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 fin...
A) What sequence (At least 8 sequences) of pseudorandom numbers is generated using the linear congruential generator xn+1 (3xn2) mod 13 with seed Xo- 1? B)Find the sequence of pseudorandom numbers generated by the power generator with p-7, d-3, and seed xo 2
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?
11. What sequence of pseudorandom numbers is generated using the linear congruential generator xn +1 (4xn + 1) mod 7 with seed Xo-37 12. Encrypt the message STOP POLLUTION by translating the letters into numbers, applying the encryption function/ P)-(p + 4) mod 26, and then translating the numbers back into letters. 13. Decrypt this message encrypted using the shift cipher f (p) (p+ 10) mod 26 CEBBOXNOBXYG 14. Let P() be the statement that 12 +22 ++n2 -n-)(en+2) for...
Problem 8 (Pseudorandom Numbers). Randomly chosen numbers are often needed for computer simulations of physical phenomena, and they are needed to generate random keys for cryptography. Different methods have been devised for generating numbers that have certain properties of randomly chosen numbers. Because numbers generated by systematic methods are not truly random, they are called pseudorandom numbers. There are fundamental properties we would like the pseudorandom sequence/pseudorandom number generator to possess for cryptographic purposes. 1. It is easy to compute...
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) x = 2 (mod 7) x = 3 (mod 9) x = 4 (mod 11)
Problem 1 Use the Chinese remainder theorem, find all integers x such that: (20 pts) x = 1 (mod 5) r = 2 (mod 7) x = 3 (mod 9) I= 4 mod 11) Answer,
Problem 3. Use the Chinese Remainder Theorem to find all congruence classes that satisfy x2 = 1 mod 77.
Please do 10 & 11 Use Intermediate Value Theorem lial p(x) = x4 +7x = 9 has two real root. 8. df Open with Google Docs Then use your calculator to find the ro 9 Let f(z)2with € [0, 00). Find a positive mumber e and two sequences {xn} and {yn} such that lim-(nn) = 0 but |f(xn)- f(Yn)| 2 e. Then conclude that f(x) = x2 is not uniformly continuous on [0, ao) [0, oo). Show that f is...
Please answer question 3 Find all (infinitely many) solutions of the system of congruence's: Use Fermata little theorem to find 8^223 mod 11. (You are not allowed to use modular exponentiation.) Show that if p f a, then a^y-2 is an inverse of a modulo p. Use this observation to compute an inverse 2 modulo 7. What is the decryption function for an affine cipher if the encryption function is 13x + 17 (mod 26)? Encode and then decode the...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...