Modular Systems

Group 4

Rossane CruzShana CueDavid De GuzmanCassandra Dela CruzRussel Dela CruzHarlyn DesinganoKeziah Dulay

Definition

● Let a and b ∈ Z. There are two operations and a relation:

a. a +n b = remainder of (a + b) ÷ n

b. a ·n b = remainder of (a · b) ÷ n

c. a ≅ b (modn) if a and b have the same remainder when divided by n

Examples

+n

1.8 +9 12

2.14 +3 711

3.90 +20 123

·n

1. 9 ·5 10

2. 12 ·11 12

3. 499 ·46 28

+n , ·n

1. 9 ·8 7 +5 13

2. 75 ·11 24 +12 9

3. 26 ·32 316 +21 1

Congruence (≅)

a ≅ b (modn)

Examples:

1. -7 = 14 (mod 3)

2. -43 = 17 (mod 4)

3. -1 = -4 (mod 3)

The Group Zn

The set Zn = {0,...,n−1} is an abelian/commutative group under +n.

<Zn, +n> a group.

<Zpx, ·p> a group where p is prime.

Examples:

1.Z4 = {0,1, 2, 3}

2.Z5 = {0, 1, 2, 3, 4}

3.Z6 = {0, 1, 2, 3, 4, 5}

The Ring ⟨Zn, +n, ·n⟩

<R, +> a group

<R, +> is commutative

<R, +> is associative

<R, +> is distributive

Examples:

1.Z4 = {0,1, 2, 3}

2.Z5 = {0, 1, 2, 3, 4}

3.Z6 = {0, 1, 2, 3, 4, 5}

The Field

<F, +, ·> should be a ring

<F*, ·> a group

<Zp, +p, ·p> are finite fields

Cryptography

-study of methods to send and receive messages

-elements: sender, receiver, adversary

-used in the past mainly for military and diplomatic communications

-used alongside technology to protect sensitive information

Methods

a.Private key cryptography - only sender and receiver knows the code

b.Public key cryptography - codes are available to public

Affine Cipher (Parts)

Plaintext - original message

Ciphertext - encoded text

Code - the key used to encode and decode text

Values - x and yplaintext C R I S P

x 3 18 9 19 16

y 4 1 8 6 17

ciphertext

D A H F Q

Encoding: y=5x-11

Decoding: x=21y-3

Encoding

Use the form y=ax+b

a ∈ {1,3,5,7,11,15,17,19,21,23,25} and b ∈ Z26

Example:

1.Encode MATH using the code y=7x+2

plaintext M A T H

x 13 1 20 8

y 15 9 12 6

ciphertext

O I L F

Decoding

Example:

1.Decode OILF using the code y= 7x+2

15(y=7x+2)15y= x+415y+22 = x+4+2215y+22=x

plaintext O I L F

x 15 9 12 6

y 13 1 20 8

ciphertext M A T H

Exercises

1)Solve the following 2) Find the value of x

a.4 + 2 (27 - 2 3) a. x2 + 2x = 0

(mod 4)

b.72 -26 2 *26 7 b. x2 - x

= 2 (mod 6)

3) Decode the passage using y = 3x + 16

QX’S VEXXER XI NSDE ZIDEB SFB ZIUX

XNSF XI FEDER NSDE ZIDEB SX SZZ NANANA.

Sources

Matematika Para sa Pangkalahatang Edukasyon (book)