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Codes, Ciphers, and Cryptography-Ch 3.1. Michael A. Karls Ball State University. Substitution and Permutation Ciphers. In Chapter 1 we looked at various examples of monoalphabetc substitution ciphers. A convenient way to describe these ciphers is via permutations !. Functions.

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Codes ciphers and cryptography ch 3 1

Codes, Ciphers, and Cryptography-Ch 3.1

Michael A. Karls

Ball State University


Substitution and permutation ciphers

Substitution and Permutation Ciphers

  • In Chapter 1 we looked at various examples of monoalphabetc substitution ciphers.

  • A convenient way to describe these ciphers is via permutations!


Functions

Functions

  • A function f is a rule that assigns to each element x in a set A exactly one element, called f(x), in a set B.

  • Notation: f: A  B; y = f(x)

f

x

f(x)

B

A


One to one function

One-to-One Function

  • We say a function f: AB is one-to-one (1-1) if f(x1)  f(x2) whenever x1 x2.

f

0

1

3

4

6

5

5

A

B

f is 1-1


Onto function

Onto Function

  • We say f: AB is onto if given y in B, there is an element x in A such that f(x) = y.

g

1

1

4

2

5

A

B

g is onto


Neither function

“Neither” Function

  • Not all functions are 1-1 or onto!

h

0

1

1

4

2

5

A

B

h is neither 1-1 nor onto


Example 1 some functions

Example 1: Some functions!

  • (a) f: AB where A = (- 1, 1), B = [0, 1), and f(x) = x2.

  • f(1) = 12 = 1

  • f(2) = 22 = 4

  • f(-2) = (-2)2 = 4

  • f is onto, not 1-1


Example 1 some functions1

Example 1: Some functions!

  • (b) f: AB where A = {books in library}, B = {possible call numbers}, and f is the rule “call number on book spine”.

  • f is 1-1, but not onto (different books have different call numbers).


Example 1 some functions2

Example 1: Some functions!

  • (c) Permutations: Let A = Zn and B = Zn where Zn = {0, 1, 2, … , n}. Then a 1-1, onto function f: ZnZn is called a permutation.

  • Example: Z6 = {0, 1, 2, 3, 4, 5}.  is the permutation given by the table below.

  • Notation: instead of (x), we use x – it will be useful later!


Cycle notation

Cycle Notation

  • We can use cycle notation to describe a permutation!

  • A cycle is a process that repeats itself.

  • As an example,  in Example 1(c) would be written as

     = (012)(3)(45),

    a 3-cycle, followed by a 1-cycle, followed by a 2-cycle.

  • Here,

    (012) represents the cycle 0120

    (3) represents the cycle 33

    (45) represents the cycle 454.

  • Notice that the cycles of  are disjoint, i.e. no symbol appears in more than one cycle.


Cycle notation1

Cycle Notation

  • Fact 1: Every permutation can be written as a product of disjoint cycles.


Cycle notation2

Cycle Notation

  • Example 2: Write the permutation : Z6Z6 given by (01)(2453) in table form.

  • Solution:

  • x (01)(2453) x

    0  1  1

    1  0  0

    2  2  4

    3  3  2

    4  4  5

    5  5  3


Cycle notation3

Cycle Notation

  • Example 2 (cont.)

  • Thus, the table form of the permutation  = (01)(2453) is given by:


Operations on permutations

Operations on Permutations

  • Given two permutations : ZnZn and : ZnZn, we can form new permutations!

  • Given a permutation , the inverse of  is the permutation -1 defined by:

    x = y^(-1) if and only if y = x.

x

y

-1

B

A


Operations on permutations1

Operations on Permutations

  • Example 3: For Example 1(c), -1 is given by the table below.

  • Therefore, -1 = (021)(3)(45) in cycle form.


Operations on permutations2

Operations on Permutations

  • Note: If  is a 1-cycle or a 2-cycle, then -1 = .

    If  = (x1 x2 … xn), then -1 = (x1 xn … x2).

  • “Proof”:

    For  = (1234), we have 12341.

    For -1, we have 14321, which is (1432) in cycle notation.


