What is the order of an element of a group, find the order of 4 (mod 7). And what is a primitive element mod n, is 5 a primitive element modulo 19? Verify or refute it.
What is the order of an element of a group, find the order of 4 (mod...
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (mod x – 5). (b) Find the total number of monic irreducible polynomials of degree 5 in Fr[c]. (c) Find a primitive root modulo 52020. (Make sure to justify your answer.) (d) Determine the total number of primitive roots modulo 52020.
4. Recall that a € ZnZ is called a primitive root modulo n, if the order of a in Z/nZ is equal to on). We have seen in class that, if p is a prime, then we can always find primitive roots modulo p. Find all elements of (Z/11Z)* that are primitive roots modulo 11. 5. Can you find primitive roots modulo 16? Explain your answer. 6. In class, we found 2 primitive roots modulo 9 = 32, namely 2...
Problem 4. Let G be a group. Recall that the order of an element g G is the smallest k such that gk = 1 (or 00, if such a k doesn't exist). (a) Find the order of each element of the symmetric group S (b) Let σ-(135)(24) and τ-(15)(23)(4) be permutations in S5. Find the cycle decompositions for (c) Let σ-(123456789). Compute ơ-i, σ3, σ-50, and σί006 (d) Find all numbers n such that Ss contains an element of...
(d) Decrypt the ciphertext message LEWLYPLUJL PZ H NYLHA ALHJOLY that was encrypted with the shift cipher f(p) (p+7) mod 26. [10 points] (e) [Extra Credit - 5 points] Encrypt the message "BA" using the RSA cryptosystem with key (ne) = (35,5), where n = p . q 5-7 and ged(e, (p-1) 1)) (5, 24) 1. 6. [5 points each (a) Is 2 a primitive root of 11? (b) Find the discrete logarithm of 3 modulo 11 to the base...
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?
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
Just #5 Find the order of every element in EIs. d the rde of every element in the symmetry gronp of thesquare T. What are all of the cyelie sthgroups of the quiatermion group. 0s2 N. Tist all of the cyclic snbgronps of (3 9. Tist eery generator of each sbgroup of order n
g 2 is a primitive root modulo 19. Use the following table to assist you in the solution of the first two questions and 4(a). The most efficient solutions involve the use of the table and the application of theory; numerically correct solutions involving long computations will not receive full credit t 1234567 8 9 10 11 12 13 14 15 16 1718 2 48 16 13 7 149 18 1715 11 3 6 12 5 10 1 (a) Find...
Exercise 4. Consider the permutation group S7. a. Show that the subgroup generated by the element (1,2,3,4,5,6) is a cyclic group of order 6. b. Show that the subgroup generated by the element (1,3, 4, 5, 6, 7) is a cyclic group of order 6. c. Show that the subgroup generated by the element (1,2,3) is a cyclic group of order 3. d. Show that the subgroup generated by the element (6, 7) is a cyclic group of order 2....
Find the additive inverse of the following value mod m. 16. 7 mod 9 17. 4 mod 12 18. 63 mod 30 19. 222 mod 3