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Exercise 9.3.18. Show that 341 is a pseudoprime base 2 but not a pseu- doprime base 3.

Exercise 9.3.18. Show that 341 is a pseudoprime base 2 but not a pseu- doprime base 3.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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Please do Exercise 9.3.18 and please show step by step and explain

Let n be an odd composite number and b be a positive integer such that ged(b, n) = 1. If bn-1 = 1 (mod n), then n is a pseudoprime base b. We can get a more accurate test for the primality of n if we test n versus a 294 CHAPTER 9 INTRODUCTION TO CRYPTOGRAPHY number of prime bases. If n is a pseudoprime for several prime bases, then we can say with high confidence that n is most probably a prime. Exercise 9.3.18. Show that 341 is a pseudoprime base 2 but not a pseu- doprime base 3.
Transcribed Image Text:Let n be an odd composite number and b be a positive integer such that ged(b, n) = 1. If bn-1 = 1 (mod n), then n is a pseudoprime base b. We can get a more accurate test for the primality of n if we test n versus a 294 CHAPTER 9 INTRODUCTION TO CRYPTOGRAPHY number of prime bases. If n is a pseudoprime for several prime bases, then we can say with high confidence that n is most probably a prime. Exercise 9.3.18. Show that 341 is a pseudoprime base 2 but not a pseu- doprime base 3.
Probabilistic methods using the "little Fermat theorem" In practice, neither the brute force nor the Fermat method is used to verify large prime numbers. Instead, probabilistic methods are used: these methods can show that it's very, very likely that n is a prime, but they don't prove for certain. The principal test of this type is the Miller-Rabin test for primality. This test uses some of the principles described below. In Exercise 18.3.15 in Section 18.3.2, we will prove the following fact (which is widely known as Fermat's little theorem): If p is any prime number and a is any nonzero integer, then a²-1 = 1 (mod p). We can use Fermat's little theorem as a screening test for primes. For example, 15 cannot be prime since 215-12¹44 (mod 15). However, 17 is a potential prime since 217-12161 (mod 17). We say that an odd composite number n is a pseudoprime if 2-1 (mod n).
Transcribed Image Text:Probabilistic methods using the "little Fermat theorem" In practice, neither the brute force nor the Fermat method is used to verify large prime numbers. Instead, probabilistic methods are used: these methods can show that it's very, very likely that n is a prime, but they don't prove for certain. The principal test of this type is the Miller-Rabin test for primality. This test uses some of the principles described below. In Exercise 18.3.15 in Section 18.3.2, we will prove the following fact (which is widely known as Fermat's little theorem): If p is any prime number and a is any nonzero integer, then a²-1 = 1 (mod p). We can use Fermat's little theorem as a screening test for primes. For example, 15 cannot be prime since 215-12¹44 (mod 15). However, 17 is a potential prime since 217-12161 (mod 17). We say that an odd composite number n is a pseudoprime if 2-1 (mod n).
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