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Part 2 of Course

Chapter 2 of Textbook

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Part 2 of Course

Objective: Application of Number Theory in Computer security.

Number theory has a long history E.g.: Chinese Remainder Theorem: 2300 years old

Regarded as useless until recently

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Part 2 of Course

Part 2 of course: Show how to make e-commerce secure using Number theory. Three logic lectures: L04-L06

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L04: Intro to Crypto and Modulus

Objective: Basic mathematical concepts for Part 2 Introduction to Cryptography

Outline Modular Arithmetic: mod n Operations on the set Introduction Cryptography

Private-Key Cryptography

Caesar cipher: Using addition mod n

Crypto using multiplication mod n

Public-Key Cryptography

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Modular Arithmetic

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Euclid’s Division Theorem

If

m = n q’ + r’, 0<= r’ <n

Then

q’=q, r’=r

Examples m=25, n=4

25 = 4 x 6 +1 q=6, r=1

m=-25, n=4 -25 = 4 x (-7) +3 q=-7, r=3

Will be proved later

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Modular Arithmetic

Applies also to the case when m is negative.

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Modular Arithmetic

Applies also to the case when m is negative.

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Modular Arithmetic/Simple Properties

Note

[-25 mod 4] = 4 - [25 mod 4]

In general

Example: 5 mod 4 = 1, (-5) mod 4 = 3

6 mod 4 = 2, (-6) mod 4 = 2

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Modular Arithmetic/Properties

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Modular Arithmetic/Properties

Examples

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Intuition

Adding multiples of n to i changes the quotient, but not the remainder.

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Lemma 2.3 has a second part

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L04: Intro to Crypto and Modulus

Modular Arithmetic: mod n Operations on the set Introduction Cryptography

Private-Key Cryptography Caesar cipher: Using addition mod n

Cryto using multiplication mod n

Public-Key Cryptography

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Modulo Arithmetic on the Set

Operations on

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Laws of Arithmetic over Real Numbers

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Properties of Operations on

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Does each

Has additive inverse? Yes. -x mod n

Has multiplicative inverse? Major question to be discussed later.

Properties of Operations on

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L04: Intro to Crypto and Modulus

Modular Arithmetic: mod n

Operations on the set

Introduction Cryptography

Private-Key Cryptography Caesar cipher: Using addition mod n

Cryto using multiplication mod n

Public-Key Cryptography

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L04: Intro to Crypto and Modulus

Modular Arithmetic: mod n

Modulo arithmetic on the set

Introduction Cryptography

Private-Key Cryptography Caesar cipher: Using addition mod n

Crypto using multiplication mod n

Public-Key Cryptography

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Private-Key Cryptography

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Caeser Cipher and Mod 26

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Caeser Cipher and Mod 26

Encrypting

Decrypting:

E.G. s=2 Plaintext message: SEA 18 4 0

Cipher text: 20 6 2

Decrypted message: 18 4 0

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Caeser Cipher and Mod 26

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Encrypting/Decrypting as Functions

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L04: Intro to Crypto and Modulus

Modular Arithmetic: mod n

Operations on the set

Introduction Cryptography

Private-Key Cryptography Caesar cipher: Using addition mod n

Crypto using multiplication mod n

Public-Key Cryptography

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Cryptography with Multiplication mod n

Also possible to implement crypto system using multiplication mod n

Need to deal with an important new issue.

Plaintext: 5 7 8

Ciphertext: 1 11 4

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Cryptography with Multiplication mod n

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Cryptography with Multiplication mod n

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Multiplicative Inverse Exists?

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Multiplicative Inverse Exists?

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Multiplicative Inverse Exists?

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Multiplicative Inverse Exists?

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L04: Intro to Crypto and Modulus

Modular Arithmetic: mod n

Operations on the set

Introduction Cryptography

Private-Key Cryptography Caesar cipher: Using addition mod n

Crypto using multiplication mod n

Public-Key Cryptography

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Drawback of Private-Key Cryptosystem

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Public-Key Cryptosystem

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Public-Key Cryptosystem

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Public-Key Cryptosystem

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Public-Key Cryptosystem

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Is Public-Key Cryptosystem Possible?

Need a function whose inverse is DIFFICULT to compute without private key. Sounds almost impossible.

In 1970’s, Rivest, Shamir and Adelman figured out how to do this using modular arithmetic

The result: RSA public-key crypto-system. L06.

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23-02-2010: RecapPage 48

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23-02-2010: RecapPage 49

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25-02-2010: Recap

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25-02-2010: Recap

Example of Private-Key cryptosystem Caeser Cipher: cryptosystem using addition mod n

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25-02-2010: Recap

L04: Examples on multiplicative inverse

L05:

When does multiplicative inverse exist?

How to find it?