1 codes, ciphers, and cryptography-ch 3.1 michael a. karls ball state university

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Codes, Ciphers, and Cryptography-Ch 3.1

Michael A. Karls

Ball State University

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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!

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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)

A B

xf(x)

f

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One-to-One Function

We say a function f: AB is one-to-one (1-1) if f(x1) f(x2) whenever x1 x2.

1

4

5

0

365

A B

f

f is 1-1

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Onto Function

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

1

4

5

1

2

A B

g

g is onto

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“Neither” Function

Not all functions are 1-1 or onto!

1

4

5

0

1

2

A B

h

h is neither 1-1 nor onto

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Example 1: Some functions!

(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

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Example 1: Some functions!

(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).

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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: 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!

x 0 1 2 3 4 5

x 1 2 0 3 5 4

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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 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.

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Cycle Notation

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

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Cycle Notation

Example 2: Write the permutation : Z6Z6 given by (01)(2453) in table form.

Solution: x (01)(2453) x

0 1 11 0 02 2 43 3 24 4 55 5 3

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Cycle Notation

Example 2 (cont.) Thus, the table form of the permutation

= (01)(2453) is given by:

x 0 1 2 3 4 5

x 1 0 4 2 5 3

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Operations on Permutations

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.

A B

x

y-1

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Operations on Permutations

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

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

x 0 1 2 3 4 5

x^(-1) 2 0 1 3 5 4

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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 12341. For -1, we have 14321, which is (1432) in cycle notation.

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Operations on Permutations

Given permutations : ZnZn and : ZnZn, the product is the permutation obtained by applying first, then .

Notation: x = (x).

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Operations on Permutations

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

0 11 1 0 01 22 2 2 42 00 0 1 13 33 3 3 24 44 5 5 35 55 4 4 5

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

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Operations on Permutations

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

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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!

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Operations on Permutations

Example 5(cont.) x (01)(2354) x^(-1) (021)(3)(45) (x^(-1))^(-1)

0 1 01 0 22 3 33 5 44 2 15 4 5

Thus, -1-1= (0)(1234)(5), so()-1 = -1-1 for this example!

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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 : Z26Z26.

Usually we just write the letters instead of the numbers!

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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

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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”.

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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!

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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

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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

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Permutation Ciphers

T H I S I S A F O

1 2 3 4 5 6 7 8 9

4 3 8 7 6 2 1 9 5

S I F A S H T O I

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Permutation Ciphers

R M O F A T R A N

1 2 3 4 5 6 7 8 9

4 3 8 7 6 2 1 9 5

F O A R T M R N A

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Permutation Ciphers

S P O S I T I O N

1 2 3 4 5 6 7 8 9

4 3 8 7 6 2 1 9 5

S O O I T P S N I

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Permutation Ciphers

C I P H E R A B C

1 2 3 4 5 6 7 8 9

4 3 8 7 6 2 1 9 5

H P B A R I C C E

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Permutation Ciphers

For Example 7, the ciphertext is SIFASHTOI FOARTMRNA SOOITPSNI

HPBARICCE

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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.