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Student_Solution_Chap_10 - CHAPTER 10 Symmetric-Key...

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1 CHAPTER 10 Symmetric-Key Cryptography (Solution to Odd-Numbered Problems) Review Questions 1. Symmetric-key cryptography is based on sharing secrecy; asymmetric-key cryp- tography is based on personal secrecy. 3. In cryptography, a trapdoor is a secret with which Bob can use a feasible algorithm to decrypt the ciphertext. If Eve does not know the trapdoor, she needs to use an algorithm which is normally infeasible. 5. RSA uses two exponents, e and d , where e is public and d is private. Alice calcu- lates C = P e mod n to create ciphertext C from plaintext P; Bob uses P = C d mod n to retrieve the plaintext sent by Alice. a. The one-way function is the C = P e mod n. Given P and e , it is easy to calculate C; given C and e , it difficult to calculate P if n is very large. b. The trapdoor in this system is the value of d , which enables Bob to use P = C d mod n. c. The public key is the tuple ( e , n ). The private key is d. d. The security of RSA mainly depend on the factorization of n . If n is very large, and the value of e and d are chosen properly, the system is secure. 7. ElGamal is based on discrete logarithm problem. The plaintext is masked with e 1 rd to create the ciphertext. Part of the mask is created by Bob and become public; the other part is created by Alice. a. The one-way function is C = mask (P). Given P and the mask, it is easy to cal- culate the C; given C is difficult to unmask P. b. The trapdoor is the value of d that enables Bob to unmask C. c. The public key is ( e 1 , e 2 and n ). The private key is d . d. The security of ElGamal depends on two points; p should be very large and Alice needs to select a new r for each encryption.
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2 9. An elliptic curve is an equation in two variables similar to the equations used to calculate the length of a curve in the circumference of an ellipse. Elliptic curves
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