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Math 5248 Lec 2 Homework 5 Due 4/30/13 Ten questions, ten points each. 100 points total. Chapter 17: Let n = 1105= 5•13•17. 1. How many numbers b are there in the range 1<b<n such that gcd(b,n) =1. 2. Use the CRT to count how many solutions there are to bn-1 ≡ 1 mod n, with gcd(b,n)=1. 3. Find an integer b such that the Fermat test shows n is definitely composite (that is, find b relatively prime to n with bn-1 not congruent to 1 mod n), OR show that no such b exists. 4. Using the procedure on page 230, (and not by guessing or brute force) find a b such that the Euler test shows n is definitely composite. 5. In Z/21x, find the set L of all the false witnesses to the primality of 21. Then verify that L is a subgroup of Z/21x. Chapter 16: 6. Find the 1000th point of the LCG given by sn+1 = (3sn + 15)%31 with s0 = 7. 7. Find the fixed point of the LCG given by sn+1 = (53sn + 8)%113. 8 Find the period of the length-eight (mod 2) LFSR with coefficients (c0, c1, …, c7) = (0, 1, 0, 1, 0, 1, 0, 1) and initial state (s0, s1, …, s7) = (0, 0, 0, 0, 1, 0, 0, 1) That is, determine the size of the loop of states that will repeat. 9. Find the 5th pseudorandom bit generated by the BBS pRNG with modulus n = (43)(11) and 0th state/seed s0 = 7. 10. Find the period length of the BBS pRNG with modulus n = (43)(11) and seed so = 7. What is the loop of pseudorandom bits?