29. Give the number of primitive roots (mod p), where p is a prime. 29. Give the number of primitive roots (mod p), where p is a prime.
Let
p be an odd prime. Prove that if g is a primitive root modulo p,
then g^(p-1)/2 ≡ -1 (mod p).
Let p be an odd prime. Prove that if g is a primitive root modulo p, then go-1)/2 =-1 (mod p) Hint: Use Lemma 2 from Chapter 28 (If p is prime and d(p 1), then cd-1 Ξ 0 (mod p) has exactly d solutions).
Let p be an odd prime. Prove that if g is a primitive...
8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of p.
8. Let g be a primitive root of an odd prime p, and suppose that p3 (mod 4). Show that -g is not a primitive root of p.
Need help!! Please help — crypto math
1. Determine L13(18) for p 19. 2. Let p be prime, and α a primitive root mod p. Prove that α(p-1)/2-_1 (mod p). 3. It can be shown that 5 is a primitive root for the prime 1223. You want to solve the discrete logarithm problem 53 (mod 1223). You know 3611 Prove it. 1 (mod 1223). Is x even or odd?
1. Determine L13(18) for p 19. 2. Let p be prime,...
7.23 Theorem. Let p be a prime congruent to 3 modulo 4. Let a be a natural number with 1 a< p-1. Then a is a quadrutic residue modulo pif and only ifp-a is a quadratic non-residue modulo p. 7.24 Theorem. Let p be a prime of the form p odd prime. Then p 3 (mod 4). 241 where q is an The next theorem describes the symmetry between primitive roots and quadratic residues for primes arising from odd Sophie...
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
3. Let p>3 be an odd prime and let {ri,r2, .r} be the set of incongruent primitive roots modulo p. Compute the product rir .r modulo p. Recall the proof of Wil- son's Theorem for inspiration
Given an elliptic curve E mod p, where p is a prime, the number
of points on the curve is denoted as #E. Also, the ECDLP is
expressed as dP = T.
Which of these statements is TRUE? (select all that apply)
Incorrect 0/0.15 pts Question 18 The image below illustrates different elliptic curves. Elliptic curve cryptosystems rely on the hardness of the generalized discrete logarithm problem. ECDLP.png Given an elliptic curve E mod p, where p is a prime,...
76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2 = -1 mod p. 77.let p be an odd prime. Prove that if Ord, (a) = 3, then 1+ a + a? = 0 mod p and Ord,(1 + a) = 6. 78.Show that 3 is a primitive root modulo 17. How many primitive roots does 17 have? Find them.
(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.
2.5. Let p be an odd prime and let g be a primitive root modulo has a square root modulo p if and only if its discrete logarithm log,(a) mod p. Prove t that is even.
negate:
(b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is called composite if it is not prime.) (c) For every integer n > 0, there exists a prime number p such that n S p < 2n.
(b) There exists a composite number n such p-11 (mod n) whenever p is a prime that doesn't divide n. (Recall that a natural number is...