Abstract
In this work, we add an additional condition to strong pseudoprime test to base 2. Then, we provide theoretical and heuristic evidence showing that the resulting algorithm catches all composite numbers. Therefore, we believe that our method provides a probabilistic primality test with a running time \(O(\log ^{2+\epsilon }n)\) for an integer n and \(\epsilon >0\). Our method is based on the structure of singular cubics’ Jacobian groups on which we also define an effective addition algorithm.
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References
Adleman, L.M., Pomerance, C., Rumely, R.S.: On distinguishing prime numbers from composite numbers. Ann. Math. 117(1), 173–206 (1983)
Agrawal, M., Kayal, N., Saxena, N.: Primes is in P. Ann. Math. 160, 781–793 (2004)
Atkin, A.O.L., Morain, F.: Elliptic curves and primality proving. Math. Comp. 61(203), 29–68 (1993)
Cohen, H., Frey, G.: Handbook of Elliptic and Hyperelliptic Curve Cryptography. Chapman & Hall/CRC, Boca Raton (2005)
Cohen, H.: A Course in Computational Algebraic Number Theory. Springer, New York (2000)
Jaeschke, G.: On strong pseudoprimes to several bases. Math. Comp. 61(204), 915–926 (1993)
Lenstra, H.W., Jr.: Factoring integers with elliptic curves. Ann. Math. 126(3), 649–673 (1987)
Miller, G.: Riemann’s hypothesis and tests for primality. J. Comput. Syst. Sci. 13, 300–317 (1976)
Nari, K., Ozdemir, E.: Group operation on nodal curves, arXiv:1904.03978 [math.NT]
Pollard, J.M.: Theorems of factorization and primality testing. Proc. Camb. Philos. Soc. 76(3), 521–528 (1974)
Pomerance, C., Selfridge, J.L., Wagstaff, S.S.: The pseudoprimes to 25 . \(10^9\). Math. Comp. 35, 1003–1026 (1980)
Rabin, M.O.: Probabilistic algorithms for testing primality. J. Number Theory 12, 128–138 (1980)
Schoof, R.: Four primality testing algorithms. Algorithmic Number Theory 44, 101–126 (2008)
The PARI Group, PARI/GP version 2.11.0, Univ. Bordeaux, (2018) http://pari.math.u-bordeaux.fr/
Washington, L.C.: Elliptic Curves: Number Theory and Cryptography, 2nd edn. Chapman & Hall/CRC, Boca Raton (2008)
Williams, H.C.: A p+1 method of factoring. Math. Comp. 39, 225–234 (1982)
Zhang, Z.: Finding strong pseudoprimes to several bases. Math. Comp. 70, 863–872 (2001)
Zhang, Z., Tang, M.: Finding strong pseudoprimes to several bases II. Math. Comp. 72, 2085–2097 (2003)
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Nari, K., Ozdemir, E. & Ozkirisci, N.A. Strong pseudoprimes to base 2. Ramanujan J 59, 1323–1332 (2022). https://doi.org/10.1007/s11139-022-00570-8
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DOI: https://doi.org/10.1007/s11139-022-00570-8