to be a Zero-Divisor.An element a in a ring R is a Zero
Divisor if a 0R and there exists a nonzero element b of R such that
either ab = 0R or ba = 0R.Now, an Integral domain is a Commutative
Ring with Identity that has no Zero-Divisors.
From an Integral Domain there is only one other refinement of the
definition that results in the strongest type of ring structure; a
Field.A field is a ring with identity 1R 0R that satisfies the
property that for every a 0R, the equation ax =1R has a
solution.Now we can analyze Z26 to see how it fits into this
structure.First of all we want to check Z26 to see if it fits the
description of a ring.
If we take any two elements of Z26 and add them by the definition
of Z26 the sum of those two elements is in Z26; hence the set Z26
is closed under addition and the first property of a ring is
met.Now if we take any three elements of Z26 and add them such that
a + (b + c) is an element of Z26 then (a + b) + c is also an
element of Z26 and the set is associative under addition; which
means it meets the second property of a ring.Now given a and b are
elements of Z26, since the set is closed under addition a + b is an
element of Z26 and we can see by the structure of Z26 that b + a is
also in the set.Hence, Z26 is commutative under addition which
meets the third requirement of the definition of a ring.
Also there is an element 0R of Z26 such that for any a in Z26, a +
0R = a = 0R + a.Thus the zero element exists in the set and the
fourth requirement for a ring is met.Next we notice that for every
a that is an element of Z26 there exists an x in Z26 such that the
equation a + x = 0R has a solution which means the fifth
requirement of the definition of a ring is met.Now, given a and b
in Z26 the product ab is also in Z26.Hence Z26 is closed under
multiplication.Also, given any a, b, and c in Z26 the product a(bc)
is in Z26 and the product (ab)c is also in Z26, thus we see that
Z26 is associative under multiplication as well and hence the
seventh property of a ring is fulfilled.
Now if we take elements a, b, and c of Z26 and given a(b + c) = ab
+ ac, with ab + ac an element of Z26 then (a + b)c = ac + bc is an
element of Z26 and since the set is closed under multiplication ac
and bc are elements of Z26.And as the set is also closed under
addition, ac + bc is also an element of Z26.Hence, since Z26 fills
all the above requirements, Z26 is a ring.Now we want to see if Z26
is more than just a regular ring.
Indeed it is.Given a and b are elements of Z26 and ab is an element
of Z26 by the closure property under multiplication, we can analyze
the commutative product ba and see that it is also an element of
Z26.Hence, Z26 is a Commutative Ring.Also, since there is an
element 1R in Z26 such that for every a in Z26 a1R = a = 1Ra; Z26
is a Commutative Ring with Identity.Now, on the other hand, there
are non-zero elements of Z26 which when multiplied return a zero
value.Thus the set Z26 is not an Integral Domain.Therefore, Z26 is
a Commutative Ring with Identity.
Now with these definitions fresh in our mind we can begin to
discuss the General Substitution Cipher.The General Substitution
Cipher is a cipher in which any letter of the alphabet can be
interchanged with any other letter of the alphabet.This means that
basically any cipher in which letters are exchanged for other
letters falls under the category of the General Substitution
Cipher; no over-riding structure for substitution need be applied
for the General Substitution Cipher.But, on the other hand, ciphers
with a specific structure to their substitutions are also members
of the General Substitution Cipher.As an example the Caesar Cipher
is a member of the class of General Substitution Ciphers.
Mathematically the General Substitution Cipher can be represented
by the equation
y = (a + x)(mod n), unless it is a random type of substitution;
then there exists no equation to represent the substitution.In the
equation for the General Substitution Cipher a represents the shift
involved to encrypt a letter, x represents the letter to be
encrypted, and n represents the number of characters in the
alphabet.In the case of English and other languages using the same
alphabet; n = 26.There are two good examples of General
Substitution Ciphers; the Caesar Cipher and the Vigenere
Cipher.
The Caesar Cipher is a simple substitution cipher obtained by
shifting the letters of the alphabet n spaces to the left.The shift
that Julius Caesar used was 3 spaces so that A becomes D, B becomes
E, and so on such that the cipher alphabet would relate to the
normal alphabet as below;
Normal
Alphabet:a-b-c-d-e-f-g-h-i-j-k-l-m-n-o-p-q-r-s-t-u-v-w-x-y-z
Cipher Alphabet:
d-e-f-g-h-i-j-k-l-m-n-o-p-q-r-s-t-u-v-w-x-y-z-a-b-c
To relate the Caesar Cipher to the above equation for the General
Substitution Cipher we can assign each letter in the alphabet an
integer value between zero and 25 such that A->0, B->1, etc.
