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Codes, Ciphers, and Cryptography-Ch 3.1

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Codes, Ciphers, and Cryptography-Ch 3.1

Michael A. Karls

Ball State University

- In Chapter 1 we looked at various examples of monoalphabetc substitution ciphers.
- A convenient way to describe these ciphers is via permutations!

- 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

- We say a function f: AB 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

- We say f: AB 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

- Not all functions are 1-1 or onto!

h

0

1

1

4

2

5

A

B

h is neither 1-1 nor onto

- (a) f: AB 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

- (b) f: AB 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).

- (c) Permutations: Let A = Zn and B = Zn where Zn = {0, 1, 2, … , n}. Then a 1-1, onto function f: ZnZn 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!

- 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 0120

(3) represents the cycle 33

(45) represents the cycle 454.

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

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

- Example 2: Write the permutation : Z6Z6 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

- Example 2 (cont.)
- Thus, the table form of the permutation = (01)(2453) is given by:

- Given two permutations : ZnZn and : ZnZn, 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

- Example 3: For Example 1(c), -1 is given by the table below.
- Therefore, -1 = (021)(3)(45) in cycle form.

- 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 12341.

For -1, we have 14321, which is (1432) in cycle notation.

- Given permutations : ZnZn and : ZnZn, the product is the permutation obtained by applying first, then .
- Notation: x = (x).

- Example 4: Find if = (012)(3)(45) and = (01)(2453).
- Solution:
- x (012)(3)(45) x (01)(2453) (x)
0 11 1 0 0

1 22 2 2 4

2 00 0 1 1

3 33 3 3 2

4 44 5 5 3

5 55 4 4 5

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

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

- 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!

- 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!

- By labeling the letters A, B, C, … , Y, Z as 0, 1, 2, … , 24, 25, any substitution cipher is equivalent to some permutation : Z26Z26.
- Usually we just write the letters instead of the numbers!

- 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

- 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”.

- 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!

- 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

- 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

- For Example 7, the ciphertext is
- SIFASHTOI FOARTMRNA SOOITPSNI HPBARICCE

- 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.