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

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

COMP 170 L2 Page 2 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Calculating exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2 Exponentiation mod n l Encryption with addition and multiplication mod n n Easy to find the way to decrypt l RSA: use exponentiation mod n

COMP 170 L2 Exponentiation mod n

COMP 170 L2

Corollary of Lemma 2.19

COMP 170 L2 Page 8 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2 Public-Key Cryptography

COMP 170 L2 RSA Algorithm l Questions to answer

COMP 170 L2 One-Way Function

COMP 170 L2 RSA Algorithm l Builds a one-way function using n Exponentiation mod n n Prime numbers n gcd n Multiplicative inverse

COMP 170 L2 RSA Algorithm

COMP 170 L2 RSA Algorithm

COMP 170 L2 RSA Example l Key generation

COMP 170 L2 RSA Example l Encryption and decryption  Try: http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/

COMP 170 L2 Page 17 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2 A Lemma

COMP 170 L2

Fermat’s Little Theorem

COMP 170 L2

l What is a is a multiple of p?

COMP 170 L2 l Simplifies computation

COMP 170 L2

Page 25 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2 Decipherability

COMP 170 L2

Decipherability

COMP 170 L2

Decipherability Proved!

COMP 170 L2

Page 35 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2

Page 38 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l The Chinese Remainder Theorem

COMP 170 L2 Exponentiation mod n efficiently Page 39

COMP 170 L2 Exponentiation mod n efficiently

COMP 170 L2 Exponentiation mod n efficiently

COMP 170 L2 Exponentiation mod n efficiently Page 42

COMP 170 L2 Complexity of Repeated Squaring Page 43

COMP 170 L2 Page 44 The RSA Algorithm l Exponentiation mod n l The RSA Cryptosystem l Correctness n Fermat’s Little Theorem n Decipherability of RSA n Security of RSA l Exponentiation mod n efficiently l 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

The Chinese Remainder Theorem

COMP 170 L2 Past Exam Question

COMP 170 L2

Past Exam Question l About Chinese remainder theorem (CRT) l Think n 36 = 3 * 13, 5 = 3 * 17; not relatively prime, so cannot use CRT n Brute-force  x = q1 * 36 + 12 => x mod 3 = 0  x = q2 * 51 + 5 => x mod 3 = 2  Cannot have solution. n What is 12 is changed 11?

COMP 170 L2 l Think: n 35 = 5 * 7; 69 = 3 * 23 n Relatively prime. Also can apply CRT. Unique solution exists. l How to find the solution?

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