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- Consider a large institution that has 9970 employees. How many keys are necessary to allow any pair of employees to securely communicate using a symmetric cipher? Suppose that we instead use an asymmetric cryptographic scheme. How many keys are required in this case? One of the one-way functions used in public key cryptography is integer multiplication/factorization. Multiplying two integers is easy, but factoring is hard. The number 2963659 is the product of two primes. What is the smaller of the two primes? What is the largest of the two primes? One of the one-way functions used in public key cryptography is the discrete logarithm. Computing r = g* mod p from g, e, and pis easy. But given only r, g and p, recovering e is hard. Suppose p = 1579, g = 2 and r = 387 What is the smallest positive integer e such that r = g mod p? Use the Euclidean algorithm to compute GCD(853969545,574290294) = GCD(125665559,519562250) =Question 15 For Questions 15.1 - 15.2 consider the following integers: In a RSA cryptosystem with public- key (3233, 59), compute: 15.1 the private-key and give you final answer as an ordered pair (n, d).In an RSA system, the public key of a given user is e = 65, n = 2881. What is the private key of this user? Hint: First use trial-and-error to determine p and q; then use the extended Euclidean algorithm to find the multiplicative inverse of 31 modulo n. Write it understandable.
- Use the two prime numbers p 5 and q =13 in the first step to give ONE integrated example to show how the five steps in the basic process of RSA Cryptography works. Based on your example, demonstrate how the number 60 as an original message is encrypted into ciphertext and decrypted back correctly to the original message. If you need to choose a number in any step, you must choose it from the set {r e R: 8 Sx< 12}. You must show all detailed steps involved.1. Let (01, 02, 03,..., 026!} be the set of permutations of the alphabet A = {A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z}, and consider the simple substitution cryptosystem with ok = A B C D E F G H I J K L M X MTAK ZQB NO LES N O P Q R S T U V W X Y Z IF G R V JH CY UDP W and key k. (a) Compute C = e(AUBUR NUNIV ERSIT YATMO NTGOM ERY, k). (b) Compute M = d(NIZFV SXHNF IHBKF VPXIA KIHVF GP, k).Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 11000101 — ХоXјX2XҙХ4X5X6X7 when encrypted by the LFSR produced the ciphertext 10010000 Уo У1 У2 Уз Уз У5 У6 Ут . What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent P3 = 0, p2 = 1, p1 = 0, po = 1).
- The answer above is NOT correct. Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 11100010 — ХоXјX2XҙX4XsX6X7 when encrypted by the LFSR produced the ciphertext 11010110 — Уo У1 У2 Уз Уз У5 У6 Ут. What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent p3 = 0, p2 = 1, p1 = 0, po = 1). 0010Consider to use RSA with a known key IK to construct a cryptographic hash function H as follow: Encrypt the first block, XOR the result with the second block and encrypt again, etc. Then, the last ciphertext block is the hash value. For example, H(M,M2) = Enc(IK, Enc(IK, M,) O M2) = h. Show that this H does not satisfy the property of second image resistance. That is, we can find N1 and N2 such that H(N,N2)=h.Suppose the two prime numbers p = 23 and q = 37 are used for an RSA encryption scheme. What are the values of the connection key n, the encryption e and the decryption key d, given that d < e? e = d =
- A ciphertext has been generated with an affine cipher. The most frequent letter of the ciphertext is “F”, and the second most frequent letter of the ciphertext is “C”. Break this code by figuring out the key numbers. Hint: Based on the letter frequency, E is the most frequent letter and T is the second most frequent letter. Therefore, F should be the ciphertext of E and C should be the ciphertext of T.What is the security requirement on password hashing? Given a password w, its hash is calculated as . This question is about the security requirement on function h. (a) Can we use any function as h for password hashing? For example, can we use the "mod 100" function, which is a common computer science function? (Given any integer value x, "mod 100" function works by "x mod 100," which is a value between [0, 99]; for example, if x = 105, x mod 100 = 5; x = 1000000, x mod 100 = 0;) (b) If not, what is the security requirement on h?Note: The notation from this problem is from Understanding Cryptography by Paar and Pelzl. We conduct a known-plaintext attack against an LFSR. Through trial and error we have determined that the number of states is m = 4. The plaintext given by 01110010 = 0X1X2X3X4X5X6X7 when encrypted by the LFSR produced the ciphertext 11000011 = YOY1Y2Y3Y3Y5Y6Y7- What are the tap bits of the LFSR? Please enter your answer as unspaced binary digits (e.g. 0101 to represent p3 = 0, p2 = 1, p1 = 0, po = 1).