Isogenies
are rational maps between elliptic curves that are also group morphisms
with respect to the group structure of the elliptic curves. The kernel
of an isogeny is always finite, and a natural way to describe an isogeny
is to give a description of its kernel. Given the kernel of an isogeny,
Velu formulas (or square root Velu formulas) allow to compute and
evaluate the isogeny. These formulas are only efficient for small degree
isogenies. Hence, in general, only smooth degrees isogenies can be
computed and evaluated exploiting these formulas. The Deuring
Correspondence allows to interpret supersingular isogenies as ideals. It
enables the computation and the evaluation of isogenies of generic
degree, provided that the endomorphism rings of the curves in play are
known. The recent SIDH attacks have proven that the images of torsion
points through an isogeny can be used to efficiently evaluate the
isogeny if these points have (power)smooth order. This enables a brand
new way to represent isogenies. This has been leveraged to design
SQISignHD, a variant of the SQISign signature. SQISignHD is currently
the most compact post-quantum signature scheme. In this talk, we will
discuss this new isogeny representation and show how it is used in
SQISignHD. We will then show how to adapt SQISignHD to obtain a
signature scheme for the CSIDH group action setting.