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?