AC-Unit1.pptx CRYPTOGRAPHIC NNNNFOR ALL

Unit -1
Introduction to Number Theory
Traditional Symmetric-Key Ciphers

• Divisibility and the Division Algorithm
• The Euclidean Algorithm
• Modular Arithmetic, Prime Numbers, Fermat’s and Euler’s Theorems,
Testing for Primality, The Chinese Remainder Theorem
• Traditional Symmetric-Key Ciphers: Substitution ciphers:
Monoalphabetic ciphers, Polyalphabetic ciphers, Transposition ciphers.
Stream Ciphers and Block Ciphers: Stream Ciphers, Block Ciphers.
Unit 1 CONTENTS

Divisors
● A non-zero Number b divides a if for some m have a=mb (a,b,m all integers)
i.e b divides into a with no remainder and is denoted as b | a and is said b is a
divisor of a.
● Ex: 1,2,3,4,6,8,12,24 divide 24
● 13 | 182, -5 | 30, 17 | 289, -3|33, 17 | 0

The Division Algorithm
● Given any positive integer n and any nonnegative integer a, if we divide a by n,
we get an integer quotient q and an integer remainder r that obey the following
relationship:
a = qn + r 0<=r<n; q= floor(a/n) (4.1)
● Equation (4.1) is referred to as the division algorithm.
● Figure 4.1a demonstrates that, given a and positive n, it is always possible to
find q and r that satisfy the preceding relationship.
● Represent the integers on the number line; a will fall somewhere on that line
Starting at 0, proceed to n, 2n, up to qn, such that qn <= a and (q + 1)n > a.
● The distance from qn to a is r, and we have found the unique values of q and r.
The remainder r is often referred to as a residue.

The Euclidean Algorithm
● One of the basic techniques of number theory is the Euclidean algorithm, which is a simple procedure
for determining the GCD of two positive integers. 2 integers are relatively prime if their only common
positive integer factor is 1.
● Greatest Common Divisor
● Non zero b is defined to be a divisor of a if a = mb for some m, where a, b, and m are integers.
● The greatest common divisor of a and b is the largest integer that divides both a and b. Also gcd(0, 0)
= 0.
● More formally, the positive integer c is said to be the greatest common divisor of a and b if
1. c is a divisor of a and of b.
2. Any divisor of a and b is a divisor of c.
● An equivalent definition is the following:
● gcd(a, b) = max[k, such that k | a and k | b]
● Because we require that the greatest common divisor be positive, gcd(a, b) = gcd(a, -b) =
gcd(-a, b) = gcd(-a,-b). In general, gcd(a, b) = gcd( a , b ).
gcd(60, 24) = gcd(60, -24) = 12

● Also, because all nonzero integers divide 0, we have gcd(a, 0) = a .
● It is stated that two integers a and b are relatively prime if their only common
positive integer factor is 1. This is equivalent to saying that a and b are relatively
prime if gcd(a, b) = 1.
● 8 and 15 are relatively prime because the positive divisors of 8 are 1, 2, 4, and
8, and the positive divisors of 15 are 1, 3, 5, and 15. So 1 is the only integer on
both lists.

Finding the Greatest Common Divisor
● Euclid for easily finding the greatest common divisor of two integers.
● Suppose we have integers a, b such that d = gcd(a, b).
● Because gcd( | a | , | b | ) = gcd(a, b), there is no harm in assuming a >= b > 0. Now dividing a by
b and applying the division algorithm, we can state:
a = q1b + r1 0 <=r1 < b (4.2)
● If it happens that r1 = 0, then b | a and d = gcd(a, b) = b. But if r1 != 0, we can state that d | r1.
This is due to the basic properties of divisibility: the relations d | a and d | b together imply that d
(a - q1b), which is the same as d | r1.
● Consider 4.2 and assume that r1!= 0. Because b > r1,
● we can divide b by r1and apply the division algorithm to obtain:
b = q2r1 + r2 0 <=r2 < r1
● As before, if r2 = 0, then d = r1 and if r2!=0, then d = gcd(r1, r2). The division
● process continues until some zero remainder appears, say, at the (n + 1)th stage where rn-1 is
divided by rn. The result is the following system of equations:

Finding the Greatest Common Divisor
● At each iteration, we have d = gcd(ri, ri+1) until finally d = gcd(rn, 0) = rn.
● Thus, GCD of two integers can be found by repetitive application of the division
algorithm. This scheme is known as the Euclidean algorithm.

