elliptic curves

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1: 27.22 Software

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Mathematica. PrimeQ combines strong pseudoprime tests for the bases 2 and 3 and a Lucas pseudoprime test. No known composite numbers pass these three tests, and Bleichenbacher (1996) has shown that this combination of tests proves primality for integers below ${10}^{16}$. Provable PrimeQ uses the Atkin–Goldwasser–Kilian–Morain Elliptic Curve Method to prove primality. FactorInteger tries Brent–Pollard rho, Pollard $p-1$, and then cfrac after trial division. See §27.19. ecm is available also, and the Multiple Polynomial Quadratic sieve is expected in a future release.

For additional Mathematica routines for factorization and primality testing, including several different pseudoprime tests, see Bressoud and Wagon (2000).

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ECMNET Project. Links to software for elliptic curve methods of factorization and primality testing.

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2: 23.20 Mathematical Applications

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§23.20(ii) Elliptic Curves

►An algebraic curve that can be put either into the form …is an example of an elliptic curve (§22.18(iv)). … ►For extensive tables of elliptic curves see Cremona (1997, pp. 84–340). …

3: 22.18 Mathematical Applications

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§22.18(i) Lengths and Parametrization of Plane Curves

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§22.18(iv) Elliptic Curves and the Jacobi–Abel Addition Theorem

►Algebraic curves of the form $y^2=P(x)$, where $P$ is a nonsingular polynomial of degree 3 or 4 (see McKean and Moll (1999, §1.10)), are elliptic curves, which are also considered in §23.20(ii). …For any two points $(x_1,y_1)$ and $(x_2,y_2)$ on this curve, their sum $(x_3,y_3)$, always a third point on the curve, is defined by the Jacobi–Abel addition law …

4: 27.18 Methods of Computation: Primes

… ►The ECPP (Elliptic Curve Primality Proving) algorithm handles primes with over 20,000 digits. …

5: 27.19 Methods of Computation: Factorization

… ►Type I probabilistic algorithms include the Brent–Pollard rho algorithm (also called Monte Carlo method), the Pollard $p-1$ algorithm, and the Elliptic Curve Method (ecm). …

6: 19.30 Lengths of Plane Curves

§19.30 Lengths of Plane Curves

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§19.30(i) Ellipse

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§19.30(ii) Hyperbola

… ►For other plane curves with arclength representable by an elliptic integral see Greenhill (1892, p. 190) and Bowman (1953, pp. 32–33).

7: Bibliography M

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Sixth Painlevé Equation, Universal Elliptic Curve, and Mirror of ${𝐏}^2$ . In Geometry of Differential Equations, A. Khovanskii, A. Varchenko, and V. Vassiliev (Eds.), Amer. Math. Soc. Transl. Ser. 2, Vol. 186, pp. 131–151.

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External Links:

MathReview (Bruce Hunt), ZentralBlatt

Referenced by:

§32.2(iv).

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BibTeX

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Elliptic Curves. Cambridge University Press, Cambridge.

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Notes:

Reprint with corrections of original edition, published in 1997 by Cambridge University Press

External Links:

ISBN 0-521-58228-8; 0-521-65817-9, MathReview (Amílcar Pacheco), ZentralBlatt

Referenced by:

§20.1, §20.11(i), §20.11(ii), §20.12(i), §20.12(i), §20.12(ii), §20.7(iii), §20.7(v), §20.9(i), §20.9(ii), Ch.20, §22.18(ii), §22.18(iv), Ch.22, §23.1, §23.15(i), §23.20(i), §23.20(ii), §23.20(ii), §23.21(ii), §23.22(ii), §23.6(iv), Ch.23, Notations T.

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8: Bibliography E

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Notes:

Links to software for elliptic curve methods of factorization and primality testing.

External Links:

Home Page

Referenced by:

4th item.

Encodings:

BibTeX

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9: 23.22 Methods of Computation

… ►Suppose that the invariants $g_2=c$, $g_3=d$, are given, for example in the differential equation (23.3.10) or via coefficients of an elliptic curve (§23.20(ii)). …

10: Bibliography K

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Introduction to Elliptic Curves and Modular Forms. 2nd edition, Graduate Texts in Mathematics, Vol. 97, Springer-Verlag, New York.

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External Links:

ISBN 0-387-97966-2, MathReview, ZentralBlatt

Referenced by:

§20.10(i), §20.12(i), §20.9(iii), §22.18(iv), §23.1, §23.20(ii), §23.20(v), Ch.23.

Encodings:

BibTeX

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