[Cfrg] Curves as groups

[Douglas Stebila <[e-mail]>]

Do members of CFRG think it important that the curve(s) we recommend be usable as arbitrary groups?  By this I mean that the implementations readily expose both a scalar point multiplication operation and a point addition operation.  

Utility as arbitrary groups would I think imply criteria R3 (Desired: re-use components between signatures and key exchange algorithms) but the reverse is not necessarily true.  

Curve25519+Ed25519 together give key exchange and signatures, but are high-level cryptographic functions, not groups.  The Curve25519 private API has scalar point multiplication functions but not point addition; the Ed25519 private API has both.  The NUMS library has public interfaces for scalar point multiplication and kP+lQ, but arbitrary point addition is only in the private API with a disclaimer on context of use.  To what extent are these a choice by implementers to only implement the functions needed to achieve their desired cryptographic scheme versus inherent to the mathematical design?

While criteria R3 will ensure we have something that we can use for key exchange and signatures immediately, I think it would be good for cryptographers to also have good, plain old groups from elliptic curves, for whatever new schemes we may invent or try to implement over the next 10 years.  Prior IETF initiatives (RFC 3526, RFC 5114) seem to have focused on groups rather than key exchange / signature functions.

Douglas