Modular Arithmetic
● The Modulus : If a is an integer and n is a positive integer, a mod n is he
remainder. when a is divided by n. The integer n is called the modulus. Thus, for
any integer a, we can rewrite Equation (4.1) as a = qn + r 0<=r<n; q = floor (a/n );
● a = floor (a/n ) * n + (a mod n)
Ex: 11 mod 7 = 4; - 11 mod 7 = 3
● Two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n).
This is written as a ≡ b (mod n)
● 73 ≡ 4 (mod 23) 21 ≡ -9 (mod 10)
● Note that if a ≡ 0 (mod n), then n | a.

Properties of Congruences
● Congruences have the following properties:
1. a ≡ b (mod n) if n (a - b).
2. a ≡ b (mod n) implies b ≡ a (mod n).
3. a ≡ b (mod n) and b ≡ c (mod n) imply a ≡ c (mod n).
● To demonstrate the first point, if n (a - b), then (a - b) = kn for some k.
● So we can write a = b + kn. Therefore, (a mod n) = (remainder when b + kn is
divided by n) = (remainder when b is divided by n) = (b mod n).
● 23 ≡ 8 (mod 5) because 23 - 8 = 15 = 5 * 3
● -11 ≡ 5 (mod 8) because -11 - 5 = -16 = 8 * (-2)
● 81 ≡ 0 (mod 27) because 81 - 0 = 81 = 27 * 3

Modular Arithmetic Operations
● By definition, the (mod n) operator maps all integers into the set of integers {0, 1, c, (n -
1)}. arithmetic operations within the confines of this set? Yes ; this technique is known as
modular arithmetic.
● Modular arithmetic exhibits the following properties:
1. [(a mod n) + (b mod n)] mod n = (a + b) mod n
2. [(a mod n) - (b mod n)] mod n = (a - b) mod n
3. [(a mod n) * (b mod n)] mod n = (a * b) mod n
1. Define (a mod n) = ra and (b mod n) = rb.
● Then we can write a = ra + jn for some integer j and b = rb + kn for some integer k. Then
(a + b) mod n = (ra + jn + rb + kn) mod n = (ra + rb + (k + j)n) mod n = (ra + rb) mod n
● = [(a mod n) + (b mod n)] mod n
● The remaining properties are proven as easily.

Examples of the three properties:
● 11 mod 8 = 3; 15 mod 8 = 7
[(11 mod 8) + (15 mod 8)] mod 8 = 10 mod 8 = 2
(11 + 15) mod 8 = 26 mod 8 = 2
● [(11 mod 8) - (15 mod 8)] mod 8 = -4 mod 8 = 4
(11 - 15) mod 8 = -4 mod 8 = 4
● [(11 mod 8) * (15 mod 8)] mod 8 = 21 mod 8 = 5
(11 * 15) mod 8 = 165 mod 8 = 5
● Exponentiation is performed by repeated multiplication, as in ordinary arithmetic.
● To find 117 mod 13,
○ 112 = 121 ≡ 4 (mod 13)
○ 114 = (112)2 ≡ 42 ≡ 3 (mod 13)
○ 117 ≡ 11 * 4 * 3 ≡ 132 ≡ 2 (mod 13)

Modular addition and multiplication modulo 8
● Looking at addition, results are straightforward & there is a
regular pattern to the matrix. Both matrices are symmetric about
the main diagonal in conformance to the commutative property
of addition and multiplication. As in ordinary addition, there is an
additive inverse, or negative, to each integer in modular
arithmetic. In this case, the negative of an integer x is the integer
y such that (x + y) mod 8 = 0. To find the additive inverse of an
integer in the left-hand column, scan across the corresponding
row of the matrix to find the value 0; the integer at the top of that
column is the additive inverse; thus, (2 + 6) mod 8 = 0.
● Similarly, there is a multiplicative inverse, or reciprocal, to each
integer. In modular arithmetic mod 8, the multiplicative inverse of
x is the integer y such that (x * y) mod 8 = 1 mod 8. Now, to find
the multiplicative inverse of an integer from the multiplication
table, scan across the matrix in the row for that integer to find the
value 1; the integer at the top of that column is the multiplicative
inverse; thus, (3 * 3) mod 8 = 1. Note that not all integers mod 8
have a multiplicative inverse;

