MA-410 Homework 3

Due at 4:59pm in my mailbox in SAS 3151, Thursday, April 12, 2018

Solutions may only be submitted in hard copy. Note my office hours on my schedule.

1. ENT, §6.2, Problem 7, page 116.
2. ENT, §8.4, Problem 3, page 167.
3. ENT, §9.1, Problem 8, page 174. Please prove part (a) for r being a quadratic non-residue.
4. ENT, §10.1, Problem 15, page 209. [Hint: use Maple's “&^ mod” procedure.]
5. Bonus problem: Let p be a prime ≡ 9 (mod 16); then p-1 ≡ 0 (mod 4) and p+7 ≡ 0 ≡ 7p+1 (mod 16). Let r be a quadratic non-residue, and let a be a quadratic residue.
   
   1. Case a^(p-1)/4 ≡ -1 (mod p): let b = x^-1 (a^(7p+1)/16 + a^(p+7)/16) mod p, where x is the squareroot of 2 modulo p computed in Problem 3 (Problem 8 on page 174 of the textbook). Please prove that b² ≡ a (mod p).
   2. Case a^(p-1)/4 ≡ 1 (mod p): let b = 2^-1 ((1+r^(p-1)/4)a^(7p+1)/16 + (1-r^(p-1)/4)a^(p+7)/16) mod p. Please prove that b² ≡ a (mod p).