5,252 research outputs found
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
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
Efficient algorithms for pairing-based cryptosystems
We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable
to that of RSA in larger characteristics.We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction
over Fpm, the latter technique being also useful in contexts other than that of pairing-based cryptography
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
Fast algorithms for computing isogenies between ordinary elliptic curves in small characteristic
The problem of computing an explicit isogeny between two given elliptic
curves over F_q, originally motivated by point counting, has recently awaken
new interest in the cryptology community thanks to the works of Teske and
Rostovstev & Stolbunov.
While the large characteristic case is well understood, only suboptimal
algorithms are known in small characteristic; they are due to Couveignes,
Lercier, Lercier & Joux and Lercier & Sirvent. In this paper we discuss the
differences between them and run some comparative experiments. We also present
the first complete implementation of Couveignes' second algorithm and present
improvements that make it the algorithm having the best asymptotic complexity
in the degree of the isogeny.Comment: 21 pages, 6 figures, 1 table. Submitted to J. Number Theor
Fast algorithms for computing isogenies between elliptic curves
We survey algorithms for computing isogenies between elliptic curves defined
over a field of characteristic either 0 or a large prime. We introduce a new
algorithm that computes an isogeny of degree ( different from the
characteristic) in time quasi-linear with respect to . This is based in
particular on fast algorithms for power series expansion of the Weierstrass
-function and related functions
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