  The Lyness map is a birational map in the plane which provides one of the
simplest discrete analogues of a Hamiltonian system with one degree of freedom,
having a conserved quantity and an invariant symplectic form. As an example of
a symmetric Quispel-Roberts-Thompson (QRT) map, each generic orbit of the
Lyness map lies on a curve of genus one, and corresponds to a sequence of
points on an elliptic curve which is one of the fibres in a pencil of
biquadratic curves in the plane. Here we present a version of the elliptic
curve method (ECM) for integer factorization, which is based on iteration of
the Lyness map with a particular choice of initial data. More precisely, we
give an algorithm for scalar multiplication of a point on an elliptic curve,
which is represented by one of the curves in the Lyness pencil. In order to
avoid field inversion, and require only field multiplication (${\bf M}$),
squaring (${\bf S}$) and addition, projective coordinates in $\mathbb{P}^1
\times \mathbb{P}^1$ are used. Neglecting multiplication by curve constants
(assumed small), each addition of the chosen point uses $2{\bf M}$, while each
doubling step requires $15{\bf M}$. We further show that the doubling step can
be implemented efficiently in parallel with four processors, dropping the
effective cost to $4{\bf M}$. Our scalar multiplication algorithm should
require, on average, roughly twice as many multiplications per bit as the
fastest state of the art methods using twisted Edwards curves with small
constants, but it can be applied to any elliptic curve over $\mathbb{Q}$,
whereas twisted Edwards curves (equivalent to Montgomery curves) correspond to
only a subset of all elliptic curves. Hence, if implemented in parallel, our
method may have potential advantages for integer factorization or elliptic
curve cryptography.