Properties of Modular Arithmetic
● Define the set Zn as the set of nonnegative integers less than n: Zn = {0, 1, c, (n - 1)}
● This is referred to as the set of residues, or residue classes (mod n).
● To be more precise, each integer in Zn represents a residue class. We can label the residue
classes (mod n) as [0], [1], [2],....., [n - 1], where [r] = {a: a is an integer, a ≡ r (mod n)}
● The residue classes (mod 4) are
[0] = {……, -16, -12, -8, -4, 0, 4, 8, 12, 16, ……}
[1] = {……, -15, -11, -7, -3, 1, 5, 9, 13, 17, ……}
[2] = {……, -14, -10, -6, -2, 2, 6, 10, 14, 18, ……}
[3] = {……, -13, -9, -5, -1, 3, 7, 11, 15, 19, ……}
● Of all the integers in a residue class, the smallest nonnegative integer is the one used to
represent the residue class. Finding the smallest nonnegative integer to which k is congruent
modulo n is called reducing k modulo n.
● If modular arithmetic is performed within Zn, the properties shown in Table 4.3 hold for integers
in Zn.

There is one peculiarity of modular arithmetic that sets it apart from ordinary arithmetic
First, (as in ordinary arithmetic) we can write, if (a + b) K (a + c) (mod n) then b K c (mod n) (4.4)
(5 + 23) ≡ (5 + 7) (mod 8); ≡ 7(mod 8)
Equation (4.4) is consistent with the existence of an additive inverse. Adding the additive inverse
of a to both sides of Equation (4.4),
((-a) + a + b) ≡ ((-a) + a + c) (mod n)
b ≡ c (mod n)
However, the following statement is true only with the attached condition:
if (a * b) ≡ (a * c)(mod n) then b ≡ c (mod n) if a is relatively prime to n (4.5)
Two integers are relatively prime if their only common positive integer factor is 1. Similar to the
case of Equation (4.4), it can be said that Equation (4.5) is consistent with the existence of a
multiplicative inverse.
Applying the multiplicative inverse of a to both sides of Equation (4.5),
((a-1
)ab) ≡((a-1)ac) (mod n)
b ≡ c (mod n)

AC-Unit1.pptx CRYPTOGRAPHIC NNNNFOR ALL

Euclidean Algorithm
 an efficient way to find the GCD(a,b)
 uses theorem that:

GCD(a,b) = GCD(b, a mod b)
 Euclidean Algorithm to compute GCD(a,b) is:
Euclid(a,b)
if (b=0) then return a;
else return Euclid(b, a mod b);

Extended Euclidean Algorithm
 calculates not only GCD but x & y:
ax + by = d = gcd(a, b)
 useful for later crypto computations
 follow sequence of divisions for GCD but
assume at each step i, can find x &y:
r = ax + by
 at end find GCD value and also x & y
 if GCD(a,b)=1 these values are inverses

Finding Inverses
EXTENDED EUCLID(m, b)
1. (A1, A2, A3)=(1, 0, m);
(B1, B2, B3)=(0, 1, b)
2. if B3 = 0
return A3 = gcd(m, b); no inverse
3. if B3 = 1
return B3 = gcd(m, b); B2 = b–1
mod m
4. Q = A3 div B3
5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q
B3)
6. (A1, A2, A3)=(B1, B2, B3)
7. (B1, B2, B3)=(T1, T2, T3)
8. goto 2

find the multiplicative inverse of 3 mod 5
using Extended Euclid’s Algorithm
Q a b r T1 T2 T
1 5 3 2 0 1 -1
1 3 2 1 1 -1 2
2 2 1 0 -1 2 -5
1 0 2 -5
T=T1-T2*Q

