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Cryptography

By:

Dr. Mohammad Shoab

Week 2

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Symmetric Encryption

or conventional / private-key / single-key
sender and recipient share a common key
all classical encryption algorithms are private-key
was only type prior to invention of public-key in 1970’s
and by far most widely used

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Some Basic Terminology

plaintext - original message
ciphertext - coded message
cipher - algorithm for transforming plaintext to ciphertext
key - info used in cipher known only to sender/receiver
encipher (encrypt) - converting plaintext to ciphertext
decipher (decrypt) - recovering plaintext from ciphertext
cryptography - study of encryption principles/methods
cryptanalysis (codebreaking) - study of principles/ methods of deciphering ciphertext without knowing key
cryptology - field of both cryptography and cryptanalysis

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Symmetric Cipher Model

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Requirements

two requirements for secure use of symmetric encryption:

a strong encryption algorithm
a secret key known only to sender / receiver

mathematically have:

Y = E(K, X)

X = D(K, Y)

assume encryption algorithm is known
implies a secure channel to distribute key

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There are two requirements for secure use of conventional encryption that mean we assume that it is impractical to decrypt a message on the basis of the cipher- text plus knowledge of the encryption/decryption algorithm, and hence do not need to keep the algorithm secret; rather we only need to keep the key secret. This feature of symmetric encryption is what makes it feasible for widespread use. It allows easy distribution of s/w and h/w implementations.

Can take a closer look at the essential elements of a symmetric encryption scheme: mathematically it can be considered a pair of functions with: plaintext X, ciphertext Y, key K, encryption algorithm E, decryption algorithm D. The intended receiver, in possession of the key, is able to invert the transformation. An opponent, observing Y but not having access to K or X, may attempt to recover X or K.

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Cryptography

can characterize cryptographic system by:

type of encryption operations used

substitution
transposition
product

number of keys used

single-key or private
two-key or public

way in which plaintext is processed

block
stream

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Cryptographic systems can be characterized along these three independent dimensions.

The type of operations used for transforming plaintext to ciphertext. All encryption algorithms are based on two general principles: substitution, in which each element in the plaintext (bit, letter, group of bits or letters) is mapped into another element, and transposition, in which elements in the plaintext are rearranged. The fundamental requirement is that no information be lost (that is, that all operations are reversible). Most systems, referred to as product systems, involve multiple stages of substitutions and transpositions.
The number of keys used. If both sender and receiver use the same key, the system is referred to as symmetric, single-key, secret-key, or conventional encryption. If the sender and receiver use different keys, the system is referred to as asymmetric, two-key, or public-key encryption.
The way in which the plaintext is processed. A block cipher processes the input one block of elements at a time, producing an output block for each input block. A stream cipher processes the input elements continuously, producing output one element at a time, as it goes along.

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Cryptanalysis

objective to recover key not just message
general approaches:

cryptanalytic attack
brute-force attack

if either succeed all key use compromised

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Typically objective is to recover the key in use rather then simply to recover the plaintext of a single ciphertext. There are two general approaches:

Cryptanalysis: relies on the nature of the algorithm plus perhaps some knowledge of the general characteristics of the plaintext or even some sample plaintext- ciphertext pairs. This type of attack exploits the characteristics of the algorithm to attempt to deduce a specific plaintext or to deduce the key being used.

Brute-force attacks try every possible key on a piece of ciphertext until an intelligible translation into plaintext is obtained. On average,half of all possible keys must be tried to achieve success.

If either type of attack succeeds in deducing the key, the effect is catastrophic: All future and past messages encrypted with that key are compromised.

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Cryptanalytic Attacks

ciphertext only

only know algorithm & ciphertext, is statistical, know or can identify plaintext

known plaintext

know/suspect plaintext & ciphertext

chosen plaintext

select plaintext and obtain ciphertext

chosen ciphertext

select ciphertext and obtain plaintext

chosen text

select plaintext or ciphertext to en/decrypt

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More Definitions

unconditional security

no matter how much computer power or time is available, the cipher cannot be broken since the ciphertext provides insufficient information to uniquely determine the corresponding plaintext

computational security

given limited computing resources (eg time needed for calculations is greater than age of universe), the cipher cannot be broken

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Brute Force Search

always possible to simply try every key
most basic attack, proportional to key size
assume either know / recognise plaintext