Then the alphabet would relate to the set Z26, the set of integers
zero through 25 as in the table below;
Next we let a = 3, and let n = 26.Then the equation of the General
Substitution Cipher that results in the Caesar Cipher is; y = (3 +
x)(mod 26).Now if we consult the table for modular addition of Z26
below and look at the row for the addition of 3 we see that it
gives the same relation of the normal alphabet to the cipher
alphabet as above; where D=3, E=4, etc.
So if we let the value of x = 0, and let 3 be the value assigned to
a, the equation becomes; y = (3 + 0)(mod 26). Now from the table
for Z26 we see that (3 + 0)(mod 26) = 3, which means that the
letter A has been mapped to the letter D as expected.Therefore, y =
(3 + x)(mod 26) is the mathematical representation of the Caesar
Cipher.
The next example of a General Substitution Cipher is the Vigenere
Cipher.The Vigenere Cipher is basically a General Substitution
Cipher that changes cipher-alphabets for each letter to be
enciphered.Hence if we were to represent the Vigenere Cipher as an
equation it would be the same as with the General Substitution
Cipher, only the value of a would change each time a new letter was
entered; thus a different row of the table for Z26 would be used
for each new letter.In this way, using 26 different alphabets, the
Vigenere Cipher is much stronger than the Caesar Cipher.
The manner of determining the system of rows would have to be
agreed upon by both the party writing the message and the party
receiving the message.One way to choose the row would be to choose
a key word with which to encipher the message; which would be known
as a repeated key.To encipher a message using a repeated key one
would write the key over and over above or below the corresponding
message.Then the key would be converted to a system of repeating
numbers by assigning an integer value to each letter in the same
manner as with assigning the alphabet to Z26.Then all one would
have to do is take each letter to be enciphered and convert it to
its corresponding integer in Z26, add the corresponding key
integer, find its congruence class in Z26, convert the value of the
congruence class back to its associated letter, and move on to the
next letter.Below is an example of this process.
Example
Another way to choose the row to use in order to encipher a message
is to use a running key.A running key is simply a key that
continues for at least as long as the message.An example of a
running key could be any story of correct length from your typical
English Textbook.For example one could use Dracula by Bram Stoker,
ignoring punctuation, as a running key as is shown below.
Example
As you can see the Vigenere Cipher is a much better cipher as far
as the security of the message goes; whether the key is repeating
or running.The Vigenere Cipher is also found in electro-mechanical
form; the Enigma.
The Enigma machine first appeared shortly after the end of WW I.The
invention and production of the Enigma is credited to Hugo
Alexander Koch and Arthur Scherbius.The Enigma machine was invented
to help businesses communicate over public lines while still being
able to maintain trade secrets.During the 1920s the Enigma shows up
all over Europe in use by businesses and, toward the end of the
decade, by the German military.During this time Enigma was
advertised as portable and easy to use yet statistically highly
secure (source) with 15x1012 substitutions possible for every
letter.
In essence the Enigma is nothing more than an electro-mechanical
version of the Vigenere Cipher.Enigma works in exactly the same
manner as the Vigenere Cipher, substituting one letter for another,
but with multiple cipher-alphabets, hence multiple permutations of
the letter to be enciphered, used for each letter
enciphered.Basically we can view the Enigma cipher as a
multi-dimensional Vigenere Cipher.
The Enigma is made up of 9 basic components; a keyboard, plugboard,
3 rotors, a reflector, a lampboard, and the wiring to connect it
all.Encrypting a message using the Enigma machine was relatively
simple.The operator would press the key corresponding to the letter
he wished to encipher to begin encryption by closing an electrical
circuit.From the keyboard the current would run into the
plugboard.The current would then exit the plugboard to the input
side of the first rotor.Once inside the first rotor the letter
would be permuted by the nearly random wiring of the rotor into
another letter.This new letter would then exit the first rotor and
enter the second rotor.In the second rotor the same process would
occur and a new letter than entered would exit to enter the third
rotor.Again, another permutation of the letter would occur inside
the third rotor before the letter entered the reflector.The
reflector performed another permutation of the letter and then sent
the letter back through the rotor column again.Finally after its
trip through the rotor column to the reflector then back through
the rotor column the current would travel through the plugboard
once more before lighting a letter on the lampboard.This lighted
letter would be the final substitution for the letter pressed on
the keyboard.
