Public Key Cryptography

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Public KeyCryptography
Remo Pillat COT4810 Spring 2005 02 February 2005
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Public Key Cryptography Remo
Pillat

• It takes two or more parties to share a secret
• but a secret is truly private only when one
  party alone knows it.

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Syllabus
Public Key Cryptography Remo
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• Mathematical Background
• Public Key Idea
• RSA and its Weaknesses
• Conclusion

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Overview of Cryptography
Public Key Cryptography Remo
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Congruences

• Two number a and b are called congruent modulo n
  ( )
• if a and b have the same remainder when divided
  by n
• Rules for calculations
• ?

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Congruences (2)
Public Key Cryptography Remo
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• Normal Logarithm can be calculated (easy)
• Discrete Logarithm (very difficult)

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Eulers Rule
Public Key Cryptography Remo
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• In general
• Eulers Function
• for all primes obvious
• (following the definition of prime numbers)
• Special case of Eulers Rule for prime numbers
  p

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Trapdoor One-Way Functions
Public Key Cryptography Remo
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• Is a one-way function f X ? Y with the
  additional property that given some extra
  information it becomes feasible to find for given
  an such that
• Example (Integer Factorization Problem)
• select primes p 48611 and q 53993 and form n
  pq 2624653723

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Trapdoor One-Way Functions (2)
Public Key Cryptography Remo
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• f(x) is relatively simple to calculate
• the reverse the procedure is much more difficult,
  even when n is known
• problem gets easier if p and q of n are known
• However
• No one has definitely proved the existence of
  such functions
• Existence of trap-door functions is unknown ? P
  NP ?
• But there are good candidates!
• Trapdoor One-Way functions are the basis for
  public-key cryptography

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Candidates
Public Key Cryptography Remo
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• Integer Factorization
• Any positive integer n can be written as product
  of primes
• RSA Problem
• Find m where n pq (p and q primes)
• Discrete Logarithm Problem
• Find x where n is prime for

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Alice is playing around
Public Key Cryptography Remo
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• Use the fact that Factorization is hard problem
• n pq , where p and q are primes

public key e and n pq ? used for
encryption private key d ? used for
decryption
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Alice Idea
Public Key Cryptography Remo
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• Encryption with
• Then decryption is unique defined as

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Public Key Cryptography
Public Key Cryptography Remo
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RSA
Public Key Cryptography Remo
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• Developed in 1978 by Rivest, Shamir and Adleman
  (RSA)
• Most popular public key crytosystem
• Based on the mathematical hard problem of
  integer factorization and RSA problem

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Key-Generation for RSA
Public Key Cryptography Remo
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• Generate two large random (and distinct) primes p
  and q, each roughly the same size.
• Compute n pq and
• Select random integer e,
• Compute unique integer d,
• Public key is (n, e) Private key is d

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Key-Generation (2)
Public Key Cryptography Remo
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• Generation of primes with 512 bits
• 1. Sieve of Erastostenes
• Write down all numbers from 2 to N
• 2 3 4 5 6 7 8 9 10
  11 12 13 ..
• 2 3 5 7 9
  11 13 ..
• 2 3 5 7
  11 13 ..
• ? primes will be the last numbers which stand
  there
• ? practically infeasible

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Key-Generation (3)
Public Key Cryptography Remo
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• Usually numbers with the right bit length are
  chosen randomly and tested for primality
• Statistical test are used to determine the
  probability that these numbers are primes
• i.e. Strassen Test
• Miller Rabin Test
• Fast and efficient, although there is always a
  insignificantly low chance that number is not
  prime
• Even with modern computers takes 5 minutes for
  1024 bits prime number

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Setting value for e
Public Key Cryptography Remo
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• Select random integer e,
• e should be prime ? gcd is trivially true
• Common values
• ? is easier to calculate

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RSA public-key encryption
Public Key Cryptography Remo
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• Encryption
• (a) Obtain As authentic public key (n, e)
• (b) Represent the message as an integer m in the
  interval 0, n-1
• (c) Compute
• (d) Send the ciphertext c to A
• Decryption
• (a) Use the private key d to recover

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Digital Signature (1)
Public Key Cryptography Remo
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Digital Signature (2)
Public Key Cryptography Remo
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• RSA was the first public system to support
  digital signatures and asymmetrical (public) key
  encryption
• Signature
• Send s over a data channel
• Verification

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Weaknesses of RSA (1)
Public Key Cryptography Remo
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• One weakness arises from the digital signature
  protocol
• Eve gets from the unsecured channel
• Eve sends message to Alice and asks for digital
  signature
• Alice is decrypting the message with her signature

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Weaknesses of RSA (2)
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• Forward search attack
• if message space is small or predictable
• Eve simply encrypts all possible plaintext
  messages until c is obtained
• Can be prohibited by salting the message

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Weaknesses of RSA (3)
Public Key Cryptography Remo
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• Quantum Computers
• RSA is based on the infeasible task to solve the
  Integer Factorization or the RSA problem in
  polynomial time
• RSA-428 was factorized after 600 users let their
  1600 computers search for the prime factors for
  almost 8 months
• ? 1024 bit keys are unbreakable for todays
  technology
• Quantum Computers are based on the laws on
  Quantum mechanics
• ? QBits instead of bits ? 5 QBits have
  States at the same time
• At least now only theoretical
• ? Basis of my next presentation

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Conclusion Advantages
Public Key Cryptography Remo
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• Only the private key must be kept secret
• The administration of keys on a network requires
  the presence of only a functionally trusted TTP
  as opposed to an unconditionally trusted TTP.
  (TTP trusted third party)
• a private key/public key pair may remain
  unchanged for considerable periods of time, e.g.,
  many sessions (even several years).
• efficient digital signature mechanisms.
• In a large network, the number of keys necessary
  may be considerably smaller than in the
  symmetric-key scenario.

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Conclusion Disadvantages
Public Key Cryptography Remo
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• Throughput rates for the most popular public-key
  encryption methods are several orders of
  magnitude slower than the best known
  symmetric-key schemes
• Key sizes are typically much larger than those
  required for symmetric-key encryption
• No public-key scheme has been proven to be secure
  
• Public-key cryptography does not have as
  extensive a history as symmetric-key encryption,
  being discovered only in the mid 1970s.

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References
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• 1 Coutinho, S. C. - The Mathematics of Ciphers
  Number Theory and RSA Cryptography. Natick, MA
  A. K. Peters, 1999.
• 2 Meijer, A. R. "Groups, Factoring, and
  Cryptography." Math. Mag. 69, 103-109, 1996.
• 3 Rivest, R. Shamir, A. and Adleman, L. "A
  Method for Obtaining Digital Signatures and
  Public Key Cryptosystems." Comm. ACM 21, 120-126,
  1978.
• 4 Menezes, A.J. et.al Handbook of Applied
  Cryptography CRC Press 1996 ISBN 0849385237
• 5 Kahn, D. The Codebreakers The Story of
  Secret Writing Macmillan USA 1974 ISBN
  0025604600
• 6 Beutelsbacher, Albrecht Cryptology The
  Mathematical Association of America 1996 ISBN
  0883855046