Prime Numbers
 prime numbers only have divisors of 1 and self

they cannot be written as a product of other numbers

note: 1 is prime, but is generally not of interest
 eg. 2,3,5,7 are prime, 4,6,8,9,10 are not
 prime numbers are central to number theory
 list of prime number less than 200 is:
2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101
103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191
193 197 199

Prime Factorization
 to factor a number n is to write it as a product
of other numbers: n=a x b x c
 note that factoring a number is relatively hard
compared to multiplying the factors together to
generate the number
 Fundamental theorem of arithmetic
 the prime factorization of a number n is when
its written as a product of primes

eg. 91=7x13 ; 3600=24
x32
x52

Relatively Prime Numbers &
GCD
 two numbers a, b are relatively prime if have no
common divisors apart from 1

eg. 8 & 15 are relatively prime since factors of 8 are 1,2,4,8
and of 15 are 1,3,5,15 and 1 is the only common factor
 conversely can determine the greatest common divisor
by comparing their prime factorizations and using least
powers

eg. 300=21
x31
x52
18=21
x32
hence
GCD(18,300)=21
x31
x50
=6

Fermat's Theorem
 ap-1
= 1 (mod p)

where p is prime and gcd(a,p)=1
 also known as Fermat’s Little Theorem
 also have: ap
= a (mod p)
 useful in public key and primality testing

Euler Totient Function ø(n)
 when doing arithmetic modulo n
 complete set of residues is: 0..n-1
 reduced set of residues is those numbers
(residues) which are relatively prime to n

eg for n=10,

complete set of residues is {0,1,2,3,4,5,6,7,8,9}

reduced set of residues is {1,3,7,9}
 number of elements in reduced set of residues
is called the Euler Totient Function ø(n)

Euler Totient Function ø(n)
 to compute ø(n) need to count number of
residues to be excluded
 in general need prime factorization, but

for p (p prime) ø(p)=p-1

for p.q (p,q prime) ø(p.q)=(p-1)x(q-1)
 eg.
ø(37) = 36
ø(21) = (3–1)x(7–1) = 2x6 = 12

Euler's Theorem
 a generalisation of Fermat's Theorem
 aø(n)
= 1 (mod n)

for any a,n where gcd(a,n)=1
 eg.
a=3;n=10; ø(10)=4;
hence 34
= 81 = 1 mod 10
a=2;n=11; ø(11)=10;
hence 210
= 1024 = 1 mod 11
 also have: aø(n)+1
= a (mod n)

Primality Testing
 often need to find large prime numbers
 traditionally sieve using trial division

ie. divide by all numbers (primes) in turn less than
the square root of the number

only works for small numbers
 alternatively can use statistical primality tests
based on properties of primes

for which all primes numbers satisfy property

but some composite numbers, called pseudo-
primes, also satisfy the property
 can use a slower deterministic primality test

Chinese Remainder Theorem
 used to speed up modulo computations
 if working modulo a product of numbers
 e.g., mod M, where M = m1m2..mk
 Chinese Remainder theorem lets us work
in each modulus mi separately
 since computational cost is proportional
to size, this is faster than working in the
full modulus M

Chinese Remainder Theorem
 can implement CRT in several ways
 to compute A(mod M)

first compute all ai = A mod mi separately

determine constants ci below, where Mi = M/mi

then combine results to get answer using:

Primitive Roots
 from Euler’s theorem have aø(n)
mod n=1
 consider am
=1 (mod n), GCD(a,n)=1

must exist for m = ø(n) but may be smaller

once powers reach m, cycle will repeat
 if smallest is m = ø(n) then a is called a
primitive root
 if p is prime, then successive powers of a
"generate" the group mod p
 these are useful but relatively hard to find

3-1 INTRODUCTION
Figure 3.1 shows the general idea behind a symmetric-key cipher. The original message from Alice to Bob is called
plaintext; the message that is sent through the channel is called the ciphertext. To create the ciphertext from the
plaintext, Alice uses an encryption algorithm and a shared secret key. To create the plaintext from ciphertext, Bob
uses a decryption algorithm and the same secret key.
3.1.1 Kerckhoff’s Principle
3.1.2 Cryptanalysis
3.1.3 Categories of Traditional Ciphers
Topics discussed in this section:

Figure 3.1 General idea of symmetric-key cipher

If P is the plaintext, C is the ciphertext, and K is the key,
We assume that Bob creates P1; we prove that P1 = P:

Figure 3.2 Locking and unlocking with the same key

3.1.1 Kerckhoff’s Principle
Based on Kerckhoff’s principle, one should always assume that the adversary, Eve, knows the
encryption/decryption algorithm. The resistance of the cipher to attack must be based only on the secrecy of the
key.

