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3. The number 2 is a primitive root modulo 19; the powers of 2 modulo 19...
g 2 is a primitive root modulo 19. Use the following table to assist you in...
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...
G-2 is a primitive root modulo 19. Use the following table to assist you in the solution of the f...
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 t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
1.32. Recall that g is called a primitive root modulo p if the powers of g...
1.32. Recall that g is called a primitive root modulo p if the powers of g give all nonzero elements of F p . (b) For which of the following primes is 3 a primitive root modulo p ? (i) p = 5 (ii) p = 7 (iii) p = 11 (iv) p = 17
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 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...
4 [20 points] Given that 491 is a prime and 26 is primitive modulo 491, use the Pohlig- Hellman a...
4 [20 points] Given that 491 is a prime and 26 is primitive modulo 491, use the Pohlig- Hellman algorithm to solve 26 192 (mod 491). Be sure to show your work. You may need the following data. 2624 490 (mod 491) 26 101 (mod 491) 2610 414 (mod 491) 414 223 (mod 491) 4141 51 (mod 491) 1922451 (mod 491) 1928 381 (mod 491) 19210 3 (mod 491) 223 153 (mod 491)
4 [20 points] Given that 491 is...
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...
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,...
Given 2 as a primitive root of 29, construct a table of discrete logarithms, and use it to solve ...
Given 2 as a primitive root of 29, construct a table of discrete logarithms, and use it to solve the following congruences b. x2− 4x − 16 ≡ 0(mod 29) c. x7≡ 17(mod 29) I need step by step calculations for b and c
Problem 2. Find a primitive root for 53. Using this, you can devise a bijection α...
Problem 2. Find a primitive root for 53. Using this, you can
devise a bijection α from the integers modulo 52 to the nonzero
integers modulo 53 with the property that α(a + b) = α(a)· α(b)
modulo 53. Explain. Does the law of exponents get involved at all?
Note: For this to work right, you can think of integers mod 52 as
{0, 1, 2, . . . , 51} or as any complete system of residues modulo
52,...
9. In Z/31Z, using Proposition 3, find a primitive root modulo 31. Proposition 3. Let a,b...
9. In Z/31Z, using Proposition 3, find a primitive root modulo 31. Proposition 3. Let a,b be elements of a finite abelian group. If a has order r, and b has order s, and (r, s) = 1, then ab has order rs.
(a) Solve the simultaneous congruences p = 1 (mod x – 3), p = 7 (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.
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...
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 t1 2 3 4 567 89 10 11 12 13 14 151617 18 g 2 481613714918 17 15 11 36125101 Question 1. (a) Find the least positive residue of 126...
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...
4 [20 points] Given that 491 is a prime and 26 is primitive modulo 491, use the Pohlig- Hellman algorithm to solve 26 192 (mod 491). Be sure to show your work. You may need the following data. 2624 490 (mod 491) 26 101 (mod 491) 2610 414 (mod 491) 414 223 (mod 491) 4141 51 (mod 491) 1922451 (mod 491) 1928 381 (mod 491) 19210 3 (mod 491) 223 153 (mod 491)
4 [20 points] Given that 491 is...
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,...
Problem 2. Find a primitive root for 53. Using this, you can
devise a bijection α from the integers modulo 52 to the nonzero
integers modulo 53 with the property that α(a + b) = α(a)· α(b)
modulo 53. Explain. Does the law of exponents get involved at all?
Note: For this to work right, you can think of integers mod 52 as
{0, 1, 2, . . . , 51} or as any complete system of residues modulo
52,...
9. In Z/31Z, using Proposition 3, find a primitive root modulo 31. Proposition 3. Let a,b be elements of a finite abelian group. If a has order r, and b has order s, and (r, s) = 1, then ab has order rs.
(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.
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