Key Size (bits)	Number of Alternative Keys	Time required at 1 decryption/µs	Time required at 10⁶ decryptions/µs
32	2³² = 4.3 × 10⁹	2³¹ µs = 35.8 minutes	2.15 milliseconds
56	2⁵⁶ = 7.2 × 10¹⁶	2⁵⁵ µs = 1142 years	10.01 hours
128	2¹²⁸ = 3.4 × 10³⁸	2¹²⁷ µs = 5.4 × 10²⁴ years	5.4 × 10¹⁸ years
168	2¹⁶⁸ = 3.7 × 10⁵⁰	2¹⁶⁷ µs = 5.9 × 10³⁶ years	5.9 × 10³⁰ years
26 characters (permutation)	26! = 4 × 10²⁶	2 × 10²⁶ µs = 6.4 × 10¹² years	6.4 × 10⁶ years

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Classical Substitution Ciphers

where letters of plaintext are replaced by other letters or by numbers or symbols
or if plaintext is viewed as a sequence of bits, then substitution involves replacing plaintext bit patterns with ciphertext bit patterns

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Caesar Cipher

earliest known substitution cipher
by Julius Caesar
first attested use in military affairs
replaces each letter by 3rd letter on
example:

meet me after the toga party

PHHW PH DIWHU WKH WRJD SDUWB

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Caesar Cipher

can define transformation as:

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

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

mathematically give each letter a number

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

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25

then have Caesar cipher as:

c = E(k, p) = (p + k) mod (26)

p = D(k, c) = (c – k) mod (26)

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Cryptanalysis of Caesar Cipher

only have 26 possible ciphers

A maps to A,B,..Z

could simply try each in turn
a brute force search
given ciphertext, just try all shifts of letters
do need to recognize when have plaintext
eg. break ciphertext "GCUA VQ DTGCM"

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Monoalphabetic Cipher

rather than just shifting the alphabet
could shuffle (jumble) the letters arbitrarily
each plaintext letter maps to a different random ciphertext letter
hence key is 26 letters long

Plain: abcdefghijklmnopqrstuvwxyz

Cipher: DKVQFIBJWPESCXHTMYAUOLRGZN

Plaintext: ifwewishtoreplaceletters

Ciphertext: WIRFRWAJUHYFTSDVFSFUUFYA

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Monoalphabetic Cipher Security

now have a total of 26! = 4 x 10²⁶ keys
with so many keys, might think is secure
but would be !!!WRONG!!!
problem is language characteristics

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Language Redundancy and Cryptanalysis

human languages are redundant
eg "th lrd s m shphrd shll nt wnt"
letters are not equally commonly used
in English E is by far the most common letter

followed by T,R,N,I,O,A,S

other letters like Z,J,K,Q,X are fairly rare
have tables of single, double & triple letter frequencies for various languages

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English Letter Frequencies

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Use in Cryptanalysis

key concept - monoalphabetic substitution ciphers do not change relative letter frequencies
discovered by Arabian scientists in 9^th century
calculate letter frequencies for ciphertext
compare counts/plots against known values
if caesar cipher look for common peaks/troughs

peaks at: A-E-I triple, NO pair, RST triple
troughs at: JK, X-Z

for monoalphabetic must identify each letter

tables of common double/triple letters help

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The simplicity and strength of the monoalphabetic substitution cipher meant it dominated cryptographic use for the first millenium AD. It was broken by Arabic scientists. The earliest known description is in Abu al-Kindi's "A Manuscript on Deciphering Cryptographic Messages", published in the 9th century but only rediscovered in 1987 in Istanbul, but other later works also attest to their knowledge of the field. Monoalphabetic ciphers are easy to break because they reflect the frequency data of the original alphabet. The cryptanalyst looks for a mapping between the observed pattern in the ciphertext, and the known source language letter frequencies. If English, look for peaks at: A-E-I triple, NO pair, RST triple, and troughs at: JK, X-Z.

Monoalphabetic ciphers are easy to break because they reflect the frequency data of the original alphabet.

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Example Cryptanalysis

given ciphertext:

UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ

VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX

EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ

count relative letter frequencies (see text)
guess P & Z are e and t
guess ZW is th and hence ZWP is the
proceeding with trial and error finally get:

