Public%20Key%20Cryptography

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Public Key Cryptography

• Bryan Pearsaul

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Outline

• What is Cryptology?
• Symmetric Ciphers
• Asymmetric Ciphers
• Diffie-Hellman
• RSA (Rivest/Shamir/Adleman)
• Moral Issues

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Outline

• Summary
• References

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What is Cryptology?

• The science of keeping data secure

• Two transformation algorithms
• Enciphering and Deciphering

• Symmetric ciphers

• Asymmetric ciphers

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Symmetric Ciphers

• Also known as private key

• Both parties must agree on the key
• in advance

• D_K(E_K(P)) P

• Not very computationally intensive

• Key must be securely sent to both parties

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Symmetric Cipher Example

• k 4

• Turn plaintext SECRET into ciphertext

• S4W, E4I, C4G, R4V, E4I, T4X

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Symmetric Cipher Example

• Much more elaborate transformations
• are available

• Some that are so complicated that
• even if the transformation was
• public a key would still be needed

• Still require a distributed key

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Asymmetric cipher

• Also known as public key

• Two keys public k, private k

• Private key not required for both
• parties

• More computationally intensive

D
E
E_K(X)
Enciphering
Deciphering
D_K(E_K(X)) X
X
K
K
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Diffie-Hellman

• One of the first public key cryptographic systems

• Developed by Martin Hellman, Ralph Merkle, and
  Whitfield Diffie at Stanford University in 1976

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Diffie-Hellman

• Based on a special case of the
• subset-sum, or knapsack, problem

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11
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6
5
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Subset-sum Problem
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Diffie-Hellman Example

• Block cipher

• Block size of 7 bits. Possible 27 combinations

• Private key (a1, a2, , an) of 7 integers
  (1, 2, 5, 11, 32, 87, 141)

• Chose two special integers, w and m, such that w
  and m are relatively prime,
• meaning gcd(w,m) 1 w 901, m 1234

• Public key (a1, a2, , an) of 7 integers using
  the equation ai w ai mod m
• (901, 568, 803, 39, 450, 645, 1173)

• Partition SECRET into 7 bit blocks each block
  consisting of xn bits (x1, x2, , xn)

S 1010011
E 1000101
C 1000011
R 1010010
E 1000101
T 1010100
n

• Bx ? xiai

i1

• S 1 X (901) 0 X (568) 1 X (803) 0 X (39)
  0 X (450) 1 X (645) 1 X (1173)

• S 3522

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Diffie-Hellman Example

• Encrypted blocks Bx received. Special version of
  subset-sum problem

• Which subset of (a1, a2, , an) sums to Bx
  where Bx Bx w-1 mod m

• w-1 is the modular inverse of w for m, w w-1
  mod m 1

• Bx 3522 X (901)-1 mod 1234
• Bx 3522 X 1171 mod 1234
• Bx 234

• 1. sum ? 0
• 2. for i n step -1 until 1 do
• if ai sum lt Bx
• then sum ? sum ai
• subset(i) ? 1
• else subset(i) ? 0
• 3. if sum Bx then exit with subset
• else exit with failure

• Private key (1, 2, 5, 11, 32, 87, 141), Bx
  234, find subset (1, 0, 1, 0, 0, 1, 1) S

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Diffie-Hellman

• Two possible points of vulnerability

• An algorithm which solves NP-complete problems
  quickly

• An algorithm that solves the particular problem
  on which a cryptographic system is based.

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RSA

• Developed by Ron Rivest, Adi Shamir, and Leonard
  Adleman at MIT in 1977.

• Based on the difficulty of factoring large numbers

• Factorization so far is unsolvable in
  polynomial-time

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RSA Example

• Find two large prime integers, p and q, and form
  product n pq

• Find a random integer, e, that is relatively
  prime to ?(n) (p-1)(q-1)

• p and q are kept private, (n,e) are the public
  key

• Message is partitioned into blocks, b, such that
  b lt n

• Each block is encrypted using the equation c
  be mod n

• For the private key, calculate integer d which
  is the modular inverse of e
• for ?(n), or e d mod ?(n) 1

• Once d is calculated it becomes your private key
  and all records of
• p and q should be destroyed

• Each encrypted block, c, is decrypted using the
  equation b cd mod n

• p 61, q 53, n 3233, ?(n) 3120, e 17, d
  2753

• encrypt(123) 12317 mod 3233 855

• decrypt(855) 8552753 mod 3233 123

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RSA

• Security of RSA relies on two assumptions

• Factoring is required to break the system

• Factorization cannot be done in polynomial-time

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Moral Issues

• Privacy

• Who does the data belong to?

• Information Theft

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Summary

• Cryptology

• Symmetric and Asymmetric ciphers
• Pros and Cons

• Diffie-Hellman and RSA

• Moral Issues

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References

• A.K. Dewdney, The New Turning Omnibus, pp.
  250-257, Henry Holt and Company, 2001.

• RSA Cryptosystem, http//primes.utm.edu/glossary/
  page.php?sortRSA.

• Cryptology FAQ, http//www.faqs.org/faqs/cryptogr
  aphy-faq/part06/.

• The Extended Euclidian Algorithm,
  http//www.grc.nasa.gov/WWW/price000/pfc/htc/zz_xe
  uclidalg.html.

• A. Shamir, A Polynomial-Time Algorithm for
  Breaking the Basic Merkle-Hellman Cryptosystem",
  Advances in Cryptology - CRYPTO '82 Proceedings,
  pp. 279-288, Plenum Press, 1983. IEEE
  Transactions on Information Theory, Vol. IT-30,
  pp. 699-704, 1984.