Cryptology
Chapter 1 – Origins, Examples, Ideas in Cryptology
 Cryptology is secret writing.
 Making transmitted information secure from eavesdroppers and tampering.
 1.1 A Crypto-Chronology
o Cryptography – the science and art of designing and using methods of
message concealment.
 Kings in antiquity shaving a slaves head, tattoo a message, and
concealing with hair growth.
 During WWI Germans used invisible ink.
 During WWII Roosevelt and Churchill communicated over
transatlantic cable using synchronization of two identical sound
recordings.
o Cryptanalysis –the activity of breaking a message concealment method.
o Cryptanalyst – person who engages in
the above activity.
o Cryptology – interaction between
cryptography and cryptanalysis.
o Data Encryption Standard (DES)
o Modified Hieroglyphics – 1900 B.C.
 Egyptian tomb writing made
different enough to cause
intrigue, rather then encryption.
o Atbash 500 B.C.
 Reversing the order of letters in a word. The word abc would become
zyx.
 A is replaced with Z, B with Y, and so forth through the entire
alphabet.
o Spartan Scytale (pronounced si’-ta-lee) 500 B.C.
 A long strip of paper (one letter wide) is wrapped
around a cylindrical tube. A message is then
written across the strips. The messenger delivers
the message to an individual who has the same
diameter cylinder which allows them to decipher the
message.
o Polybius’ Checkerboard, 205 – 123 B.C.
 The letters of the alphabet are arranged in a table
whose outer edges are numbered. Each letter is
represented by a two digit number. (Ex: 35 is P).
The first digit represents the row, and the second digit the column.
o Caesar Cipher 50 B.C.
 A shift cipher. A becomes D, B becomes E, etc.
o Nomenclators 1400
 A Nomenclator is a code book. It is divided into two parts. One
converts words into code. The second converts code into words.


An example of the first part is a phone book.
It converts names to phone numbers.
o Alberti’s Cipher Disk
 Two movable disks are positioned one inside
the other both containing the letters of the
alphabet.
o Polyalphabetic substitution
 Like the shift cipher, except that after a few
words the wheel is moved. Part of the
message contains where to place the wheel.
Early Writings on Cryptology
o Shihab al-Din abu etc. in an encyclopedia included work on cryptanalysis
(15th century)
o Johannes Trithemius, published 1518 posthumously, discussed a form of the
polyalphabet substitution
o Porta’s Digraphic System, 1535-1615
 Special symbols were used to represent pairs of letters.
 It obscures letter frequency
o Hill Cipher – Pairs of letters are substituted for other pairs of letters.
o Block Ciphers – A block of any
number of letters is used to represent
a single character or another block of
characters.
o Stream Ciphers – Each single
character of plaintext is transformed
into a corresponding ciphertext
character.
o Blaise de Vigenere, 1523-1596
 The picture to the right is an
example of a Vigenere
Square.
 To encipher a message: For
each letter in the message
find its column. Then go down along the side to the row that begins
with that letter. Then proceed across the row to the selected column.
The letter is coded with that entry.
 Deciphering, you find the letter in the row. Then go along the side to
the column that is that letter. Then proceed down until you reach the
row. The letter is coded with that entry.
o Francis Bacon’s Bilateral Cipher, 1623
 The letters of the alphabet are represented by binary. Instead of 0 and
1, Bacon uses a and b.
 A is aaaaa, B is aaaab, C
is aaaba, etc.
o Thomas Jefferson’s Wheel
Cypher

o
o
o
o
Consisted of 36 concentric wooden disks, each about 1/6 inch thick
and 2 inches in diameter. Each disk consisted of the alphabet in
random order. Each disk moves independently. Move the disks so
that the message you want to encode appears. Then copy down
another line of letters. Send this message. The receiver aligns the
wheel according to this line of code,
then finds the row that makes sense.
The Telegraph, 1844: A Glimpse of Error
correcting Codes
 A table consisting of rows and
columns contains the code. Each
letter of the message is placed into
one box of the table. A route, shown
is 3, is selected. The coded message
can then be decoded by knowing the
correct route and decoding the message.
The Vernam One-Time Tape, 1917
 Plaintext is combined with a
random key. The sender and
receiver are the only ones that
have both of these. The two must
be of equal length. When
combined, letter by letter, then
divided by some number, the
remainder gives the coded
message.
ADFGVX, 1918
 The rows and columns of a 6x6 table are
named by these six letters. The entries in
each box are filled with letters and digits (36
total).
 Here, the digit 2 is represented by FG.
 A message is then written out so that each
line of the message contains the same number of letters.
 Suppose we put the letters in 4 per row. Above all the letters
place a word, for example spam. Now write a permutation of
this word and move the columns according to the permutation.
Then write out the entire code as a string of letters involving
ADFGVX.
Cryptology and Mathematics Linked, 1920s
 Cryptology got financial support from George Fabyan in 1917 at
Riverbank Laboratory in Geneva, Illinois.
 Lester Hill’s “Cryptography in Algebraic Alphabet” (1929) indicated
the link between mathematics and cryptology.
 During WW II cryptologists broke both the Japanese and German
secret codes.

