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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

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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

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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

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COMP 170 L2 Exponentiation mod n

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COMP 170 L2

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Corollary of Lemma 2.19

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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

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COMP 170 L2 Public-Key Cryptography

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COMP 170 L2 RSA Algorithm l Questions to answer

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COMP 170 L2 One-Way Function

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

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COMP 170 L2 RSA Algorithm

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COMP 170 L2 RSA Algorithm

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COMP 170 L2 RSA Example l Key generation

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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/

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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

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COMP 170 L2 A Lemma

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COMP 170 L2

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Fermat’s Little Theorem

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COMP 170 L2

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l What is a is a multiple of p?

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COMP 170 L2 l Simplifies computation

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COMP 170 L2

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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

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COMP 170 L2 Decipherability

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COMP 170 L2

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Decipherability

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COMP 170 L2

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Decipherability Proved!

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COMP 170 L2

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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

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COMP 170 L2

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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

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COMP 170 L2 Exponentiation mod n efficiently Page 39

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COMP 170 L2 Exponentiation mod n efficiently

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COMP 170 L2 Exponentiation mod n efficiently

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COMP 170 L2 Exponentiation mod n efficiently Page 42

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COMP 170 L2 Complexity of Repeated Squaring Page 43

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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

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COMP 170 L2 The Chinese Remainder Theorem

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COMP 170 L2 The Chinese Remainder Theorem

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COMP 170 L2 The Chinese Remainder Theorem

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COMP 170 L2 The Chinese Remainder Theorem

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COMP 170 L2 The Chinese Remainder Theorem

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COMP 170 L2

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The Chinese Remainder Theorem

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COMP 170 L2 Past Exam Question

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COMP 170 L2

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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?

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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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Copyright © Zeph Grunschlag, 2001-2002. RSA Encryption Zeph Grunschlag.

Copyright © Zeph Grunschlag, 2001-2002. RSA Encryption Zeph Grunschlag.

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