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COMP 170 L2 Page 1 L06: The RSA Algorithm Objective: Present the RSA Cryptosystem Prove its correctness Discuss related issues
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L06: The RSA Algorithm

Jan 15, 2016

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L06: The RSA Algorithm. Objective: Present the RSA Cryptosystem Prove its correctness Discuss related issues. The RSA Algorithm. Exponentiation mod n The RSA Cryptosystem Correctness Fermat’s Little Theorem Decipherability of RSA Security of RSA - PowerPoint PPT Presentation
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Page 1: L06: The RSA Algorithm

COMP 170 L2Page 1

L06: The RSA Algorithm

Objective:

Present the RSA Cryptosystem

Prove its correctness

Discuss related issues

Page 2: L06: The RSA Algorithm

COMP 170 L2Page 2

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Calculating exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 3: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n

Encryption with addition and multiplication mod n

Easy to find the way to decrypt

RSA: use exponentiation mod n

Page 4: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n

Page 5: L06: The RSA Algorithm

COMP 170 L2

Page 6: L06: The RSA Algorithm

COMP 170 L2

Page 7: L06: The RSA Algorithm

COMP 170 L2

Corollary of Lemma 2.19

Page 8: L06: The RSA Algorithm

COMP 170 L2Page 8

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 9: L06: The RSA Algorithm

COMP 170 L2

Public-Key Cryptography

Page 10: L06: The RSA Algorithm

COMP 170 L2

RSA Algorithm

Questions to answer

Page 11: L06: The RSA Algorithm

COMP 170 L2

One-Way Function

Page 12: L06: The RSA Algorithm

COMP 170 L2

RSA Algorithm

Builds a one-way function using

Exponentiation mod n

Prime numbers

gcd

Multiplicative inverse

Page 13: L06: The RSA Algorithm

COMP 170 L2

RSA Algorithm

Page 14: L06: The RSA Algorithm

COMP 170 L2

RSA Algorithm

Page 15: L06: The RSA Algorithm

COMP 170 L2

RSA Example

Key generation

Page 16: L06: The RSA Algorithm

COMP 170 L2

RSA Example

Encryption and decryption

Try: http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/

Page 17: L06: The RSA Algorithm

COMP 170 L2Page 17

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 18: L06: The RSA Algorithm

COMP 170 L2

A Lemma

Page 19: L06: The RSA Algorithm

COMP 170 L2

Page 20: L06: The RSA Algorithm

COMP 170 L2

Fermat’s Little Theorem

Page 21: L06: The RSA Algorithm

COMP 170 L2

Page 22: L06: The RSA Algorithm

COMP 170 L2

What is a is a multiple of p?

Page 23: L06: The RSA Algorithm

COMP 170 L2

Simplifies computation

Page 24: L06: The RSA Algorithm

COMP 170 L2

Page 25: L06: The RSA Algorithm

COMP 170 L2Page 25

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 26: L06: The RSA Algorithm

COMP 170 L2

Decipherability

Page 27: L06: The RSA Algorithm

COMP 170 L2

Page 28: L06: The RSA Algorithm

COMP 170 L2

Page 29: L06: The RSA Algorithm

COMP 170 L2

Decipherability

Page 30: L06: The RSA Algorithm

COMP 170 L2

Page 31: L06: The RSA Algorithm

COMP 170 L2

Page 32: L06: The RSA Algorithm

COMP 170 L2

Page 33: L06: The RSA Algorithm

COMP 170 L2

Decipherability Proved!

Page 34: L06: The RSA Algorithm

COMP 170 L2

Page 35: L06: The RSA Algorithm

COMP 170 L2Page 35

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 36: L06: The RSA Algorithm

COMP 170 L2

Page 37: L06: The RSA Algorithm

COMP 170 L2

Page 38: L06: The RSA Algorithm

COMP 170 L2Page 38

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 39: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n efficientlyPage 39

Page 40: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n efficiently

Page 41: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n efficiently

Page 42: L06: The RSA Algorithm

COMP 170 L2

Exponentiation mod n efficientlyPage 42

Page 43: L06: The RSA Algorithm

COMP 170 L2

Complexity of Repeated SquaringPage 43

Page 44: L06: The RSA Algorithm

COMP 170 L2Page 44

The RSA Algorithm

Exponentiation mod n

The RSA Cryptosystem

Correctness

Fermat’s Little Theorem

Decipherability of RSA

Security of RSA

Exponentiation mod n efficiently

The Chinese Remainder Theorem

Page 45: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 46: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 47: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 48: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 49: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 50: L06: The RSA Algorithm

COMP 170 L2

Page 51: L06: The RSA Algorithm

COMP 170 L2

The Chinese Remainder Theorem

Page 52: L06: The RSA Algorithm

COMP 170 L2

Past Exam Question

Page 53: L06: The RSA Algorithm

COMP 170 L2

Page 54: L06: The RSA Algorithm

COMP 170 L2

Past Exam Question

About Chinese remainder theorem (CRT)

Think

36 = 3 * 13, 5 = 3 * 17; not relatively prime, so cannot use CRT

Brute-force x = q1 * 36 + 12 => x mod 3 = 0

x = q2 * 51 + 5 => x mod 3 = 2

Cannot have solution.

What is 12 is changed 11?

Page 55: L06: The RSA Algorithm

COMP 170 L2

Think: 35 = 5 * 7; 69 = 3 * 23 Relatively prime. Also can apply CRT. Unique solution exists.

How to find the solution?