POWER ANALYSIS ATTACKS

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RSA
Public Key Algorithm
Prepared by Y. Sinan Hanay
05/06/08
UMass
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Outline

• Cryptography
• Encryption Schemes
• RSA
• Attacks
• Summary

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Cryptography

• Secure Communication over Insecure Channel
• The Need for Cryptography
• Past
• Military Communication
• Present
• ATM Cards, Internet, Digital Signatures

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Cryptography
MESSAGE (PLAIN TEXT)
Sender
Receiver
Unsecure Channel
SOLUTION
Sender
Receiver
CIPHER TEXT
MESSAGE (PLAIN TEXT)
MESSAGE (PLAIN TEXT)
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Encryption Schemes

• Symmetric Key Cryptography
• Caesar Cipher
• DES, AES, Triple DES
• Faster
• Problem Keeping Key Secret
• DES broken in 22 hours in 1999.

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Publick Key Cryptography

• Symmetric Key Cryptography
• DES, AES, Triple DES
• Faster
• Problem Keeping Key Secret
• Public Key Cryptography
• RSA, El-Gamal, Elliptic Curve
• Slower
• Hybrid Cryptosystem
• Distribute Symmetric Keys with an Asymmetric
  Cryptosystem

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PKC History

• Public Key Cryptography ( Diffie, Hellman, Merkel
  1976)
• D(E(M)) M
• Both E and D are easy to compute
• One Way and Trapdoor Function
• E(D(M)) M
• One Way Function Easy to compute, Hard to
  invert
• Factoring
• Subset Sum
• RSA

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RSA

• By Rivest, Shamir, Adleman in 1977
• After working more than 40 candidates
• Distribution of Symmetric Keys
• Digital Signature
• Trapdoor Function

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RSA Key Generation

• 1) Choose Two Prime numbers (p and q)?
• 2) Compute N p q (N is the first
  element of key)?
• 3) Compute ? ( p 1) ( q - 1)?
• 4) Find e s.t. e and ? are coprime (elt?).
• 5) Find d s.t. , d e 1 (mod?)
• (e, N) Public Key
• (d, N) Private Key

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RSA Key Generation - Example

• 1) Choose Two Prime numbers (p and q)?
• p 179 , q 431
• 2) Compute N p q (N is the first
  element of key)?
• N 77149
• 3) Compute ? ( p 1) ( q 1)?
• ? 76540
• 4) Find e s.t. e and ? are coprime (elt?).
• e 281
• 5) Find d s.t. , d e 1 (mod?)
• d 3541
• (e, N) Public Key (281, 77149)?
• (d, N) Private Key (3541,77149)?

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RSA Encryption Decryption

• ENCRYPTION ( e 281, d 3541 , N 77149, m
  21)?
• m Message, c cipher text
• c 21281
  mod(77149) 65625
• DECRYPTION
• m 656253541
  mod(77149) 21
• Interactive Applet
• www-unix.ecs.umass.edu/hanay/RSA

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RSA Applications

• Digital Signature

Bob
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RSA Vulnerabilities

• When
• RSA can be inverted via Chinese Remainder theorem
  if n can be factored out
• Dixon Factorization
• Chosen Plain Text Attack
• Timing Attack

m lt n1/e then eth root of c m
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Summary

• Security Depends on One-Way, Trapdoor Function
• Factorization problem
• RSA-100 was factored in a few days in 1991
• RSA-640 was factored in 30 2.2GHz-Opteron-CPU
  years in 2005 ( 5 months calendar time)

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Thank You
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