o
o
o
o
William F. Friedman broke the Japanese code. Especially
helpful at Midway which turned the war in our favor.
 German engineer Arthur Scherbius designed the enigma
machine. The allies broke the code and it is estimated that 2
years were saved in the European war theater. The allies never
broke the code. They were eventually able to read the code
because of operator mistakes, procedural flaws, and obtaining
code books and an enigma machine.
Information Theory: The Mathematics of Language and Cryptology, 1949
 In 1949, Claude Shannon, in the Bell Systems Technical Journal,
defined ad examined the idea of information capacity and entropy of a
signal. He defined the concept of perfect security. He looked at
languages and the probability of breaking security.
Data Encryption Standard, 1977
 With the growth of industry and commerce after WW II and the use of
telephones and other electronic devices to transmit data in binary form,
security became a serious issue.
 In 1975 the National Bureau of Standards obtained proposals for
standards, the NSA modified them, and in 1977 the data encryption
standard was set.
 The standard has been efficient and is difficult to break. It is based on
a 56 bit key.
Public-Key Cryptography, 1978
 DES – Data Encryption Standard
 How do you distribute the keys among the parties using the
cryptosystem?
 In 1976, Merkle and Hellman proposed the method.
 The basic idea is this: You have a large number that is the
product of two primes. The large number is given to everyone
that wants it. The two primes are kept secret. The primes are
what makes the information secret.
 Here is the basic idea again: Two parties want to exchange
encrypted information.
o Party A selects an encryption key and a decryption key.
o Party A publishes the encryption key, but keeps the
decryption key secret.
o Party B does the same.
o Party A calls party B and says I want to use encryption
X with key K.
o Party B agrees and they transmit information.
o Anyone else will have to find the answer by trial and
error which in essence takes a very long time.
The Beginning of the Twenty-First Century
 Obviously security is important today: ATMs, purchasing over
internet, email, etc.
 Cryptographic Protocols

Integrity: Determine whether a message between parties has
been altered.
 Authenticity: Determine whether a message is in fact from the
party it claims to be.
 Nonrepudiation: Ensuring agreements are not repudiated (brake
agreements)
o The Future Y2K and Beyond
 The Advanced Encryption Standard (AES) will replace the DES. It
meets the following requirements:
 Use symmetric (secret-key) cryptography.
 It would be a block cipher.
In cryptography, a block cipher is a symmetric key cipher which operates on fixedlength groups of bits, termed blocks, with an unvarying transformation. When encrypting,
a block cipher might take (for example) a 128-bit block of plaintext as input, and output a
corresponding 128-bit block of ciphertext. The exact transformation is controlled using a
second input — the secret key. Decryption is similar: the decryption algorithm takes, in
this example, a 128-bit block of ciphertext together with the secret key, and yields the
original 128-bit block of plaintext.
To encrypt messages longer than the block size (128 bits in the above example), a mode
of operation is used.
Block ciphers can be contrasted with stream ciphers; a stream cipher operates on
individual digits one at a time, and the transformation varies during the encryption. The
distinction between the two types is not always clear-cut: a block cipher, when used in
certain modes of operation, acts effectively as a stream cipher.



