# elliptic curves

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## 1—10 of 27 matching pages

##### 1: 27.22 Software

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).

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

##### 2: 23.20 Mathematical Applications

###### §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

###### §22.18(i) Lengths and Parametrization of Plane Curves

… ►###### §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

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

##### 5: 27.19 Methods of Computation: Factorization

*Brent–Pollard rho algorithm*(also called

*Monte Carlo method*), the

*Pollard $p-1$ algorithm*, and the

*Elliptic Curve Method*(ecm). …