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:
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 * => x mod 3 = 0 x = q2 * => 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?