it was disclosed yesterday that several informal but

direct contacts have been made with political

representatives of the viet cong in moscow

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Illustrate the process with this example from the text in Stallings section 2.2. Comparing letter frequency breakdown with Figure 2.5, it seems likely that cipher letters P and Z are the equivalents of plain letters e and t, but it is not certain which is which. The letters S, U, O, M, and H are all of relatively high frequency and probably correspond to plain letters from the set {a, h, i, n, o, r, s}. The letters with the lowest frequencies (namely, A, B, G, Y, I, J) are likely included in the set {b, j, k, q, v, x, z}. A powerful tool is to look at the frequency of two-letter combinations, known as digrams. A table similar to Figure 2.5 could be drawn up showing the relative frequency of digrams. The most common such digram is th. In our ciphertext, the most common digram is ZW, which appears three times. So we make the correspondence of Z with t and W with h. Then, by our earlier hypothesis, we can equate P with e. Now notice that the sequence ZWP appears in the ciphertext, and we can translate that sequence as "the." This is the most frequent trigram (three- letter combination) in English, which seems to indicate that we are on the right track. Next, notice the sequence ZWSZ in the first line. We do not know that these four letters form a complete word, but if they do, it is of the form th_t. If so, S equates with a. Only four letters have been identified, but already we have quite a bit of the message. Continued analysis of frequencies plus trial and error should easily yield a solution from this point. The complete plaintext, with spaces added between words, is shown on slide.

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Playfair Cipher

not even the large number of keys in a monoalphabetic cipher provides security
one approach to improving security was to encrypt multiple letters
the Playfair Cipher is an example
invented by Charles Wheatstone in 1854, but named after his friend Baron Playfair

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Playfair Key Matrix

a 5X5 matrix of letters based on a keyword
fill in letters of keyword (sans duplicates)
fill rest of matrix with other letters
eg. using the keyword MONARCHY

M	O	N	A	R
C	H	Y	B	D
E	F	G	I/J	K
L	P	Q	S	T
U	V	W	X	Z

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Encrypting and Decrypting

plaintext is encrypted two letters at a time

if a pair is a repeated letter, insert filler like 'X’
if both letters fall in the same row, replace each with letter to right (wrapping back to start from end)
if both letters fall in the same column, replace each with the letter below it (wrapping to top from bottom)
otherwise each letter is replaced by the letter in the same row and in the column of the other letter of the pair

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Plaintext is encrypted two letters at a time,according to the rules as shown. Note how you wrap from right side back to left, or from bottom back to top.

if a pair is a repeated letter, insert a filler like 'X', eg. "balloon" encrypts as "ba lx lo on"
if both letters fall in the same row, replace each with letter to right (wrapping back to start from end), eg. “ar" encrypts as "RM"
if both letters fall in the same column, replace each with the letter below it (again wrapping to top from bottom), eg. “mu" encrypts to "CM"
otherwise each letter is replaced by the one in its row in the column of the other letter of the pair, eg. “hs" encrypts to "BP", and “ea" to "IM" or "JM" (as desired)

Decrypting of course works exactly in reverse. Can see this by working the example pairs shown, backwards.

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Security of Playfair Cipher

security much improved over monoalphabetic
since have 26 x 26 = 676 digrams
would need a 676 entry frequency table to analyse (verses 26 for a monoalphabetic)
and correspondingly more ciphertext
was widely used for many years

eg. by US & British military in WW1

it can be broken, given a few hundred letters
since still has much of plaintext structure

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The Playfair cipher is a great advance over simple monoalphabetic ciphers, since there are 26*26=676 digrams (vs 26 letters), so that identification of individual digrams is more difficult. Also,the relative frequencies of individual letters exhibit a much greater range than that of digrams, making frequency analysis much more difficult. The Playfair cipher was for a long time considered unbreakable. It was used as the standard field system by the British Army in World War I and still enjoyed considerable use by the U.S.Army and other Allied forces during World War II. Despite this level of confidence in its security, the Playfair cipher is relatively easy to break because it still leaves much of the structure of the plaintext language intact. A few hundred letters of ciphertext are generally sufficient.

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Polyalphabetic Ciphers

polyalphabetic substitution ciphers
improve security using multiple cipher alphabets
make cryptanalysis harder with more alphabets to guess and flatter frequency distribution
use a key to select which alphabet is used for each letter of the message
use each alphabet in turn
repeat from start after end of key is reached

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Vigenère Cipher

simplest polyalphabetic substitution cipher
effectively multiple caesar ciphers
key is multiple letters long K = k₁ k₂ ... k_d
i^th letter specifies i^th alphabet to use
use each alphabet in turn
repeat from start after d letters in message
decryption simply works in reverse

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Example of Vigenère Cipher

write the plaintext out
write the keyword repeated above it
use each key letter as a caesar cipher key
encrypt the corresponding plaintext letter
eg using keyword deceptive

key: deceptivedeceptivedeceptive

plaintext: wearediscoveredsaveyourself

ciphertext:ZICVTWQNGRZGVTWAVZHCQYGLMGJ

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Security of Vigenère Ciphers

have multiple ciphertext letters for each plaintext letter
hence letter frequencies are obscured
but not totally lost
start with letter frequencies

see if look monoalphabetic or not

if not, then need to determine number of alphabets, since then can attach each

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Kasiski Method

method developed by Babbage / Kasiski
repetitions in ciphertext give clues to period
so find same plaintext an exact period apart
which results in the same ciphertext
of course, could also be random fluke
eg repeated “VTW” in previous example
suggests size of 3 or 9
then attack each monoalphabetic cipher individually using same techniques as before

