Cryptography

Day 0 - Digital System Representations

Today’s Schedule

10:30 - 11:00 → Binary, bits, bytes

11:10 - 11:45 → Intro to Crypto

-- Lunch --

1:00-2:00 Intro to Cryptography (cont)

2:00-3:30 One Time Pad (if there is time!)

System

• Set of components that work together to achieve a certain task

Why Binary?

• Simplicity

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Now we can pretty much represent anything by:

1. Real-world signals → Binary
2. Binary → Decimal Numbers?
3. Binary → Characters?

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1

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Bit & Byte

• A bit is the smallest unit of information
• It represents one 2-way decision or a 2-way choice
• yes / no, true / false, on / off …
• Typical value: either 0 or 1
• All information in a computer is stored and processed as bits
• A byte is 8 bits that are treated as a unit

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Bits

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Byte

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Bitwise Operators

AND (&)

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OR (|)

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XOR (^)

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NOT (~)

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1 & 1 = 1

1 & 0 = 0

0 & 1 = 0

0 & 0 = 0

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1 | 1 = 1

1 | 0 = 1

0 | 1 = 1

0 | 0 = 0

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1 ^ 1 = 0

1 ^ 0 = 1

0 ^ 1 = 1

0 ^ 0 = 0

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~ 1 = 0

~ 0 = 1

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Bitwise Operations

AND (&) 0 0 1 1� 1 0 1 0� ------------------------

0 0 1 0

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OR (|) 0 0 1 1� 1 0 1 0� ------------------------

1 0 1 1

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XOR (^) 0 0 1 1� 1 0 1 0� ------------------------

1 0 0 1

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NOT (~) 0 1 � -----------

1 0

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What does this byte represent?

Decimal Numbers Review

• Deci == 10: Decimal is a Base-10 system
• Base-10: group of 10

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Decimal Numbers Review

• Deci == 10: Decimal is a Base-10 system
• Base-10: position represents a power of 10
• Each number to the right of another, is ~10 bigger

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4 4 4, 4 4 4, 4 4 4

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* 10 ⁸

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* 10 ⁷

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* 10 ⁶

* 10 ⁵

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* 10 ⁴

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* 10 ³

* 10 ²

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* 10 ¹

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* 10 ⁰

Base-10 Decimal System

• Decimal numbers are sums of powers of 10
• Formula:
• (Digit * 10^n) + …. + (Digit * 10^3) + (Digit * 10^2) + (Digit + 10^1) + (Digit + 10^0)
• 1234?

1 thousands + 2 hundreds + 3 tens + 4 ones

1 * 1000 + 2 * 100 + 3 * 10 + 4 * 1

1 * 10³ + 2 * 10² + 3 * 10¹ + 4 * 10⁰

Base-2 Binary Number System

• Base-2 → What does this mean?
• Groups of 2!

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1

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10

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11

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111

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1111

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2⁴ 2³ 2² 2¹ 2⁰

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Base-2 Binary Number System

• Base-2 → powers of 2

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• 1 0 1 ₂ → (1 * 2 ² ) + (0 * 2 ¹ ) + (1 * 2 ⁰ ) == 5₁₀
• 22₁₀

→ 16 + 4 + 2 → (1 * 16) + (0 * 8) + (1 * 4) + (1 * 2) + (0 * 1)

→ (1 * 2⁴ ) + (0 * 2³ ) + (1 * 2² ) + (1 * 2¹ ) + (0 * 2⁰ )

⇒ 22₁₀ → 1 0 1 1 0 ₂

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Convert from Binary to Decimal

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� �1 0 1 1

1 * 2³ + 0 * 2² + 1 * 2¹ + 1 * 2⁰

1 * 8 + 0 * 4 + 1 * 2 + 1 * 1

8 + 0 + 2 + 1

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Convert from Decimal to Binary

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__ * 2³ + __ * 2² + __* 2¹ + __ * 2⁰

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1 * 2³ + 0 * 2² + 1 * 2¹ + 1 * 2⁰

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1 0 1 1

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Decimal → Binary

1. Start with a number (n)
2. Divide n by 2
3. Remainders: 0/1 → append to Binary number on the left side
4. Quotient → new ‘n’
5. Continue from Step 2 until quotient becomes zero
6. Final result is your binary number!

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Decimal → Binary

• Convert n = 97 to binary

---------------------------------------------------------------------------------�n ÷ 2 quotient remainder num�---------------------------------------------------------------------------------

97 ÷ 2 48 1 1

48 ÷ 2 24 0 01

24 ÷ 2 12 0 001

12 ÷ 2 6 0 0001

6 ÷ 2 3 0 00001

3 ÷ 2 1 1 100001

1 ÷ 2 0 (Stop) 1 1100001

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Rules:

1. Start with a number (n)
2. Divide n by 2
3. Remainders: 0/1 → append to Binary number on the left side
4. Quotient → new ‘n’
5. Continue from Step 2 until quotient becomes zero
6. Final result is your binary number!

Decimal → Binary

• Convert n = 97 to binary

------------------------------------------------------------------�n ÷ 2 quotient remainder�------------------------------------------------------------------

97 ÷ 2 48 1

48 ÷ 2 24 0

24 ÷ 2 12 0

12 ÷ 2 6 0

6 ÷ 2 3 0

3 ÷ 2 1 1

1 ÷ 2 0 (Stop) 1

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Summary

Binary → Language of Computers

Decimal → Language of Humans

• Base conversions to communicate back and forth with computers
• Decimal → Binary
• Binary → Decimal

Breakout Rooms:

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Convert 73 into binary

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Convert 11011010 into decimal

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How do we represent “Codebreakers”?

Characters

• How could I represent the text “Codebreakers” with in a digital system?

Sequencing with integers [a→ 0, b→ 1, c→ 2, …., z→ 25]

C: 2, o: 14, d: 3, e: 4, �b: 1, r: 17, e: 4, a: 0, k: 10, e: 4, r: 17, s: 18

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ASCII

• American Standard Code for Information Interchange
• Number representation of a character
• Arbitrary but agreed upon representation in US and rest of the world

Character → ASCII

Note that UPPER and lower letters are considered to be different characters

What’s the ASCII representation of

H E L L O ?

72 69 76 76 79 63

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ASCII → Character

Note that UPPER and lower letters are considered to be different characters

What does this spell?

72 69 76 76 79 33

H E L L O !

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Character <=> Binary?

New Functions! ord() and chr()

ord(<character>)

Takes in a single character input and returns its decimal ASCII code

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>>> ord(‘Q’)

81

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chr(<integer>)

Reverses ord(), gives you a character for a decimal input

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>>> chr(81)

‘Q’