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?
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.Wh...
Let n = rs, where r and s are distinct odd primes. Show that there is not a primitive root modulo n. (Solve through letting L = LCM(r − 1, s − 1). Show that the order of any element of (Z/nZ)∗ is at most L, and that L < φ(n).)
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.
8) (Problem 17 (a) on page 49) Let p and q be two distinct primes. Show that for any integer a, pq|(a p+q − a p+1 − a q+1 + a 2 ). Hint: Find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo p, and then find the least residue of a p+q − a p+1 − a q+1 + a 2 modulo q. After that, use the following result: Suppose x,...
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
1. For n-pg, where p and q are distinct odd primes, define (p-1)(q-1) λ(n) gcd(-1-1.411) Suppose that we modify the RSA cryptosystem by requiring that ed 1 mod X(n). a. Prove that encryption and decryption are still inverse operations in this modified cryptosystem. RSA cryptosystem.
Suppose that Bob, very concerned with security, selects an encrypting modulus n=pq where p and q are large primes. Suppose he also chooses two encrypting exponents e1 and e2. He asks people sending him messages to “double encrypt” their messages as follows. For each plain text (an element of Zn), he asks them to encrypt it using RSA with modulus n and encrypting exponent e1, and then encrypt the result of that using RSA with modulus n and encrypting exponent...
Activity: A Journey Through Calculus from A to Z sin(x-1) :- 1) x< h(x) kr2 - 8x + 6. 13x53 Ver-6 – x2 +5, x>3 Consider f'(x), the derivative of the continuous functionſ defined on the closed interval -6,7] except at x 5. A portion of f' is given in the graph above and consists of a semicircle and two line segments. The function (x) is a piecewise defined function given above where k is a constant The function g(x)...