Conclusions (Assuming GRH)

•DLOG has roughly equivalent difficulty on
elliptic curves over Fq whose endomorphism rings are “comparable” in size.

•There is
a random polynomial time reduction (equivalence) between the DLOG problems on such elliptic curves.

•NIST and
IPSec international standards curves were not chosen as to foist
cryptographically weak
curves upon an unsuspecting public.

•Method gives a new elementary construction of
expander
graphs.