Cryptanalysis
As cryptography is the science and art of creating secret codes, cryptanalysis is the science and art
of breaking those codes.
Figure 3.3 Cryptanalysis attacks

Figure 3.4 Ciphertext-only attack
Ciphertext-Only Attack

Figure 3.5 Known-plaintext attack
Known-Plaintext Attack

Figure 3.6 Chosen-plaintext attack
Chosen-Plaintext Attack

Figure 3.7 Chosen-ciphertext attack
Chosen-Ciphertext Attack

3-2 SUBSTITUTION CIPHERS
A substitution cipher replaces one symbol with another. Substitution ciphers can be categorized as either
monoalphabetic ciphers or polyalphabetic ciphers.
3.2.1 Monoalphabetic Ciphres
3.2.2 Polyalphabetic Ciphers
A substitution cipher replaces one symbol with another.
Note

Monoalphabetic Ciphers
In monoalphabetic substitution, the relationship between a symbol in
the plaintext to a symbol in the ciphertext is always one-to-one.
Note

The following shows a plaintext and its corresponding ciphertext. The cipher is probably monoalphabetic because
both l’s (els) are encrypted as O’s.
Example 3.1
The following shows a plaintext and its corresponding ciphertext. The cipher is not monoalphabetic because each
l (el) is encrypted by a different character.
Example 3.2

The simplest monoalphabetic cipher is the additive cipher. This cipher is sometimes called a shift cipher and
sometimes a Caesar cipher, but the term additive cipher better reveals its mathematical nature.
Additive Cipher
Figure 3.8 Plaintext and ciphertext in Z26

Figure 3.9 Additive cipher
When the cipher is additive, the plaintext, ciphertext, and key are
integers in Z26.
Note

Use the additive cipher with key = 15 to encrypt the message “hello”.
We apply the encryption algorithm to the plaintext, character by character:
Solution

Use the additive cipher with key = 15 to decrypt the message “WTAAD”.
Example 3.4
We apply the decryption algorithm to the plaintext character by character:
Solution

Historically, additive ciphers are called shift ciphers. Julius Caesar used an additive cipher to communicate with
his officers. For this reason, additive ciphers are sometimes referred to as the Caesar cipher. Caesar used a key of
3 for his communications.
Shift Cipher and Caesar Cipher
Additive ciphers are sometimes referred to as shift ciphers or Caesar
cipher.
Note

Eve has intercepted the ciphertext “UVACLYFZLJBYL”. Show how she can use a brute-force attack to break the
cipher.
Example 3.5
Eve tries keys from 1 to 7. With a key of 7, the plaintext is “not very secure”, which makes sense.
Solution

Table 3.1 Frequency of characters in English
Table 3.2 Frequency of diagrams and trigrams

Eve has intercepted the following ciphertext. Using a statistical attack, find the plaintext.
Example 3.6
When Eve tabulates the frequency of letters in this ciphertext, she gets: I =14, V =13, S =12, and so on. The most
common character is I with 14 occurrences. This means key = 4.
Solution

Multiplicative Ciphers
In a multiplicative cipher, the plaintext and ciphertext are integers in
Z26; the key is an integer in Z26*.
Note
Figure 3.10 Multiplicative cipher

What is the key domain for any multiplicative cipher?
Example 3.7
The key needs to be in Z26*. This set has only 12 members: 1, 3, 5, 7, 9, 11, 15, 17, 19, 21, 23, 25.
Solution
We use a multiplicative cipher to encrypt the message “hello” with a key of 7. The ciphertext is “XCZZU”.
Example 3.8

Affine Ciphers
Figure 3.11 Affine cipher

The affine cipher uses a pair of keys in which the first key is from Z26* and the second is from Z26. The size of
the key domain is
26 × 12 = 312.
Example 3.09
Use an affine cipher to encrypt the message “hello” with the key pair (7, 2).
Example 3.10