Figure 1 (Colossus, 17)
At this point one might ask; just how many substitutions were
possible for each letter enciphered using the Enigma machine?Well,
in order to answer this question we make a few observations and
then use those observations to make our calculations.First of all
we must look at the way in which the messages enciphered by Enigma
were transmitted.Once a message was enciphered by Enigma it was
sent over the air as a message broken up into 4-letter
blocks.Secondly, we must take note of the manner in which our
calculations must be done.
In the matter of the plugboard there were 26 jacks and 6 cables;
which gives us a chance to interchange 12 letters of the
alphabet.Now we must decide whether the order in which these cables
were plugged into the plugboard would cause any change in the
overall outcome of the operation because this fact will determine
the difference between a combination and a permutation.To review, a
permutation of n symbols is an arrangement of those symbols in a
definite order while a combination is the number of distinct
subsets, each of size r, that can be constructed from a set with n
elements.Here we will assume that the order in which the jacks,
which corresponded to letters, were chosen mattered because if, for
example, the letter A was chosen as the first jack and W was chosen
as the second; those letter could not be used anymore during later
choices.By this assumption we are able to see that the plugboard
settings are in fact a permutation which gives us our method for
calculating the number of arrangements possible.
Given there were 26 jacks and 6 double ended cables we can now
calculate our number for the plugboard.We know we are going to
chose first one jack and then another, and then another; so on and
so forth, until our cables are used up and we can make no more
connections.For our first connection there are 26 possibilities
from which to choose.For the second there will be 25, the third 24;
etc.Therefore, we end up with;
26*25*24*23*22*21*20*19*18*17*16*15=4.62605375232*1015 for the
number of possible permutations in the plugboard.Now we can move
along to our calculation of the rotor and reflector
permutations.
In each of the rotors there was a permanent wiring which was
handmade to look as random as possible.Therefore, in each rotor
there are 26 possible letters that any one letter can map to.Now,
since there were three of these rotors and the reflector takes the
input from the third rotor and puts it back through the entire
rotor column; we can calculate the number of possible permutations
of any letter as being 266=308,915,776.Since for any letter the
reflector will act as another rotor we can count it as an extra run
through a rotor and recalculate the number of possible
substitutions from two runs through the rotor column and a pass
through the reflector as 267 = 8,031,810,176.
Now, if we assign variables to each of the components of the Enigma
machine letting P represent the plugboard, R1 represent the first
rotor, R2 represent the second rotor, R3 represent the third rotor,
and R* represent the reflector we can come up with the equation
T=PR1R2R3R*R3R2R1P, where T is the total number of substitutions
possible for the entire enciphering process of the Enigma
machine.Then substituting the appropriate numbers for the variable
we obtain the following equation;
T=(4.62605375232*1015)(26)(26)(26)(26)(26)(26)(26)(4.62605375232*1015).Simplifying
this equation gives; T=(4.62605375232*1015)2(26)7, and simplifying
even more gives T=3.71555856062*1025 for the total number of
possible substitutions for any letter.Now given that message
encrypted with the Enigma machine were sent in 4-letter blocks we
can calculate the total number of possibilities for a 4-letter
block as; B=(3.7155585606225)4 which gives B=1.90588390342*10102.As
you can see there were an astronomical number of possibilities for
every four letter block of each message sent in the Enigma
cipher.
Now that we have covered the encryption of messages we can move on
to the other side of the coin; decryption.When one starts to think
of decryption; one must think first of the number of ways to attack
the problem.In the case of the Caesar Cipher, there are only 25
different possibilities for each letter in the cipher-text.So, all
one has to do is replace every letter of the message at most 25
times to find the original message.Although this method of
deciphering a message of any length may be time consuming it is no
way difficult.
However, when a different sort of substitution cipher, such as the
Vigenere Cipher, is used; the number of possible ways to encrypt a
letter increases.For each letter of the cipher-text in the Vigenere
cipher there are 26 different cipher alphabets from which to
choose.This simple fact raises the security of the cipher
incredibly.Now, instead of just being able to replace each letter
individually to find the alphabet for the entire message, one must
try each of the 26 possible alphabets for each letter.So, assuming
that the key does not consist of just one repeated letter, because
that would turn the cipher into a mere one-alphabet shift-cipher,
and there is no repetition of any letter in the key; for a message
only five letters long one must try 26*25*24*23*22 = 7,893,600
possible combinations in order to decipher the message.