Operations on permutations3

Operations on Permutations

  • Given permutations : ZnZn and : ZnZn, the product is the permutation obtained by applying  first, then .

  • Notation: x = (x).


Operations on permutations4

Operations on Permutations

  • Example 4: Find  if  = (012)(3)(45) and  = (01)(2453).

  • Solution:

  • x (012)(3)(45) x (01)(2453) (x)

    0  11  1  0  0

    1  22  2  2 4

    2  00  0  1 1

    3  33  3  3 2

    4  44  5  5 3

    5  55  4  4 5

    Thus,  = (0)(1432)(5) (or we could write (1432)).


Operations on permutations5

Operations on Permutations

  • Fact 2: The inverse of a product of permutations is given by ()-1 =  -1 -1.


Operations on permutations6

Operations on Permutations

  • Example 5: Let  and  be as in the last example.

  • Then  = (0)(1432)(5), so it follows from the Note above that ()-1 = (0)(1234)(5).

  • Now, -1 = (021)(3)(45) and -1 = (01)(2354), so by Fact 2,

    ()-1 = -1-1 = (01)(2354) (021)(3)(45).

  • Check that we get the same result!


Operations on permutations7

Operations on Permutations

  • Example 5(cont.)

  • x (01)(2354) x^(-1) (021)(3)(45) (x^(-1))^(-1)

    0  1  0

    1  0  2

    2  3  3

    3  5  4

    4  2  1

    5  4  5

  • Thus, -1-1= (0)(1234)(5), so

    ()-1 = -1-1 for this example!


Substitution ciphers revisited

Substitution Ciphers (Revisited)

  • By labeling the letters A, B, C, … , Y, Z as 0, 1, 2, … , 24, 25, any substitution cipher is equivalent to some permutation : Z26Z26.

  • Usually we just write the letters instead of the numbers!


Substitution ciphers revisited1

Substitution Ciphers (Revisited)

  • Example 6: Use the substitution cipher:

     = (APHITX)(BERC)(DNZFVM)(GJKWLOYQSU)

    to encipher the plaintext “BSUMATH”.

  • x (APHITX)(BERC)(DNZFVM)(GJKWLOYQSU) x

    B  B  E  E  E

    S  S  S  S  U

    U  U  U  U  G


Substitution ciphers revisited2

Substitution Ciphers (Revisited)

  • Example 6 (cont.)

  • x (APHITX)(BERC)(DNZFVM)(GJKWLOYQSU) x

    M  M  M  D  D

    A  P  P  P  P

    T  X  X  X  X

    H  I  I  I  I

  • Thus, “BSUMATH” is encrypted as “EUGDPXI”.


Permutation ciphers

Permutation Ciphers

  • Another way to make a cipher with a permutation is to use a fixed-length permutation to rearrange blocks of text of the same length.

  • We illustrate this method with the next example!


Permutation ciphers1

Permutation Ciphers

  • Example 7 (a permutation cipher)

  • First, choose a permutation of the numbers 1-9.

    For example,  = (147)(238956).

  • Next, break up plaintext into blocks of length 9:

  • this is a form of a transposition cipher


Permutation ciphers2

Permutation Ciphers

  • Example 7 (a permutation cipher)

  • First, choose a permutation of the numbers 1-9.

    For example,  = (147)(238956).

  • Next, break up plaintext into blocks of length 9:

  • this is a fo|rm of a tran|sposition| cipherabc


Permutation ciphers3

Permutation Ciphers


Permutation ciphers4

Permutation Ciphers


Permutation ciphers5

Permutation Ciphers


Permutation ciphers6

Permutation Ciphers


Permutation ciphers7

Permutation Ciphers

  • For Example 7, the ciphertext is

  • SIFASHTOI FOARTMRNA SOOITPSNI HPBARICCE


Permutation ciphers8

Permutation Ciphers

  • Remark: The cipher in Example 7 is known as a stream cipher.

  • Such ciphers can be used for high speed encryption with computers.

  • Flaw: The Friedman Test can be used to guess it is a transposition cipher.


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