Cryptography Tutorial 7
1. Prove that any composite number n has a prime factor p such that p ≤
√
n.
2. Use Trial Division to factorize
(a) n = 9361;
(b) n = 84019;
(c) n = 924521;
3. Use Fermat Factorization to factorize
(a) n = 145;
(b) n = 517;
(c) n = 4717;
4. Show that the worst case for Fermat’s factorization requires
n+1 √
− n steps.
2
5. Use the Lucas-Lehmer test to determine if each of the following (Mersenne) numbers is prime
(i) 31 (ii) 127 (iii) 2047 (iv) 8191
6. Use the Lucas’ converse of Fermat’s Little Theorem to show that each of the following numbers is prime:
(i) 211 (ii) 233 (iii) 2071723, with a = 2 (Note that n − 1 = 2 × 3 × 17 × 19 × 1069.)
7. Suppose that for each prime p|(n − 1), p > 2, there exists a positive integer a such that
n−1
n−1
a 2 ≡ −1 (mod n) but a p 6≡ 1 (mod n). Show that n is prime.
8. In order to exchange a secrete key Ayanda and Bongani choose a number p = 97
and a primitive root of p say g = 5. If they chose random numbers a = 6 and b = 11
respectively, find the common secret key.
9. Use Shank’s algorithm to solve
(a) 2x ≡ 35 (mod 37);
(b) 59x ≡ 63 (mod 71).
10. Suppose p is a prime and g is a primitive root modulo p. Determine the Discrete log
of −1.