MA410 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.

Due Thursday April 4, 11:59pm.
ENT, §7.3, Problem 10, page 141:
For any nonnegative integer n and any integer a,
show that a and a^{4n+1} have the same last (decimal) digit.

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
a^{e} mod n.]

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 
ind_{3}(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) x^{12} ≡ 13 (mod 17).
(b) 8x^{5} ≡ 10 (mod 17).
(c) 9x^{8} ≡ 8 (mod 17).
(d) 7^{x} ≡ 7 (mod 17).

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
x^{2} ≡ 2 (mod p) are given by
x ≡ ±
(r
^{
7(p1)/8
}
+ r
^{
(p1)/8
}
) (mod p).
[Hint: first confirm that
r
^{
3(p1)/2
}
≡ – 1 (mod p).]
(b) Use part (a) to find all solutions to the two congruences
x^{2} ≡ 2 (mod 17) and
x^{2} ≡ 2 (mod 41).
Please prove part (a) for r being a quadratic nonresidue.
For part (b) use r=3 as a quadratic nonresidue modulo
both 17 and 41.

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 nonresidue, and let a be a quadratic residue.

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 b^{2} ≡ a (mod p).

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 b^{2} ≡ a (mod p).