If you want to implement elliptic curves, this book is a must read. It’s essentially a comprehensive guide on all of the arithmetic needed to implement curves, from finite fields, to various approaches to adding points, to scalar multiplication, etc.

A lot of less popular kinds of elliptic curves are also prominently feature in this book, such as optimal curves over optimal extension fields. This probably reflects the point in time where this book was written, where we hadn’t yet settled on a few curves over prime fields.

I still use this book as a reference quite a bit. Most recently, I needed to implement a binary field, and I referenced the algorithms in this book heavily.

One downside of this book is its age, so it doesn’t feature some of the newer constant time ladders for scalar multiplication, among other things. There’s been some interesting work here in recent years, like “Complete addition formulas for prime order elliptic curves”. Obviously, this book couldn’t have referenced a paper written 10 years in the future, but an updated version of this book might be interesting to incorporate these aspects. Some material on pairings would be nice in an updated version as well.

If you want to implement elliptic curves from scratch, this book is a valuable reference.