Secrets from Primes

Sources and simplifications

Every reference cited on this site, and every place a figure idealizes the real system.

References

  1. Victor Shoup (2008). A Computational Introduction to Number Theory and Algebra, 2nd edition (Version 2, free PDF). Cambridge University Press. ISBN 9780521516440. textbook (Prime numbers)
  2. C. K. Caldwell and Y. Xiong (2012). What is the smallest prime?. Journal of Integer Sequences 15, Article 12.9.7. paper (Prime numbers)
  3. J. J. O'Connor and E. F. Robertson (1999). Eratosthenes of Cyrene (biography). MacTutor History of Mathematics, University of St Andrews. history (Prime numbers)
  4. Euclid (c. 300 BC); D. E. Joyce (online edition) (1998). Euclid's Elements, Book IX, Proposition 20: there are infinitely many primes. Clark University. primary (Prime numbers)
  5. Great Internet Mersenne Prime Search (GIMPS) (2024). List of known Mersenne prime numbers. Mersenne Research, Inc.. database (Prime numbers)
  6. C. Bays and R. H. Hudson (2000). A new bound for the smallest x with π(x) > li(x). Mathematics of Computation 69 (231), 1285–1296. doi:10.1090/S0025-5718-99-01104-7. paper (Prime numbers)
  7. The OEIS Foundation (2025). A006880: Number of primes < 10^n (π(10^n)). The On-Line Encyclopedia of Integer Sequences. database (Prime numbers)
  8. R. D. Carmichael (1910). Note on a new number theory function. Bulletin of the American Mathematical Society 16 (5), 232–238. doi:10.1090/S0002-9904-1910-01892-9. paper (Prime numbers)
  9. The OEIS Foundation (2025). A002997: Carmichael numbers. The On-Line Encyclopedia of Integer Sequences. database (Prime numbers)
  10. W. R. Alford, A. Granville and C. Pomerance (1994). There are infinitely many Carmichael numbers. Annals of Mathematics 139 (3), 703–722. doi:10.2307/2118576. paper (Prime numbers)
  11. G. L. Miller (1976). Riemann's hypothesis and tests for primality. Journal of Computer and System Sciences 13 (3), 300–317. doi:10.1016/S0022-0000(76)80043-8. paper (Prime numbers)
  12. M. O. Rabin (1980). Probabilistic algorithm for testing primality. Journal of Number Theory 12 (1), 128–138. doi:10.1016/0022-314X(80)90084-0. paper (Prime numbers)
  13. The OEIS Foundation (2025). A014233: Smallest odd number for which the Miller–Rabin test with the first n prime bases fails (ψ_n). The On-Line Encyclopedia of Integer Sequences. database (Prime numbers)
  14. C. Pomerance, J. L. Selfridge and S. S. Wagstaff Jr. (1980). The pseudoprimes to 25·10^9. Mathematics of Computation 35 (151), 1003–1026. doi:10.1090/S0025-5718-1980-0572872-7. paper (Prime numbers)
  15. J. P. Sorenson and J. Webster (2017). Strong pseudoprimes to twelve prime bases. Mathematics of Computation 86 (304), 985–1003. doi:10.1090/mcom/3134. paper (Prime numbers)
  16. The OEIS Foundation (2025). A001262: Strong pseudoprimes to base 2. The On-Line Encyclopedia of Integer Sequences. database (Prime numbers)
  17. F. N. Cole (1903). On the factoring of large numbers. Bulletin of the American Mathematical Society 10 (3), 134–137. doi:10.1090/S0002-9904-1903-01079-9. paper (Prime numbers)
  18. J. M. Pollard (1975). A Monte Carlo method for factorization. BIT Numerical Mathematics 15, 331–334. doi:10.1007/BF01933667. paper (Prime numbers)
  19. R. P. Brent (1980). An improved Monte Carlo factorization algorithm. BIT Numerical Mathematics 20, 176–184. doi:10.1007/BF01933190. paper (Prime numbers)
  20. F. Boudot, P. Gaudry, A. Guillevic, N. Heninger, E. Thomé and P. Zimmermann (2020). Factorization of RSA-250 (announcement, 28 February 2020). CARAMBA team, LORIA / Inria Nancy. primary (Prime numbers)
  21. W. Diffie and M. E. Hellman (1976). New directions in cryptography. IEEE Transactions on Information Theory 22 (6), 644–654. doi:10.1109/TIT.1976.1055638. paper (Clock arithmetic)
  22. E. Barker (2020). Recommendation for Key Management: Part 1 – General (SP 800-57 Part 1 Rev. 5), §5.6.1.1 and Table 2. National Institute of Standards and Technology. doi:10.6028/NIST.SP.800-57pt1r5. standard (Clock arithmetic)
  23. R. L. Rivest, A. Shamir and L. Adleman (1978). A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21 (2), 120–126. doi:10.1145/359340.359342. paper (Public-key cryptography)
  24. National Institute of Standards and Technology (2024). Module-Lattice-Based Key-Encapsulation Mechanism Standard (FIPS 203). National Institute of Standards and Technology. doi:10.6028/NIST.FIPS.203. standard (Public-key cryptography)

Simplifications