Abstract
We will investigate two well-known square root finding algorithms which return the roots of some quadratic residue modulo a prime p. Instead of running the mechanisms modulo p we will investigate their behaviour when applied modulo any integer n. In most cases the results will not be the square roots, when n is composite. Since the results obtained can easily be verified for correctness we obtain a very rapid probable prime test. Based on the square root finding mechanisms we will introduce two pseudoprimality tests which will be shown to be extremely fast and very efficient. Moreover, the proposed test for n ≡1 mod 4 will be proven to be even more efficient than Grantham’s suggestion in [5].
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References
Arnault, F.: Rabin-Miller primality test: Composite numbers which pass it. Math. Comp. 64(209), 355–361 (1995)
Baillie, R., Wagstaff Jr., S.: Lucas pseudoprimes. Math. Comp. 35, 1391–1417 (1980)
Bleichenbacher, D.: Efficiency and Security of Cryptosystems based on Number Theory. Dissertation ETH Zürich (1996)
Carmichael, R.D.: On Sequences of Integers Defined by Recurrence Relations. Quart. J. Pure Appl. Math. 48, 343–372 (1920)
Grantham, J.: A Probable Prime Test with High Confidence. J. Number Theory 72, 32–47 (1998)
Grantham, J.: Frobenius Pseudoprimes (1998) (preprint)
Jaeschke, G.: On strong pseudoprimes to several bases. Math. Comp. 61, 915–926 (1993)
Koblitz, N.: A Course in Number Theory and Cryptography. Springer, Heidelberg (1994)
Menezes, A., Oorschot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1996)
Montgomery, P.: Evaluating recurrences of form Xm + n = f(Xm, Xn, Xm − n) via Lucas chains (preprint)
More, W.: The LD Probable Prime Test. In: Mullin, R.C., Mullen, G. (eds.) Contemporary Mathematics, vol. 225, pp. 185–191 (1999)
Müller, S.: On the Combined Fermat/Lucas Probable Prime Test. In: Walker, M. (ed.) Cryptography and Coding 1999. LNCS, vol. 1746, pp. 222–235. Springer, Heidelberg (1999)
Müller, W.B., Oswald, A.: In: Davies, D.W. (ed.) EUROCRYPT 1991. LNCS, vol. 547, pp. 512–516. Springer, Heidelberg (1991)
Ribenboim, P.: The New Book of Prime Number Records. Springer, Heidelberg (1996)
Somer, L.: Periodicity Properties of kth Order Linear Recurrences with Irreducible Characteristic Polynomial Over a Finite Field. In: Mullen, G.L., Shiue, P.J.S. (eds.) Finite Fields, Coding Theory and Advances in Communications and Computing, pp. 195–207. Marcel Dekker Inc., New York (1993)
Somer, L.: On Lucas d-Pseudoprimes. In: Bergum, G., Philippou, A., Horadam, A. (eds.) Applications of Fibonacci Numbers, vol. 7, pp. 369–375. Kluwer Academic Publishers, Dordrecht (1998)
Joye, M., Quisquater, J.J.: Efficient computation of full Lucas sequences. IEE Electronics Letters 32(6), 537–538 (1996)
Williams, H.C.: Éduard Lucas and primality Testing. John Wiley & Sons, Chichester (1998)
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Müller, S. (2000). On Probable Prime Testing and the Computation of Square Roots mod n . In: Bosma, W. (eds) Algorithmic Number Theory. ANTS 2000. Lecture Notes in Computer Science, vol 1838. Springer, Berlin, Heidelberg. https://doi.org/10.1007/10722028_27
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DOI: https://doi.org/10.1007/10722028_27
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