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Erschienen in:
2006 | OriginalPaper | Buchkapitel
Pairing-Friendly Elliptic Curves of Prime Order
verfasst von : Paulo S. L. M. Barreto, Michael Naehrig
Erschienen in: Selected Areas in Cryptography
Verlag: Springer Berlin Heidelberg
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Previously known techniques to construct pairing-friendly curves of prime or near-prime order are restricted to embedding degree
$k \leqslant 6 $
. More general methods produce curves over
${\mathbb F}_{p}$
where the bit length of
p
is often twice as large as that of the order
r
of the subgroup with embedding degree
k
; the best published results achieve
ρ
≡ log(
p
)/log(
r
) ~ 5/4. In this paper we make the first step towards surpassing these limitations by describing a method to construct elliptic curves of prime order and embedding degree
k
= 12. The new curves lead to very efficient implementation: non-pairing operations need no more than
${\mathbb F}_{p^4}$
arithmetic, and pairing values can be compressed to one third of their length in a way compatible with point reduction techniques. We also discuss the role of large CM discriminants
D
to minimize
ρ
; in particular, for embedding degree
k
= 2
q
where
q
is prime we show that the ability to handle log(
D
)/log(
r
) ~ (
q
–3)/(
q
–1) enables building curves with
ρ
~
q
/(
q
–1).