課程名稱︰密碼學 課程性質︰選修 課程教師︰雷欽隆 開課學院：電資學院 開課系所︰電機系 考試日期（年月日）︰2011/11/10 考試時限（分鐘）：120min 是否需發放獎勵金：是 （如未明確表示，則不予發放） 試題 : 密碼學期中考 11/10/2011 1.(10 points)If a function f statisfies the following properties: ● f(x) is polynomial time computable , and ● Computing f^-1 (x) is NP-hard Explain why f is typically not necessary a good candidate for a one-way function? 2.(10 points) Describe how to break the Vigenere Cipher. (Recall that : In Vigenere Cipher , the secrect key is a short sequence of letters , and encryption involves adding the plaintext letter to a key letter with the key letters used in rotation.) 3.(10 points) Do Z_9 ^* and Z_15 ^* have common generator(s)? If yes, what is/are the common generator(s)? Do Z_9 ^* and Z_14 ^* have commom generator(s) ? if yes , what is/are the common generator(s)? 4.(10 points) Is 16 quadratic residue in Z_21 ^* ? If yes , what are the square roots of 16 ? List all the quadratic residues in Z_21 ^*. 5.(10 points)If a random X takes at most 3 values , and we know the following 0.2 <= P(X=i) <= 0.6 ,for I=1,2,3. What are the maximum and minimum values for its entropy function H(X), respectively? 6.(10 points)Let P,K,C be the set of possible messages,keys and ciphertexts with associated random variables P,K,C.Explain why H(P|K,C) = 0 and H(C|P,K) = 0 , then , prove that H(K) + H(P) = H(K,C). 7.(10 points) Describe the operations of a 2-key triple-DES.Explain why 2-key triple-DES does not use three consecutive encryptions. 8.(10 points) The “Mix Columns” operations in AES corresponds to 4 polynomial multiplies each of degree 4 using GF(2^8) as the coefficients , and reduced , and then reduced by M(x) = (x^4 + 1). That is for a column ┌ ┐ ┌ ┐ │a│ │a'│ │b│ , the resulting new column │b'│ after the “Mix Columns”operation │c│ │c'│ │d│ │d'│ └ ┘ └ ┘ will be ┌ ┐ ┌ ┐┌ ┐ │a'│ │ 2 3 1 1 ││a│ │b'│ = │ 1 2 3 1 ││b│,. │c'│ │ 1 1 2 3 ││c│ │d'│ │ 3 1 1 2 ││d│ └ ┘ └ ┘└ ┘ For example , a' = (ax^3 + bx^2 + cx + d)(2x^3 + 3x^2 + x + 1) mod (x^4 + 1) If a=2b , b=d4 , c=de , and d=ad. What is the value of b' ? 9.(10 points) Consider the ECB , CBC , CFB , OFB , and Counter modes , which of them will result in a self-synchronizing cipher ? Which mode is most suitable for high-speed netwwork eneryption? Which mode is usually used for short message encryption ? 10. (10 points)If we design a message authentication code scheme by MAC_k (m) = h(k||m) , where h is a hash function , and k is a secret key. Explain why this scheme is not secure. --

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