  Dimension 4 isogenies have first been introduced in cryptography for the
cryptanalysis of Supersingular Isogeny Diffie-Hellman (SIDH) and have been used
constructively in several schemes, including SQIsignHD, a derivative of SQIsign
isogeny based signature scheme. Unlike in dimensions 2 and 3, we can no longer
rely on the Jacobian model and its derivatives to compute isogenies. In
dimension 4 (and higher), we can only use theta-models. Previous works by
Romain Cosset, David Lubicz and Damien Robert have focused on the computation
of $\ell$-isogenies in theta-models of level $n$ coprime to $\ell$ (which
requires to use $n^g$ coordinates in dimension $g$). For cryptographic
applications, we need to compute chains of $2$-isogenies, requiring to use
$\geq 3^g$ coordinates in dimension $g$ with state of the art algorithms.
  In this paper, we present algorithms to compute chains of $2$-isogenies
between abelian varieties of dimension $g\geq 1$ with theta-coordinates of
level $n=2$, generalizing a previous work by Pierrick Dartois, Luciano Maino,
Giacomo Pope and Damien Robert in dimension $g=2$. We propose an implementation
of these algorithms in dimension $g=4$ to compute endomorphisms of elliptic
curve products derived from Kani's lemma with applications to SQIsignHD and
SIDH cryptanalysis. We are now able to run a complete key recovery attack on
SIDH when the endomorphism ring of the starting curve is unknown within a few
seconds on a laptop for all NIST SIKE parameters.
