COMP 170 L2Page 1

L06: The RSA Algorithm

Objective:

Present the RSA Cryptosystem

Prove its correctness

Discuss related issues

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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

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Exponentiation mod n

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Corollary of Lemma 2.19

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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

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Public-Key Cryptography

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RSA Algorithm

Questions to answer

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One-Way Function

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RSA Algorithm

Builds a one-way function using

Exponentiation mod n

Prime numbers

gcd

Multiplicative inverse

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RSA Algorithm

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RSA Algorithm

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RSA Example

Key generation

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RSA Example

Encryption and decryption

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

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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

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A Lemma

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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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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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Decipherability

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Decipherability Proved!

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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

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Exponentiation mod n efficientlyPage 42

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Complexity of Repeated SquaringPage 43

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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

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The Chinese Remainder Theorem

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The Chinese Remainder Theorem

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Past Exam Question

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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?