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8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4. (You will need Wilson's Theorem for one (mod p). Prove: a 2--1 mod p has a solution if and only if p dircction of the proof.) 8. Let p be an odd prime. In this exercise, we prove a famous result that characterizes precisely when -1 has a sqare root 1 mod 4....
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...
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,...
Let p be an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are the quadratic residues of p that lie between 1 and p - 1. Prove that 1,0 (P-1)/2 i- 1 Hint: If a is a quadratic residue less than or equal to (p-1)/2 then what is p - ai? 5. Let p be a prime with p Ξ 1 (mod 4). Suppose that ai, a2, . . . ,a(p-1)/2 are...
9.3. Exercise 9.2 asked you to determine the value of (p-1)! (mod p) when p is a prime (a) Compute the value of (m 1)l (mod m) for some small values of m that are no (b) If you know the val ue of (n -1)! (mod n), how can you use the value to definitely ber. prime. Do you find the same pattern as you found for primes? distinguish whether n is prime or composite?
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod 19). (Hint: Corresponding to any given x , points (x,y) and (x,-y) can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is a quadratic residue mod 19. Recall that a value v ∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p). If v is indeed a quadratic residue, we can calculate the two...
Let p be a prime. Show that Zp(X)/(X2+1) is a field iff the equation x2=-1 has no solution (mod p).
this is number theory i need help with thanks alsonlls show all work Assume a, b,...are integers, r, s, t > 1, m > 2, p =prime> 2. 1. Write c= (m) and let 91, 92,...,q* be all the distinct prime factors of c. Suppose that (a,m) = 1 and ac/4 # 1(mod m), 1sisk. Prove that a is a primitive root (mod m). Prove that 2 is a primitive root (mod 11). 3. Find the indices of 3, 4...
Let p be an odd prime. Write p in the form p = 2k + 1 for some k E N. Prove that kl-(-1)* mod p. Hint: Each j e Z satisfies j (p-od p.