Use the affine cipher to decrypt the message “ZEBBW” with the key pair (7, 2) in modulus 26.
Example 3.11
Solution
The additive cipher is a special case of an affine cipher in which
k1 = 1. The multiplicative cipher is a special case of affine cipher in which k2 = 0.
Example 3.12

Because additive, multiplicative, and affine ciphers have small key domains, they are very vulnerable to brute-
force attack.
Monoalphabetic Substitution Cipher
A better solution is to create a mapping between each plaintext character and the corresponding ciphertext
character. Alice and Bob can agree on a table showing the mapping for each character.
Figure 3.12 An example key for monoalphabetic substitution cipher

We can use the key in Figure 3.12 to encrypt the message
Example 3.13
The ciphertext is

3.2.2 Polyalphabetic Ciphers
In polyalphabetic substitution, each occurrence of a character may have a different substitute.
The relationship between a character in the plaintext to a character in the ciphertext is one-to-
many.
Autokey Cipher

Assume that Alice and Bob agreed to use an autokey cipher with initial key value k1 = 12. Now Alice wants to
send Bob the message “Attack is today”. Enciphering is done character by character.
Example 3.14

Playfair Cipher
Figure 3.13 An example of a secret key in the Playfair cipher
Let us encrypt the plaintext “hello” using the key in Figure 3.13.
Example 3.15

Vigenere Cipher
We can encrypt the message “She is listening” using the 6-character keyword “PASCAL”.
Example 3.16

Let us see how we can encrypt the message “She is listening” using the 6-character keyword “PASCAL”. The
initial key stream is (15, 0, 18, 2, 0, 11). The key stream is the repetition of this initial key stream (as many times
as needed).
Example 3.16

Vigenere cipher can be seen as combinations of m additive ciphers.
Example 3.17
Figure 3.14 A Vigenere cipher as a combination of m additive ciphers

Using Example 3.18, we can say that the additive cipher is a special case of Vigenere cipher in which m = 1.
Example 3.18
Table 3.3
A Vigenere Tableau

Vigenere Cipher (Crypanalysis)
Let us assume we have intercepted the following ciphertext:
Example 3.19
The Kasiski test for repetition of three-character segments yields the results shown in Table 3.4.

Let us assume we have intercepted the following ciphertext:
Example 3.19
The Kasiski test for repetition of three-character segments yields the results shown in Table 3.4.

The greatest common divisor of differences is 4, which means that the key length is multiple of 4. First try m = 4.
Example 3.19 (Continued)
In this case, the plaintext makes sense.

Hill Cipher
Figure 3.15 Key in the Hill cipher
The key matrix in the Hill cipher needs to have a multiplicative
inverse.
Note

For example, the plaintext “code is ready” can make a 3 × 4 matrix when adding extra bogus character “z” to the
last block and removing the spaces. The ciphertext is “OHKNIHGKLISS”.
Example 3.20
Figure 3.16 Example 3.20

Assume that Eve knows that m = 3. She has intercepted three plaintext/ciphertext pair blocks (not necessarily
from the same message) as shown in Figure 3.17.
Example 3.21

She makes matrices P and C from these pairs. Because P is invertible, she inverts the P matrix and multiplies it by
C to get the K matrix as shown in Figure 3.18.
Example 3.21
Now she has the key and can break any ciphertext encrypted with that key.
(Continued)

One of the goals of cryptography is perfect secrecy. A study by Shannon has shown that perfect
secrecy can be achieved if each plaintext symbol is encrypted with a key randomly chosen from a
key domain. This idea is used in a cipher called one-time pad, invented by Vernam.
One-Time Pad

Rotor Cipher
Figure 3.19 A rotor cipher

3.2.2 Continued
Enigma Machine
Figure 3.20 A schematic of the Enigma machine

3-3 TRANSPOSITION CIPHERS
A transposition cipher does not substitute one symbol for another, instead it changes the location of the
symbols.
3.3.1 Keyless Transposition Ciphers
3.3.2 Keyed Transposition Ciphers
3.3.3 Combining Two Approaches
A transposition cipher reorders symbols.
Note

Keyless Transposition Ciphers
Simple transposition ciphers, which were used in the past, are keyless.
A good example of a keyless cipher using the first method is the rail fence cipher. The ciphertext is created
reading the pattern row by row. For example, to send the message “Meet me at the park” to Bob, Alice writes
Example 3.22
She then creates the ciphertext “MEMATEAKETETHPR”.

