Let p be a prime number larger than 2. Suppose we uniformly sample for two times with replacement from {0,.., p-1}. Then multiply these two numbers with each other in (mod p) space. What is the probability of getting (p-1)/2?

Q: For which odd primes p is 3 a square mod p?

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Q: Let n = pq, where p and q are distinct primes such that p = q = 3 (mod 4). Suppose a E Z+ with a <…

A: We use the properties and definitions of Jacobi symbols to solve this problem. The detailed solution…

Q: Least residue of 8^68 modulo 17

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Q: 7) Let p be an odd prime, find if a) p =1 mod 8

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Q: Let n and m be two integers such that n> m > 0. If a bit string of length n is randomly selected,…

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Q: When all the numbers in a given data series appear only once, then the series is said to: a. be…

A: We know that, Mode:     The value occurs with the highest frequency. Uni mode: When there is only…

Q: Fermat's "Little" Theorem states that whenver n is prime and a is an integer, a"-l = 1 = 1 mod n a)…

A: Given a =463 and  n=953 We need to find x such that                   463952 ≡ x (mod 953).

Q: The n-th random number is generated by the function below. If xO=2, x2 is * Xn = (7Xn-1 + 9) mod 11…

A: Given Xn = ( 7 Xn-1 + 9) mod 11 X0 = 2 X2 = ?

Q: Prove with induction the sum of first n Fibonacci numbers with even indices, i.e., f2 +f4 +···+f2n…

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Q: 7. A sequence of binary digits (0's and 1's) representing the outcome of a se- quence of fair coin…

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Q: 13. Compute 41023 (mod 11). a. 14. Prove that for any integer n, n31 = n(mod7).

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Q: For which odd primes p is 5 a square mod p?

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Q: 1. Suppose pseudo-random numbers are produced by using: xn+1 = (3xn + 11) mod 13. If x3= 5, find x2…

A: Given that xn=3xn+11mod13x3=5

Q: Hello, I need help for the problem in the picture. Thank you!

A: We are given that α ∈ Z is a primitive root modulo an odd prime p. So,    αϕ(p) ≡ 1 mod p =&gt; αp-1…

Q: Let n be a fixed positive integer greater than 1. If a mod n = a' andb mod n = b', prove that (a +…

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Q: Let o (n) be the sum of the divisors of n and let (n) be the Euler phi functions. Let n = = pq be…

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Q: (a) Show that, for any odd prime p > 5, 1 if p = 1, –5 (mod 12) (주) -1 if p= -1,5 (mod 12)

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Q: Find the sequence of pseudorandom numbers generated by the linear congruential method with modulus m…

A: Given that, the recurrence relation is xn+1=axn+c mod m, and m=9, a=89, c=2 and x0=5. Obtain the…

Q: prove (1+ )" in <4 for all n21

A: In the given question we have to prove that  1+1nn&lt;4∀n≥1 . we can prove this by two methods one…

Q: Use the LCM to generate 40-non-repeated random numbers with any value of xo, a, c, and m (m must be…

A: # Function to generate random numbers def linearCongruentialMethod(Xo, m, a, c,…

Q: 6. Prove by contradiction: On the outside rim of a circular disk the integers from 1 through 30 are…

A: We need to prove the given statement.

Q: 9 The residue of 6 (mod 5) is (Round to the nearest whole number.)

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Q: 4 If a = b (mod n), then a" = b™ (mod n) for every positive integer m.

A: It is given that  a≡b (mod n), then am≡bm (mod n) for every positive integer m. It will prove by…

Q: 6. Show that ifn = pq, a product of distinct primes, then a (n)+1 (mod n) for all a. = a

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Q: Prove bey wnduc hiou that for n22.

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Q: Given any odd prime p, prove that: 2(p−3)! ≡−1 mod p. (Hint: Start with 2(p−3)!≡x mod p. Then work…

A: Let p be any odd prime number.  So we start with  2p-3!≡x mod p Also we have p-2≡-2 mod pp-1≡-1 mod…

Q: Prove that min (a) If a = b (mod n) and mln ,then a = b (mod m) (b) If a = b (mod n) and c =d (mod…

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Q: Let p be an odd prime. Suppose p5 mod 8. Show p-1 p? – 1 mod 2. 5

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Q: Suppose RAND is a function that uses an algorithm to generate random numbers between 0 and 2 with…

A: Let X be a random variable such that X: A number selected at random between numbers 0 and 2…

Q: Let p > 3 be a prime. Show that if p2+ p + 1 is also prime, then p =5 mod 6.

A: We prove the result by the method of contradiction.

Q: The pigeonhole principle states: if n things are put in fewer than n holes, some hole has at least…

A: The pigeonhole principle states:  If n things are put in fewer than n holes then some hole has at…

Q: (4) Let p be a prime number. Then r = 1 mod p has exactly two solutions modulo p.

A: 4) To prove, x2≡1mod p has exactly 2 roots such that p is a prime number.

Q: Show that if n is prime and b is a positive integer with n ̸∣ b, then n passes Miller’s test to the…

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Q: What sequence of pseudorandom numbers is generated using the linear congruential generator xn+1 =…

A: According to the given information, it is required to find the sequence of pseudorandom numbers.

Q: 2. Generate six random numbers by using Congruential or Residual Generator.Xi +1 ≡ (aXi + c) (mod…

A: 2)   Given that,   Xi+1=(aXi+c)(mod m)   Here,   a=5 X0=3 c=1 m=8

Q: 6. What sequence of pseudorandom numbers is gener- ated using the linear congruential generator xn+1…

A: Given: Linear congruential generator xn+1=4xn+1 mod 7, with x0=3 To find: Sequence of psedorandom…

Q: (c) Use Fermat little theorem to show that (a + b)® = a® + ® mod prime p and any positive integer a.…

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Q: Let p be prime. Show that C) mod p = C for 0 <k< p What does this imply about the binomial…

A: See the following below.

Q: If p is a prime satisfying n < p < 2n, show that 2n = 0 (mod p)

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Q: 3. Let's flip a fair coin independently an infinite number of times. Show with (one of the two…

A: Given the experiment that flipping a fair coin independently an infinite number of times.

Q: (This exercise constructs another proof of the infinitude of pimes) Show that the integer Qn = n! +…

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Q: 3. Find all primes p < 500 such that there are exactly 18 elements that have order 19 modulo p.

A: Step:-1 Since, we know for a prime p, modulo p forms a cyclic group. And, also according to a known…

Q: (a) gcd(q, n) = 1 if and only if kq # 0 mod (n) for all 1 < k< n. %3D

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Q: Use mathematical induction to prove the following ormulae for every positive interger n   1^3 +…

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Q: • Let p be the probability of a success in a Bernoulli experiment and Pn de- note the probability…

A: The Bernoulli trials have either success with probability p or failure with probability 1-p. The…

Q: Suppose that Bob has used two distinct primes p and q to create the public modulus n = pq and public…

A: RSA algorithm: 1. Choose two distinct prime numbers p and q. Take n = pq. 2. Compute λ(n), where λ…