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For some centuries the Vigenère cipher was le chiffre indéchiffrable (the unbreakable cipher). As a result of a challenge, it was broken by Charles Babbage (the inventor of the computer) in 1854 but kept secret (possibly because of the Crimean War - not the first time governments have kept advances to themselves!). The method was independently reinvented by a Prussian, Friedrich Kasiski, who published the attack now named after him in 1863. However lack of major advances meant that various polyalphabetic substitution ciphers were used into the 20C. One very famous incident was the breaking of the Zimmermann telegram in WW1 which resulted in the USA entering the war. The important is that if two identical sequences of plaintext letters occur at a distance that is an integer multiple of the keyword length, they will generate identical ciphertext sequences.

In general the approach is to find a number of duplicated sequences, collect all their distances apart, look for common factors, remembering that some will be random flukes and need to be discarded. Now have a series of monoalphabetic ciphers, each with original language letter frequency characteristics. Can attack these in turn to break the cipher.

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Autokey Cipher

ideally want a key as long as the message
Vigenère proposed the autokey cipher
with keyword is prefixed to message as key
knowing keyword can recover the first few letters
use these in turn on the rest of the message
but still have frequency characteristics to attack
eg. given key deceptive

key: deceptivewearediscoveredsav

plaintext: wearediscoveredsaveyourself

ciphertext:ZICVTWQNGKZEIIGASXSTSLVVWLA

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Taking the polyalphabetic idea to the extreme, want as many different translation alphabets as letters in the message being sent. One way of doing this with a smallish key, is to use the Autokey cipher.

The example uses the keyword "DECEPTIVE" prefixed to as much of the message "WEAREDISCOVEREDSAV" as is needed. When deciphering, recover the first 9 letters using the keyword "DECEPTIVE". Then instead of repeating the keyword, start using the recovered letters from the message "WEAREDISC". As recover more letters, have more of key to recover later letters.

Problem is that the same language characteristics are used by the key as the message. ie. a key of 'E' will be used more often than a 'T' etc hence an 'E' encrypted with a key of 'E' occurs with probability (0.1275)2 = 0.01663, about twice as often as a 'T' encrypted with a key of 'T' have to use a larger frequency table, but it exists given sufficient ciphertext this can be broken.

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Vernam Cipher

ultimate defense is to use a key as long as the plaintext
with no statistical relationship to it
invented by AT&T engineer Gilbert Vernam in 1918
originally proposed using a very long but eventually repeating key

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One-Time Pad

if a truly random key as long as the message is used, the cipher will be secure
called a One-Time pad
is unbreakable since ciphertext bears no statistical relationship to the plaintext
since for any plaintext & any ciphertext there exists a key mapping one to other
can only use the key once though
problems in generation & safe distribution of key

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The One-Time Pad is an evolution of the Vernham cipher. An Army Signal Corp officer, Joseph Mauborgne, proposed an improvement using a random key that was truly as long as the message, with no repetitions, which thus totally obscures the original message. It produces random output that bears no statistical relationship to the plaintext. Because the ciphertext contains no information whatsoever about the plaintext, there is simply no way to break the code, since any plaintext can be mapped to any ciphertext given some key.

The one-time pad offers complete security but, in practice, has two fundamental difficulties:

There is the practical problem of making large quantities of random keys.
And the problem of key distribution and protection, where for every message to be sent, a key of equal length is needed by both sender and receiver.

Because of these difficulties, the one-time pad is of limited utility, and is useful primarily for low-bandwidth channels requiring very high security. The one-time pad is the only cryptosystem that exhibits what is referred to as perfect secrecy.

