76.Let p be an odd prime. Prove that if Ord, (a) = his even, then a/2...
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...
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.
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...
Let g be a primitive root modulo to the odd prime p. Prove that: 2)=-1 2)=-1
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,...
please prove proofs and do 7.4 7.2 Theorem. Let p be a prime, and let b and e be integers. Then there exists a linear change of variahle, yx+ with a an integer truns- farming the congruence xbx e0 (mod p) into a congruence of the farm y (mod p) for some integer 8 Our goal is to understand which integers are perfect squares of other inte- gers modulo a prime p. The first theorem below tells us that half...
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.
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
g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p) g(p+1)/2 (a) Suppose 9 is a p rimitive root of an odd prime p. Prove that- (mod p)
(b) Let p be a prime that is congruent to 3 modulo 4. Let b ∈ Z. Let a = b (p+1)/4 . Show that a 2 ≡ ±b (mod p). (c) Give an algorithm to compute square roots of something modulo p, when p ≡ 3 (mod 4). Note: Not all things are square modulo p, so the algorithm should return the square root or inform you there is no square.