Abstract
In August 2002, Agrawal, Kayal, and Saxena described an unconditional, deterministic algorithm for proving the primality of an integer N. Though of immense theoretical interest, their technique, even incorporating the many improvements that have been proposed since its publication, remains somewhat slow for practical application. This paper describes a new, highly efficient method for certifying the primality of an integer \(N \equiv 1 \pmod 3\), making use of quantities known as Eisenstein pseudocubes. This improves on previous attempts, including the peudosquare-based approach of Lukes et al., and the pseudosquare improvement proposed by Berrizbeitia, et al.
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References
Lukes, R.F., Patterson, C.D., Williams, H.C.: Some results on pseudosquares. Mathematics of Computation 65(213), S25–S27, 361–372 (1996)
Hall, M.: Quadratic residues in factorization. Bulletin of the American Mathematical Society 39, 758–763 (1933)
Williams, H.C.: Édouard Lucas and Primality Testing. Canadian Mathematical Society Series of Monographs and Advanced Texts, vol. 22. Wiley Interscience, Hoboken (1998)
Williams, H.C.: Primality testing on a computer. Ars Combinatoria 5, 127–185 (1978)
Berrizbeitia, P., Müller, S., Williams, H.C.: Pseudocubes and primality testing. In: Buell, D.A. (ed.) ANTS 2004. LNCS, vol. 3076, pp. 102–116. Springer, Heidelberg (2004)
Wooding, K., Williams, H.C.: Doubly-focused enumeration of pseudosquares and pseudocubes. In: Hess, F., Pauli, S., Pohst, M. (eds.) ANTS 2006. LNCS, vol. 4076, pp. 208–221. Springer, Heidelberg (2006)
Sorenson, J.P.: Sieving for pseudosquares and pseudocubes in parallel using doubly-focused enumeration and wheel datastructures. In: Hanrot, G., Morain, F., Thomé, E. (eds.) ANTS-IX. LNCS, vol. 6197, pp. 331–339. Springer, Heidelberg (2010)
Ireland, K., Rosen, M.: A Classical Introduction to Modern Number Theory, 2nd edn. Graduate Texts in Mathematics, vol. 84. Springer, Heidelberg (1990)
Williams, H.C.: Some properties of a special set of recurring sequences. Pacific Journal of Mathematics 77(1), 273–285 (1978)
Wooding, K.: The Sieve Problem in One- and Two-Dimensions. PhD thesis, The University of Calgary, Calgary, AB (April 2010), http://math.ucalgary.ca/~hwilliam/files/wooding10thesis.pdf
Bernstein, D.J.: Detecting perfect powers in essentially linear time. Mathematics of Computation 67, 1253–1283 (1998)
Cohen, H.: A Course in Computational Algebraic Number Theory, 4th edn. Springer, Heidelberg (1993)
Williams, H.C.: An m 3 public-key encryption scheme. In: Williams, H.C. (ed.) CRYPTO 1985. LNCS, vol. 218, pp. 358–368. Springer, Heidelberg (1986)
Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1979)
Crandall, R., Pomerance, C.: Prime numbers: A computational Perspective, 2nd edn. Springer, New York (2005)
Lehmer, D.H.: The sieve problem for all-purpose computers. Mathematical Tables and Other Aids to Computation 7(41), 6–14 (1953)
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Wooding, K., Williams, H.C. (2010). Improved Primality Proving with Eisenstein Pseudocubes. In: Hanrot, G., Morain, F., Thomé, E. (eds) Algorithmic Number Theory. ANTS 2010. Lecture Notes in Computer Science, vol 6197. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-14518-6_29
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DOI: https://doi.org/10.1007/978-3-642-14518-6_29
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