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Transposition Ciphers

now consider classical transposition or permutation ciphers
these hide the message by rearranging the letter order
without altering the actual letters used
can recognise these since have the same frequency distribution as the original text

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Rail Fence cipher

write message letters out diagonally over a number of rows
then read off cipher row by row
eg. write message out as:

m e m a t r h t g p r y

e t e f e t e o a a t

giving ciphertext

MEMATRHTGPRYETEFETEOAAT

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Row Transposition Ciphers

is a more complex transposition
write letters of message out in rows over a specified number of columns
then reorder the columns according to some key before reading off the rows

Key: 4312567

Column Out 3 4 2 1 5 6 7

Plaintext: a t t a c k p

o s t p o n e

d u n t i l t

w o a m x y z

Ciphertext: TTNAAPTMTSUOAODWCOIXKNLYPETZ

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Product Ciphers

ciphers using substitutions or transpositions are not secure because of language characteristics
hence consider using several ciphers in succession to make harder, but:

two substitutions make a more complex substitution
two transpositions make more complex transposition
but a substitution followed by a transposition makes a new much harder cipher

this is bridge from classical to modern ciphers

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Rotor Machines

before modern ciphers, rotor machines were most common complex ciphers in use
widely used in WW2

German Enigma, Allied Hagelin, Japanese Purple

implemented a very complex, varying substitution cipher
used a series of cylinders, each giving one substitution, which rotated and changed after each letter was encrypted
with 3 cylinders have 26³=17576 alphabets

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The next major advance in ciphers required use of mechanical cipher machines which enabled to use of complex varying substitutions.

A rotor machine consists of a set of independently rotating cylinders through which electrical pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal wiring that connects each input pin to a unique output pin. If we associate each input and output pin with a letter of the alphabet, then a single cylinder defines a monoalphabetic substitution. After each input key is depressed, the cylinder rotates one position, so that the internal connections are shifted accordingly. The power of the rotor machine is in the use of multiple cylinders, in which the output pins of one cylinder are connected to the input pins of the next, and with the cylinders rotating like an “odometer”, leading to a very large number of substitution alphabets being used, eg with 3 cylinders have 26³=17576 alphabets used.

They were extensively used in world war 2, and the history of their use and analysis is one of the great stories from WW2.

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Hagelin Rotor Machine

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Rotor Machine Principles

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The basic principle of the rotor machine is illustrated in Figure 2.8. The machine consists of a set of independently rotating cylinders through which electrical pulses can flow. Each cylinder has 26 input pins and 26 output pins, with internal wiring that connects each input pin to a unique output pin. If we associate each input and output pin with a letter of the alphabet, then a single cylinder defines a monoalphabetic substitution. If an operator depresses the key for the letter A, an electric signal is applied to the first pin of the first cylinder and flows through the internal connection to the twenty-fifth output pin. Consider a machine with a single cylinder. After each input key is depressed, the cylinder rotates one position, so that the internal connections are shifted accordingly. Thus, a different monoalphabetic substitution cipher is defined. After 26 letters of plaintext, the cylinder would be back to the initial position. Thus, we have a polyalphabetic substitution algorithm with a period of 26.

A single-cylinder system is trivial and does not present a formidable cryptanalytic task. The power of the rotor machine is in the use of multiple cylinders, in which the output pins of one cylinder are connected to the input pins of the next. Figure 2.8 shows a three-cylinder system. With multiple cylinders, the one closest to the operator input rotates one pin position with each keystroke. The right half of Figure 2.8 shows the system's configuration after a single keystroke. For every complete rotation of the inner cylinder, the middle cylinder rotates one pin position. Finally, for every complete rotation of the middle cylinder, the outer cylinder rotates one pin position. The result is that there are 26 " 26 " 26 = 17,576 different substitution alphabets used before the system repeats.

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Steganography

an alternative to encryption
hides existence of message

using only a subset of letters/words in a longer message marked in some way
using invisible ink
hiding in LSB in graphic image or sound file

has drawbacks

high overhead to hide relatively few info bits

advantage is can obscure encryption use

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The End

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Exercise

Q1. What is symmetric encryption?

Q2. What are the requirements of symmetric encryption?

Q3. Explain cryptography and cryptanalysis.

Q4. Explain unconditional security and computational security.

Q5. What is a monoalphabetic cipher? Explain it’s security.

Q6. Explain Playfair cipher with it’s matrix.

Q7. What is a polyalphabetic cipher?

Q8. Explain Vigenere cipher.

Q9. What is Kasiski method?

Q10. What is Vernam cipher?

Q11. What is Steganography?

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Q12. In which encryption sender and recipient share a common key?

Asymmetric encryption
Symmetric encryption
DES
None of the above

Q13. Converting plaintext to ciphertext is called.

Encipher
Decipher
Cryptanalysis
Encoding

Q14. Recovering ciphertext from plaintext is called.

Encipher
Decipher
Cryptanalysis
Encoding

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Q15. Human languages are.

Common
Uncommon
Redundant
Classical

Q16. If a truly random key as long as the message is used, the cipher will be?

Unsecure
Complex
Continuous
Secure

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