ENCODING and DECODING

ENCODING and DECODING

Math 1

Mathematics for General Education

Institute of Mathematics University of the Philippines Diliman

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modulo 12 Multiplication 

a 

b = (a · b) mod 12

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modulo 12 Multiplication 

a 

b = (a · b) mod 12

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modulo 12 Multiplication 

a 

b = (a · b) mod 12

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modulo 12 Multiplication 

a 

b = (a · b) mod 12

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modulo 12 Multiplication 

a 

b = (a · b) mod 12

Modular Systems

Recall:

congruence modulo 12 (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by 12

⇐⇒ a and b have the same remainder when divided by 12

Modulo 12 Addition ⊕

a ⊕

b = (a + b) mod 12

Modular Systems

congruence modulo n (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by n

⇐⇒ a and b have the same remainder when divided by n

Modulo n Addition ⊕

a ⊕

b = (a + b) mod n

Modulo n Multiplication 

a 

b = (a · b) mod n

Modular Systems

congruence modulo n (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by n

⇐⇒ a and b have the same remainder when divided by n

Modulo n Addition ⊕

a ⊕

b = (a + b) mod n

Modulo n Multiplication 

a 

b = (a · b) mod n

Modular Systems

congruence modulo n (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by n

⇐⇒ a and b have the same remainder when divided by n

Modulo n Addition ⊕

a ⊕

b = (a + b) mod n

Modulo n Multiplication 

a 

b = (a · b) mod n

Modular Systems

congruence modulo n (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by n

⇐⇒ a and b have the same remainder when divided by n

Modulo n Addition ⊕

a ⊕

b = (a + b) mod n

Modulo n Multiplication 

a 

b = (a · b) mod n

Modular Systems

congruence modulo n (≡

)

a ≡

b ⇐⇒ (a − b) is divisible by n

⇐⇒ a and b have the same remainder when divided by n

Modulo n Addition ⊕

a ⊕

b = (a + b) mod n

Modulo 2

a ≡

b ⇐⇒ (a − b) is divisible by 2

⇐⇒ a and b have the same remainder when divided by 2

The only possible remainders when we divide an integer by 2 are 0

and 1.

Modulo 2

a ≡

b ⇐⇒ (a − b) is divisible by 2

⇐⇒ a and b have the same remainder when divided by 2

The only possible remainders when we divide an integer by 2 are 0

and 1.

Modulo 2

a ≡

b ⇐⇒ (a − b) is divisible by 2

⇐⇒ a and b have the same remainder when divided by 2

The only possible remainders when we divide an integer by 2 are 0

and 1.

Modulo 2

a ≡

b ⇐⇒ (a − b) is divisible by 2

⇐⇒ a and b have the same remainder when divided by 2

The only possible remainders when we divide an integer by 2 are 0

and 1.

Modulo 2

Example 4 ≡

2 7 ≡

3 2 6≡ 5

We let

Z

= {0, 1}. Note:

0 represents all even integers

1 represents all odd integers

Modulo 2

Example 4 ≡

2 7 ≡

3 2 6≡ 5 We let

Z

= {0, 1}.

Note:

0 represents all even integers

1 represents all odd integers

Modulo 2

Example 4 ≡

2 7 ≡

3 2 6≡ 5 We let

Z

= {0, 1}.

Note:

0 represents all even integers

1 represents all odd integers

Addition and Multiplication Modulo 2

Addition and Multiplication Modulo 3

Z

= {0, 1, 2}

Note that the possible remainders when we divide a whole number by

3 are 0, 1, and 2.

Addition and Multiplication Modulo 3

Z

= {0, 1, 2}

Note that the possible remainders when we divide a whole number by

3 are 0, 1, and 2.

Addition and Multiplication Modulo 3

Z

= {0, 1, 2}

Note that the possible remainders when we divide a whole number by

3 are 0, 1, and 2.

Addition and Multiplication Modulo 4

Z

= {0, 1, 2, 3}

Addition and Multiplication Modulo 4

Z

= {0, 1, 2, 3}

Addition and Multiplication Modulo 5

Z

= {0, 1, 2, 3, 4}

Addition and Multiplication Modulo 5

Z

= {0, 1, 2, 3, 4}

Consider Z

= {0, 1, · · · , 24, 25}.

Assign to each letter an element of Z

.

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10

M = 13

A = 1

T = 20

H = 8

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10

M = 13

A = 1

T = 20

H = 8

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10

To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 =

W

To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W

To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 =

K

To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K

To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 =

D To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D

To encode H: Let x = 8.

=⇒ y = 8 ⊕ 10 = 18 = R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D To encode H: Let x = 8.

R

Encode: MATH

ENCODING FORMULA: y = x ⊕ 10 To encode M: Let x = 13.

=⇒ y = 13 ⊕ 10 = 23 = W To encode A: Let x = 1.

=⇒ y = 1 ⊕ 10 = 11 = K To encode T: Let x = 20.

=⇒ 20 ⊕ 10 = 30 ≡

4 = D

To encode H: Let x = 8.

Encoding Formula: y = x ⊕ 10

Table:Encoding

Encode M A T H

x 13 1 20 8

y 23 11 4 18

Encoded W K D R

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x

DECODING FORMULA: x = y ⊕ 16. W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =

M K: 11 ⊕ 16 = 27 ≡

1 =A D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M

K: 11 ⊕ 16 = 27 ≡

1 =A D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =

A D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A

D: 4 ⊕ 16 = 20 =T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A

T

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A

R: 18 ⊕ 16 = 34 ≡

8 =H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M K: 11 ⊕ 16 = 27 ≡

1 =A

H

Decode: WKDR

W= 23 K= 11 D= 4 R= 18

To obtain DECODING FORMULA:

Encoding Formula: y = x ⊕ 10

=⇒ y ⊕ 16 = x ⊕ 10 ⊕ 16

=⇒ y ⊕ 16 = x DECODING FORMULA: x = y ⊕ 16.

W: 23 ⊕ 16 = 39 ≡

13 =M

K: 11 ⊕ 16 = 27 ≡

1 =A

Decoding Formula: x = x ⊕ 16

Table: Decoding

Decode W K D R

y 23 11 4 18

x 13 1 20 8

Encoded M A T H

Exercise:

Encoding Formula: y = x ⊕ 20

1

Encode: LOVE

2

Find the Decoding Formula.

3

Decode: U W Y