947 research outputs found
Isomorphism classes of Edwards curves over finite fields
Edwards curves are an alternate model for elliptic curves, which have attracted notice in cryptography. We give exact formulas for the number of \Fq-isomorphism classes of Edwards curves and twisted Edwards curves. This answers a question recently asked by R. Farashahi and I. Shparlinski
On isogeny classes of Edwards curves over finite fields
We count the number of isogeny classes of Edwards curves over finite fields,
answering a question recently posed by Rezaeian and Shparlinski. We also show
that each isogeny class contains a {\em complete} Edwards curve, and that an
Edwards curve is isogenous to an {\em original} Edwards curve over \F_q if
and only if its group order is divisible by 8 if , and 16
if . Furthermore, we give formulae for the proportion of
d \in \F_q \setminus \{0,1\} for which the Edwards curve is complete or
original, relative to the total number of in each isogeny class.Comment: 27 page
The Q-curve construction for endomorphism-accelerated elliptic curves
We give a detailed account of the use of -curve reductions to
construct elliptic curves over with efficiently computable
endomorphisms, which can be used to accelerate elliptic curve-based
cryptosystems in the same way as Gallant--Lambert--Vanstone (GLV) and
Galbraith--Lin--Scott (GLS) endomorphisms. Like GLS (which is a degenerate case
of our construction), we offer the advantage over GLV of selecting from a much
wider range of curves, and thus finding secure group orders when is fixed
for efficient implementation. Unlike GLS, we also offer the possibility of
constructing twist-secure curves. We construct several one-parameter families
of elliptic curves over equipped with efficient
endomorphisms for every p \textgreater{} 3, and exhibit examples of
twist-secure curves over for the efficient Mersenne prime
.Comment: To appear in the Journal of Cryptology. arXiv admin note: text
overlap with arXiv:1305.540
Families of fast elliptic curves from Q-curves
We construct new families of elliptic curves over \FF_{p^2} with
efficiently computable endomorphisms, which can be used to accelerate elliptic
curve-based cryptosystems in the same way as Gallant-Lambert-Vanstone (GLV) and
Galbraith-Lin-Scott (GLS) endomorphisms. Our construction is based on reducing
\QQ-curves-curves over quadratic number fields without complex
multiplication, but with isogenies to their Galois conjugates-modulo inert
primes. As a first application of the general theory we construct, for every
, two one-parameter families of elliptic curves over \FF_{p^2}
equipped with endomorphisms that are faster than doubling. Like GLS (which
appears as a degenerate case of our construction), we offer the advantage over
GLV of selecting from a much wider range of curves, and thus finding secure
group orders when is fixed. Unlike GLS, we also offer the possibility of
constructing twist-secure curves. Among our examples are prime-order curves
equipped with fast endomorphisms, with almost-prime-order twists, over
\FF_{p^2} for and
On arithmetic Zariski pairs in degree 6
We define a topological invariant of complex projective plane curves. As an
application, we present new examples of arithmetic Zariski pairs.Comment: 18 pages, a correction in Introductio
- …