课程名称︰密码学 课程性质︰选修 课程教师︰雷钦隆 开课学院：电资学院 开课系所︰电机系 考试日期（年月日）︰2011/11/10 考试时限（分钟）：120min 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : 密码学期中考 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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