Improvement to "Description of the challenge (main aim)"

[defeo]

Proposed additions by Eleni Agathocleous in bold.

To advance and deploy isogeny-based cryptosystems we require

1. An effective, constructive, and algorithmic understanding of mathematical structures (such as quaternion algebras which, while common in number theory, are poorly-known and understood among ECC reseachers) and of existing analogies between number theoretic phenomena and elliptic curves (for example the strong analogy between capitulation in number fields (principalization of ideals in higher extensions) and visible elements of the Tate-Shafarevic group of Elliptic Curves or twists of elliptic curves).

2. A better understanding of quantum algorithms, both established and new, and their application to fundamental problems with isogenies and more broadly in number theory.

3. The development of tools and algorithms and utilization/adaptation of existing ones from algebraic number theory, such as class-group computations, that have not been used in ECC research before, and of which knowledge elsewhere in cryptography has grown stale.

[abelian]

I've added (an equivalent change to) for 3.

I'm not really convinced that 1. adds much to the argument (for the heavy technical cost it imposes on the reader). Besides, for me, elliptic curves over finite fields are number-theoretic phenomena.