In the following, the moduli are not pairwise relatively prime, so the Chinese Remainder Theorem does...
Problem 6: There are three users with pairwise relatively prime moduli n, n and n3. Suppose that their encryption exponents are all e3. The same message m is sent to each of them and you intercept the three ciphertexts ci mrs (mod n.), for i-1, 2, 3. (a) Show that 0 m3< nin2n (b) Show how to use the CRT to find m3 (as an exact integer, not only as m3 (mod ninns)) and, therefore also m c) Suppose that...
Problem 2 (Chinese Remaindering Theorem) [20 marks/ Let m and n be two relatively prime integers. Let s,t E Z be such that sm+tn The Chinese Remaindering Theorem states that for every a, b E Z there exists c E Z such that r a mod m (Va E Z) b mod nmod mn (3) where a convenient c is given by 1. Prove that the above c satisfies both ca mod m and cb mod n 2. LetxEZ. Prove...
3. (16 points) Solve the system of linear congruences using the Chinese Remainder Theorem. 4 (mod 11) a 11 (mod 12) x=0 (mod 13) b. (6 pts) Find the inverses n (mod 11), n21 (mod 12), and nz1 (mod 13). Using these ingredients find the common solution a (mod N) to the system. c. (4 pts) 4. (8 points) What is 1!+ 23+50! congruent to modulo 14?
9. Use the construction in the proof of the Chinese remainder theorem to find a solution to the system of congruences X 1 mod 2 x 2 mod 3 x 3 mod 5 x 4 mod 11 10. Use Fermats little theorem to find 712 mod 13 11. What sequence of pseudorandom numbers is generated using the linear congruential generator Xn+1 (4xn + 1) mod 7 with seed xo 3? 9. Use the construction in the proof of the Chinese...
Problem 1. Solve the following simultaneous congruence using the Chinese Remainder or the substitution method. a: 2 (mod 5) a: 0 (mod 7) a: = 1 Problem 1. Solve the following simultaneous congruence using the Chinese Remainder or the substitution method. x = 2 (mod 5) x = 0 (mod 7) El mod 17)
5. Chinese Remainder Theorem, 10pt] Use the method of the Chinese Remainder Theorem to solve the following problems a) [6pt] Find x (between 0 and 2063*6947) such that x 1480 (2063) and x-5024 (6947) b) [4pt] Find x (between 0 and 2063*6947*8233) such that x 1480 (2063), x 5024 (6947) and x- 7290 (8233) 5. Chinese Remainder Theorem, 10pt] Use the method of the Chinese Remainder Theorem to solve the following problems a) [6pt] Find x (between 0 and 2063*6947)...
3. If the integers mi, i = 1,..., n, are relatively prime in pairs, and a1,..., an are arbitrary integers, show that there is an integer a such that a = ai mod mi for all i, and that any two such integers are congruent modulo mi ... mn. 4. If the integers mi, i = 1,..., n, are relatively prime in pairs and m = mi...mn, show that there is a ring isomorphism between Zm and the direct product...
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...
For this problem, the Central Limit Theorem does apply. Use the sample shown below (n = 20). Determine the following confidence intervals, indicating the Cl endpoints to 2 decimal places: Confidence Interval low end high end 95% Cl for population mean 95% CI for population variance This sample was collected from a population having an unknown oʻ: 5 9 10 28.33 16.59 26.37 18.20 14.64 12.58 20.12 18.25 25.75 20.29 25.06 20.79 19.78 15.29 21.96 24.05 11.42 11.06 15.36 19.17...
The following function is_prime() is not a very efficient prime number test: #include <stdio.h> int is_prime(int n) { int d; for (d = 2; d < n; d++) { if (!(n % d)) return 0; } return 1; } int main() { if (is_prime(7)) printf("Seven is Prime!\n"); return 0; } It is unnecessary to divide n by all numbers between 2 and n - 1 to determine if n is prime. Only divisors up to and including need to be...