On the other hand, one must keep in mind that with substitution
ciphers, in general, a letter changes only in face value.It still
keeps its normal characteristics as far as its interaction with the
rest of the alphabet.For instance, in the English language the
letter U is always found after each and every Q.Therefore, when one
finds a Q while deciphering a message one knows exactly which
letter will be next; U.
Also, by applying frequency analysis to the cipher-text the number
of possibilities for each letter drops; leading the analyst to the
correct cipher-alphabet more quickly.Frequency analysis is the
practice of analyzing the frequency of use of each individual
letter of the alphabet in the language of encryption in order to
know the normal amount of use of those letters.The frequencies for
the letters of the alphabet used in English are in the table
below;
As you can see the letter E is the most used letter in the English
language with a frequency of 12.702%.Thus, when analyzing an
enciphered message if you found a letter in the text with a
frequency greater than 10%; the letter E would be a good guess for
the actual letter intended.Going back to the previous examples; we
will put this method of analysis to the test.
From the first example, the one with the repeated key, the
cipher-text was: KLXWILEVOGPVXBISLEJOJMICDVRXCHCSLCC.Below is a
table with the frequencies of the letters in the cipher-text.
As you can see frequency analysis, at least in this manner, is not
much help when dealing with a short message.Also the use of several
different cipher-alphabets by the Vigenere Cipher removes the
ability of frequency analysis to return reliable values for the
frequencies by a blunt-force attack alone.Instead, when trying to
use frequency analysis to solve the Vigenere Cipher one has to try
and figure out the length of the key.Once one has found the length
of the key, one can better apply frequency analysis to the
message.
The way to find the length of this key is to look for repeated
strings of letters in the cipher-text; then find the distance
between those repeated strings.Once the distances between all
repeated strings are known one can then find all common divisors of
the distances and make a guess as to the length of the key using
the Greatest Common Divisor.Of course, this guess gets more
reliable if there are several different repeated strings of letters
and there are several occurrences of each of these strings.
Another method used to make that decryption of enciphered messages
easier is the use of a crib.Like the blunt-force attacks of before
this method of analyzing an encrypted message can also be tedious
and time consuming but can lead to an answer faster that just
attacking the problem without a structured method.A crib, as
mentioned earlier, is a selection of letters from the cipher-text
for which the true meaning is guessed.The crib is then compared to
the cipher-text by taking bits of the cipher-text and subtracting
the guessed meaning of the crib.This process will give strings of
letters as long as the crib.These strings of letters can then be
analyzed and those strings that make logical sense can be kept
while those that are illogical can be discarded.Now with those
strings of letters that make sense one could think of a set of
words that include these letters and further analyze the
cipher-text as a whole.
As far as the decryption of the Enigma Cipher goes, there were many
different methods used to break the code.Some of those methods
involved complex machines that could run through blunt-force
attacks at a much higher speed than a human; others relied on the
capture of Enigma machines and the code books.After some time the
actual protocols of operating the machine were known and the key
could be captured along with the message.Then all one had to know
was the initial setup for that particular day to use a captured
machine to decrypt the message.The actual processes of decrypting
the Enigma cipher are far too complex for this particular paper but
follow along closely to those described above.
As you can see there is much involved in the protection of
information.The methods for encrypting and decrypting messages
mentioned in this paper are only a few of those available; and new
ways are being developed everyday.With the world as it is today one
can rest assured that the art and science of cryptology will
continue to change throughout the foreseeable future.
References Cited:
- Copeland, B. Jack.Colossus: The Secrets of Bletchley Parks
Codebreaking Computers. New York: Oxford, 2006.
- 2. Hungerford, Thomas W.Abstract Algebra: An Introduction.U.S.:
Thomson Learning, 1997.
3. Loepp, Susan, and William K. Wooters.Protecting Information:
From Classical Error Correction to Quantum Cryptography.New York:
Cambridge University Press, 2006. 4. Ratcliff, Rebecca
Ann.Delusions of Intelligence. New York: Cambridge University
Press, 2006. 5. Dictionary-MSN Encarta.MSN Encarta.MSN.
03-15-2008.. 6. Gallic Revolts-Caesars Gallic Wars Gallic
Revolts.About.com.05-02-2008. .