It would operate on 128 bit blocks of plaintext and allow for
three sizes of key: 128, 192, 256 bit.
 The replacement is currently underway as of 2002. (When
book was written)
Number theory plays an important role in cryptography.
Zero-Knowledge protocols. Here is a story that
describes this protocol.
In this story, Peggy has uncovered the secret word used to open a
magic door in a cave. The cave is shaped like a circle, with the
entrance on one side and the magic door blocking the opposite side.
Victor says he'll pay her for the secret, but not until he's sure that she
really knows it. Peggy says she'll tell him the secret, but not until she
receives the money. They devise a scheme by which Peggy can
prove that she knows the word without telling it to Victor.
First, Victor waits outside the cave as Peggy goes in. We label the
left and right paths from the entrance A and B. She randomly takes
either path A or B. Then, Victor enters the cave and shouts the name of the path he wants
her to use to return, either A or B, chosen at random. Providing she really does know the
magic word, this is easy: she opens the door, if necessary, and returns along the desired
path. Note that Victor does not know which path she has gone down.
However, suppose she did not know the word. Then, she would only be able to return by
the named path if Victor were to give the name of the same path that she had entered by.
Since Victor would choose A or B at random, he would have a 50% chance of guessing
correctly. If they were to repeat this trick many times, say 20 times in a row, her chance
of successfully anticipating all of Victor's requests would become vanishingly small.
Thus, if Peggy reliably appears at the exit Victor names, he can conclude that she is very
likely to know the secret word.
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
Oblivious Transfer Channel: You send two messages at a time, and a
coin flip decides which of the two messages to transmit to the receiver.
For the receiver, it makes it very unlikely that the sender is fraudulent.
 Elliptic cure cryptography: based on algebraic abstractions of certain
types of geometric curves.
 Quantum cryptography: Use atomic states for encryption. No one as
of this date has been able to create a device to do this.
 Biometrics: Use machines that recognize unique characteristic features
of individuals.
Cryptology and Mathematics: Functions
o The concept of function is fundamental to cryptology: defining, evaluating,
and inverting functions.
o Definition 1.2.1 – A function is a rule by which each element of one set,
called the domain, is associated with exactly one element of another set.
 f used as function notation.
 x is representative of element of domain.
 f(x) is the corresponding value in what is called the range.
o The domain can represent many things: numbers, strings of characters,
collection of functions, etc.
o The function can be: a verbal description, a graph, an algorithm, a formula,
etc.
 Example: suppose x is a string of characters and f is a function that
shifts all the characters one letter right. Then:
 f(arm) = bsn
 f(string) = tusjoh
o It is possible that some things are not in the domain: Example: The domain
consists of all words that have an even number of characters, and f is the
function of switching ad joint letters 1 and 2, 3 and 4, etc.
o Example: A function can have more than one variable. Example: f(x, n)
where x is the string and n is the number of letters to shift right.
 ex: f(“amessage”, 2) = coguucig.

o Sometimes a function may depend on two variables, but dependence on one is
more important than the other. The less important one is written as a
subscript.
o A function is 1-1 if each element in range is paired with only one element in
the domain.
 All 1-1 functions have inverses: (f-1)
 The range of the inverse function is the domain of the original.
o The fundamental connection between math and cryptology:
 For each key (128, 192, or 256 bit), an encryption method defines a 11 function and the decryption is its inverse.
 Encryption – evaluating a function for a given key.
 Decryption – evaluating an inverse function for the same key.
o The composition of functions is the process of performing encryption twice.
 The inverse function would require two decryptions.
o Permutations
 A permutation of n ordered objects is a way of reordering them.
Crypto: Models, Maxims, and Mystique
o General Concepts and Terminology
 Often Alive and Bob are used as names of two things that want to
communicate.
 Eve and Oscar are their opponents.
 The key permits encipherment of messages between Alive and Bob.
 Eve and Oscar try to determine the key from the message.
 Codes are messages that are generally not human readable. Ex: JPEG
encoding, ASCII, scan codes, etc.
 The purpose of a code is not for concealment but to make it
easily transmittable.
 The term steganography is the activity of hiding the existence
of a message. Ex: Invisible ink. (use lemon juice. Message
becomes visible when paper is heated.)
 The Mona Lisa method is hiding information in a picture by
altering its bit information slightly, not enough for the human
eye to detect.
 When you apply cryptographic methods you make people aware of a
message. This may cause one to determine what the message is
concealing.
 Genearlly, a public key cryptographic algorithm requires one key for
encipherment and another key for decipherment.
 The decryption key cannot be deduced from the encryption key
in a reasonable amount of time.
 Each entity using the algorithm generates its own
encryption/decryption key par and publicizes the encryption key.
o A Maxim of Cryptography and Methods of Attack
 Kerckhoffs’ maxim: The strength of a cipher system depends on
keeping the key information secret, not the algorithm.
 Ciphertext-Only Attack
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
The attacker has access to ciphertexts only. They then try to
deduce the message.
Known-Plaintext Attack
