L06: The RSA Algorithm

COMP 170 L2Page 1

L06: The RSA Algorithm

Objective:

Present the RSA Cryptosystem

Prove its correctness

Discuss related issues

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

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

COMP 170 L2

Exponentiation mod n

COMP 170 L2

COMP 170 L2

COMP 170 L2

Corollary of Lemma 2.19

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

COMP 170 L2

Public-Key Cryptography

COMP 170 L2

RSA Algorithm

Questions to answer

COMP 170 L2

One-Way Function

COMP 170 L2

RSA Algorithm

Builds a one-way function using

Exponentiation mod n

Prime numbers

gcd

Multiplicative inverse

COMP 170 L2

RSA Algorithm

COMP 170 L2

RSA Algorithm

COMP 170 L2

RSA Example

Key generation

COMP 170 L2

RSA Example

Encryption and decryption

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

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

COMP 170 L2

A Lemma

COMP 170 L2

COMP 170 L2

Fermat’s Little Theorem

COMP 170 L2

COMP 170 L2

What is a is a multiple of p?

COMP 170 L2

Simplifies computation

COMP 170 L2

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

COMP 170 L2

Decipherability

COMP 170 L2

COMP 170 L2

COMP 170 L2

Decipherability

COMP 170 L2

COMP 170 L2

COMP 170 L2

COMP 170 L2

Decipherability Proved!

COMP 170 L2

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

COMP 170 L2

COMP 170 L2

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

COMP 170 L2

Exponentiation mod n efficientlyPage 39

COMP 170 L2

Exponentiation mod n efficiently

COMP 170 L2

Exponentiation mod n efficiently

COMP 170 L2

Exponentiation mod n efficientlyPage 42

COMP 170 L2

Complexity of Repeated SquaringPage 43

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

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

COMP 170 L2

The Chinese Remainder Theorem

COMP 170 L2

Past Exam Question

COMP 170 L2

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?

COMP 170 L2

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

How to find the solution?