Download - 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
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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
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COMP 170 L2
Fermat’s Little Theorem
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What is a is a multiple of p?
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Simplifies computation
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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
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COMP 170 L2
COMP 170 L2
Decipherability
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COMP 170 L2
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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
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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?