Abstract Algebra Theory and Applications - Computer Science ...

EXERCISES 1074. Write a program to determine all primes less than 2000 using trial division.Write a second program that will determine all numbers less than 2000 thatare either primes or pseudoprimes. Compare the speed of the two programs.How many pseudoprimes are there below 2000?There exist composite numbers that are pseudoprimes for all bases to whichthey are relatively prime. These numbers are called Carmichael numbers.The first Carmichael number is 561 = 3 · 11 · 17. In 1992, Alford,Granville, and Pomerance proved that there are an infinite number ofCarmichael numbers [4]. However, Carmichael numbers are very rare. Thereare only 2163 Carmichael numbers less than 25×10 9 . For more sophisticatedprimality tests, see [1], [6], or [7].References and Suggested Readings[1] Bressoud, D. M. Factorization and Primality Testing. Springer-Verlag, NewYork, 1989.[2] Diffie, W. and Hellman, M. E. “New Directions in Cryptography,” IEEETrans. Inform. Theory 22 (1976), 644–54.[3] Gardner, M. “A New Kind of Cipher that Would Take a Million Years toBREAK,” Scientific American 237 (1977), 120–24.[4] Granville, A. “Primality Testing and Carmichael Numbers,” Notices of theAmerican Mathematical Society 39(1992), 696–700.[5] Hellman, M. E. “The Mathematics of Public Key Cryptography,” ScientificAmerican 241 (1979), 130–39.[6] Koblitz, N. A Course in Number Theory and Cryptography. Springer-Verlag,New York, 1987.[7] Pomerance, C., ed. Cryptology and Computational Number Theory. Proceedingsof Symposia in Applied Mathematics, vol. 42. American MathematicalSociety, Providence, RI, 1990.[8] Rivest, R. L., Shamir, A., and Adleman, L., “A Method for Obtaining Signaturesand Public-key Cryptosystems,” Comm. ACM 21(1978), 120–26.