267 research outputs found
Endomorphisms for faster elliptic curve cryptography on a large class of curves
Efficiently computable homomorphisms allow elliptic curve point
multiplication to be accelerated using the Gallant-Lambert-Vanstone
(GLV) method.
We extend results of Iijima, Matsuo, Chao and Tsujii which give
such homomorphisms
for a large class of elliptic curves by working over quadratic extensions
and demonstrate that these results can be applied to the
GLV method.
Our implementation runs in between 0.70 and 0.84 the time
of the previous best methods for elliptic
curve point multiplication on curves without small class number
complex multiplication. Further speedups are
possible when using more special curves
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
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
Easy scalar decompositions for efficient scalar multiplication on elliptic curves and genus 2 Jacobians
The first step in elliptic curve scalar multiplication algorithms based on
scalar decompositions using efficient endomorphisms-including
Gallant-Lambert-Vanstone (GLV) and Galbraith-Lin-Scott (GLS) multiplication, as
well as higher-dimensional and higher-genus constructions-is to produce a short
basis of a certain integer lattice involving the eigenvalues of the
endomorphisms. The shorter the basis vectors, the shorter the decomposed scalar
coefficients, and the faster the resulting scalar multiplication. Typically,
knowledge of the eigenvalues allows us to write down a long basis, which we
then reduce using the Euclidean algorithm, Gauss reduction, LLL, or even a more
specialized algorithm. In this work, we use elementary facts about quadratic
rings to immediately write down a short basis of the lattice for the GLV, GLS,
GLV+GLS, and Q-curve constructions on elliptic curves, and for genus 2 real
multiplication constructions. We do not pretend that this represents a
significant optimization in scalar multiplication, since the lattice reduction
step is always an offline precomputation---but it does give a better insight
into the structure of scalar decompositions. In any case, it is always more
convenient to use a ready-made short basis than it is to compute a new one
Distortion maps for genus two curves
Distortion maps are a useful tool for pairing based cryptography. Compared
with elliptic curves, the case of hyperelliptic curves of genus g > 1 is more
complicated since the full torsion subgroup has rank 2g. In this paper we prove
that distortion maps always exist for supersingular curves of genus g>1 and we
construct distortion maps in genus 2 (for embedding degrees 4,5,6 and 12).Comment: 16 page
Computing cardinalities of Q-curve reductions over finite fields
We present a specialized point-counting algorithm for a class of elliptic
curves over F\_{p^2} that includes reductions of quadratic Q-curves modulo
inert primes and, more generally, any elliptic curve over F\_{p^2} with a
low-degree isogeny to its Galois conjugate curve. These curves have interesting
cryptographic applications. Our algorithm is a variant of the
Schoof--Elkies--Atkin (SEA) algorithm, but with a new, lower-degree
endomorphism in place of Frobenius. While it has the same asymptotic asymptotic
complexity as SEA, our algorithm is much faster in practice.Comment: To appear in the proceedings of ANTS-XII. Added acknowledgement of
Drew Sutherlan
Point compression for the trace zero subgroup over a small degree extension field
Using Semaev's summation polynomials, we derive a new equation for the
-rational points of the trace zero variety of an elliptic curve
defined over . Using this equation, we produce an optimal-size
representation for such points. Our representation is compatible with scalar
multiplication. We give a point compression algorithm to compute the
representation and a decompression algorithm to recover the original point (up
to some small ambiguity). The algorithms are efficient for trace zero varieties
coming from small degree extension fields. We give explicit equations and
discuss in detail the practically relevant cases of cubic and quintic field
extensions.Comment: 23 pages, to appear in Designs, Codes and Cryptograph
Easy decision-Diffie-Hellman groups
The decision-Diffie-Hellman problem (DDH) is a central computational problem
in cryptography. It is known that the Weil and Tate pairings can be used to
solve many DDH problems on elliptic curves. Distortion maps are an important
tool for solving DDH problems using pairings and it is known that distortion
maps exist for all supersingular elliptic curves. We present an algorithm to
construct suitable distortion maps. The algorithm is efficient on the curves
usable in practice, and hence all DDH problems on these curves are easy. We
also discuss the issue of which DDH problems on ordinary curves are easy
- …