MA-410 Homework 3, Spring 2024, due as indicated for each problem.

All solutions must be submitted on the Moodle web site for the class at wolfware.ncsu.edu. You may upload a photo of your handwritten solution or a file of your typed solution.
Note my office hours on my schedule.

  1. Due Thursday April 4, 11:59pm.
    ENT, §7.3, Problem 10, page 141: For any non-negative integer n and any integer a, show that a and a4n+1 have the same last (decimal) digit.
  2. Due Thursday April 4, 11:59pm.
    ENT, §10.1, Problem 15, page 209: Decrypt the ciphertext
    1030   1511   0744   1237   1719
    that was encrypted by the RSA algorithm with key (n,k) = (2623, 869). Note that every pair of decimal digits corresponds to a letter: A = 00, … Z = 25, blank space = 26. [Hint: the recovery exponent is j = 29.]
    [My hint: use Maple's “a&^e mod n” procedure or PowerMod[a,e,n] in wolframalpha.com for computing ae mod n.]
  3. Due Tuesday, April 9, 11:59pm.
    ENT, §8.4, Problem 3, page 167: The following is a table of indices for the prime 17 relative to the primitive root 3:
    a 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
    ind3(a) 16 14 1 12 5 15 11 10 2 3 7 13 4 9 6 8
    With the aid of this table, solve the following congruences:
    (a) x12 ≡ 13 (mod 17).
    (b) 8x5 ≡ 10 (mod 17).
    (c) 9x8 ≡ 8 (mod 17).
    (d) 7x ≡ 7 (mod 17).
  4. Due Thursday, April 11, 11:59pm.
    ENT, §9.1, Problem 8, page 174: Assume that the integer r is a primitive root of the prime p, where p ≡ 1 (mod 8).
    (a) Show that the solutions of the quadratic congruence x2 ≡ 2 (mod p) are given by
    x ≡ ± (r 7(p-1)/8 + r (p-1)/8 ) (mod p).
    [Hint: first confirm that r 3(p-1)/2 ≡ – 1 (mod p).]
    (b) Use part (a) to find all solutions to the two congruences x2 ≡ 2 (mod 17) and x2 ≡ 2 (mod 41).
    Please prove part (a) for r being a quadratic non-residue. For part (b) use r=3 as a quadratic non-residue modulo both 17 and 41.
  5. Due Thursday, April 11, 11:59pm.
    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).