Alice and Bob can agree on the number of columns and use the second method. Alice writes the same plaintext,
row by row, in a table of four columns.
Example 3.23
She then creates the ciphertext “MMTAEEHREAEKTTP”.

The cipher in Example 3.23 is actually a transposition cipher. The following shows the permutation of each
character in the plaintext into the ciphertext based on the positions.
Example 3.24
The second character in the plaintext has moved to the fifth position in the ciphertext; the third character has
moved to the ninth position; and so on. Although the characters are permuted,
there is a pattern in the permutation: (01, 05, 09, 13), (02, 06, 10, 13), (03, 07, 11, 15), and (08, 12). In each section,
the difference between the two adjacent numbers is 4.

3.3.2 Keyed Transposition Ciphers
The keyless ciphers permute the characters by using writing plaintext in one way and reading it in
another way The permutation is done on the whole plaintext to create the whole ciphertext.
Another method is to divide the plaintext into groups of predetermined size, called blocks, and
then use a key to permute the characters in each block separately.

Alice needs to send the message “Enemy attacks tonight” to Bob..
Example 3.25
The key used for encryption and decryption is a permutation key, which shows how the character are permuted.
The permutation yields

3.3.3 Combining Two Approaches
Example 3.26
Figure 3.21

Figure 3.22 Encryption/decryption keys in transpositional ciphers
Keys
In Example 3.27, a single key was used in two directions for the column exchange: downward for encryption,
upward for decryption. It is customary to create two keys.

Figure 3.23 Key inversion in a transposition cipher

Using Matrices
We can use matrices to show the encryption/decryption process for a transposition cipher.
Figure 3.24 Representation of the key as a matrix in the transposition cipher
Example 3.27

Figure 3.24 Representation of the key as a matrix in the transposition cipher
Figure 3.24 shows the encryption process. Multiplying the 4 × 5 plaintext matrix by the 5 × 5 encryption key gives
the 4 × 5 ciphertext matrix.
Example 3.27

Double Transposition Ciphers
Figure 3.25 Double transposition cipher

STREAM AND BLOCK CIPHERS
The literature divides the symmetric ciphers into two broad categories: stream ciphers and block ciphers.
Although the definitions are normally applied to modern ciphers, this categorization also applies to
traditional ciphers.
3.4.1 Stream Ciphers
3.4.2 Block Ciphers
3.4.3 Combination

Stream Ciphers
Call the plaintext stream P, the ciphertext stream C, and the key stream K.
Figure 3.26 Stream cipher

3.4.1 Continued
Additive ciphers can be categorized as stream ciphers in which the key stream is the repeated value of the key. In
other words, the key stream is considered as a predetermined stream of keys or
K = (k, k, …, k). In this cipher, however, each character in the ciphertext depends only on the corresponding
character in the plaintext, because the key stream is generated independently.
Example 3.30
The monoalphabetic substitution ciphers discussed in this chapter are also stream ciphers. However, each value of
the key stream in this case is the mapping of the current plaintext character to the corresponding ciphertext
character in the mapping table.
Example 3.31

Vigenere ciphers are also stream ciphers according to the definition. In this case, the key stream is a repetition of
m values, where m is the size of the keyword. In other words,
Example 3.32
We can establish a criterion to divide stream ciphers based on their key streams. We can say that a stream cipher
is a monoalphabetic cipher if the value of ki does not depend on the position of the plaintext character in the
plaintext stream; otherwise, the cipher is polyalphabetic.
Example 3.33

Combination
In practice, blocks of plaintext are encrypted individually, but they use a stream of keys to encrypt
the whole message block by block. In other words, the cipher is a block cipher when looking at the
individual blocks, but it is a stream cipher when looking at the whole message considering each
block as a single unit.

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