Sample Quiz 2 Solution

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Sample Quiz 2 Solution

• 22 March 2005

2
Question 1

• 1. Modular arithmetic
• Show the addition and multiplication tables in
  GF(23) (mod m(x) where m(x) x3x1).
• Show the addition and multiplication tables under
  modulo 8.
• Compare the two sets, which is a field?

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Addition table in GF(23)
4
Multiplication table in GF(23)
5
Addition table under mod 8
6
Multiplication table under mod 8
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which is a field?

• The set formed by mod 8 is not a field, since not
  every number has multiplicative inverse (see
  multiplication table for 2 and 4). GF(23) is a
  field. As it obeys all the following rules
• Closure under addition and multiplication
• Associative of addition and multiplication
• Additive and multiplicative identity
• Additive and multiplicative inverse
• Commutative of addition and multiplication
• Distributive laws
• No zero divisors.

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Question 2

• In a public-key system using RSA, you intercept
  the ciphertext C 10 sent to a user whose public
  key is e 23, n 143. What is the plaintext M?
  Verify the correctness of your decryption.

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Calculate secret key d

• Factor n p q
• p 11 and q 13
• ?(n) 120
• 120 523 5
• 23 45 3
• 5 13 2
• 3 2 1

• 1 3 2
• 3 (5 3)
• 5 23
• 5 2(23 45)
• 223 - 95
• 223 9(120 523)
• 4723 9120 mod 120
• 4723 mod 120
• ? d 47

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Compute plaintext

• Compute M
• M (C)d mod n
• (10)47 mod 143
• 43

• Verify
• C (M)e mod n
• (43)23 mod 143
• 10
• C

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Question 3

• Given M 32, user As private key KRA 5,
  public key KUA (e 173, n 323) and H(x)
  x2 mod 103, verify if 237 is the right signature
  of M signed by user A (i.e. EKR_AH(M))?

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Compute signature

• H(M) (25)2 210
• 1024 mod 1000 24
• EKR_AH(M) (H(M))KR_A mod n
• (24)5 mod 323 28
• Therefore, the correct signature should be 28.
  237 is not a valid signature of user A.

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Question 4

• It is possible to use a hash function to
  construct a block cipher with a structure similar
  to DES. Because a hash function is one way and a
  block cipher must be reversible (to decrypt), how
  is it possible?

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• DES
• Encryption
• Li Ri-1
• Ri Li-1 ? F(Ri-1,Ki)
• Rearranging terms for Decryption
• Ri-1 Li
• Li-1 Ri ? F(Ri-1,Ki)
• Ri ? F(Li,Ki)
• Replace all F functions with hash functions we
  could construct a block cipher.