1. For n-pg, where p and q are distinct odd primes, define (p-1)(q-1) λ(n) gcd(-1-1.411) Suppose...
p=3, q=7 Suppose that Bob wants to create an example of an RSA public-key cryptosystem by using the two primes p ??? and q ???. He chooses public encryption key e He was further supposed to compute the private decryption key d such that ed 1 mod A(pq)). However, he confuses A and and computes instead d' such that ed' =1 (mod P(pq)). (i) Prove that d' works as a decryption key, even though it is not necessarily the same...
6. Let n be any positive integer which n = pq for distinct odd primes p. q for each i, jE{p, q} Let a be an integer with gcd(n, a) 1 which a 1 (modj) Determine r such that a(n) (mod n) and prove your answer.
7. Let p and q be distinct odd primes. Let a є Z with god(a, M) = 1. Prove that if there exists b E ZM such that b2 a] in Zp, then there are exactly four distinct [r] E Zp such that Zp
6. [Marks 3] Suppose p and q are distinct primes. Find the general solution to the set of equations: x= -1 mod p x = -1 mod q. Show all the steps/details.
For the RSA encryption algorithm , do the following (a) Use p=257,q=337(n=pq=86609),b=(p-1)(q-1)=86016. Gcd(E,b)=1, choose E=17, find D, the number which has to be used for decryption, using Extended Euclidean Algorithm (b) One would like to send the simple message represented by 18537. What is the message which will be sent? (c) Decrypt this encrypted message to recover the original message.
Question 29 1 pts In an application of the RSA cryptosystem, Bob selects positive integers p, q, e, and d, where p and a are prime. He publishes public key (e, N), where N =p'q. the number d is the decryption key. 0 = (p-1)(q-1). Select all the statements that are correct. Ifm is not equal to por q, then (m) mod N=m It must be the case that d'e mod 0 - 1 If mis not equal to por...
(i) State Sylow's theorems. (ii) Suppose G is a group with IGI pr where p, q and r are distinct primes. Let np, nq and nr, denote, respectively, the number of Sylow p, q- and r-subgroups of G. Show that Hence prove that G is not a simple group. (iii) Prove that a group of order 980 cannot be a simple group.
N=pq with p,q distinct odd primes. Give an expression for the order of (Z/NZ)x in terms of p and q. Then, give an expression for the maximum order of a single element in (Z/NZ)x in terms of p and q.Why does that imply that there does not exist a primitive root modulo N?
Use C++ forehand e receiver creates a public key and a secret key as follows. Generate two distinct primes, p andq. Since they can be used to generate the secret key, they must be kept hidden. Let n-pg, phi(n) ((p-1)*(q-1) Select an integer e such that gcd(e, (p-100g-1))-1. The public key is the pair (e,n). This should be distributed widely. Compute d such that d-l(mod (p-1)(q-1). This can be done using the pulverizer. The secret key is the pair (d.n)....
2. Alice is a student in CSE20. Having learned about the RSA cryptosystem in class, she decides to set-up her own public key as follows. She chooses the primes p=563 and q = 383, so that the modulus is N = 21 5629. She also chooses the encryption key e-49. She posts the num- bers N = 215629 and e-49 to her website. Bob, who is in love with Alice, desires to send her messages every hour. To do so,...