Abstract Algebra Theory and Applications - Computer Science ...

106 CHAPTER 6 INTRODUCTION TO CRYPTOGRAPHY(b) Write a program to implement the following factorization algorithmbased on the observation in part (a).x ← ⌈ √ n ⌉y ← 11: while x 2 − y 2 > n doy ← y + 1if x 2 − y 2 < n thenx ← x + 1y ← 1goto 1else if x 2 − y 2 = 0 thena ← x − yb ← x + ywrite n = a ∗ bThe expression ⌈ √ n ⌉ means the smallest integer greater than or equalto the square root of n. Write another program to do factorization usingtrial division and compare the speed of the two algorithms. Whichalgorithm is faster and why?2. Primality Testing. Recall Fermat’s Little Theorem from Chapter 5. Let pbe prime with gcd(a, p) = 1. Then a p−1 ≡ 1 (mod p). We can use Fermat’sLittle Theorem as a screening test for primes. For example, 15 cannot beprime since2 15−1 ≡ 2 14 ≡ 4 (mod 15).However, 17 is a potential prime since2 17−1 ≡ 2 16 ≡ 1 (mod 17).We say that an odd composite number n is a pseudoprime if2 n−1 ≡ 1 (mod n).Which of the following numbers are primes and which are pseudoprimes?(a) 342(c) 601(e) 771(b) 811(d) 561(f) 6313. Let n be an odd composite number and b be a positive integer such thatgcd(b, n) = 1. If b n−1 ≡ 1 (mod n), then n is a pseudoprime base b.Show that 341 is a pseudoprime base 2 but not a pseudoprime base 3.